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					                          MICROWAVE AND RADIO FREQUENCY ENGINEERING
                                                                                            INDEX
admittance ...........................4            gamma rays .......................20          phase velocity................ 2, 10           transverse electromagnetic
AM ....................................20          Gauss' law ...........................8       phasor notation.................. 18             waves ................................ 9
Ampere's law.......................8               general math ......................18         plane waves....................... 10          transverse plane................. 17
an parameter.......................15              glossary .............................21      polar notation .................... 18         TV..................................... 20
anisotropic .........................21            grad operator .....................18         power .......................... 12, 18        UHF .................................. 20
attenuation constant.............6                 gradient .............................18          network......................15            ULF................................... 20
B magnetic flux density .....8                     graphing ............................19       propagation constant                           ultraviolet.......................... 20
B susceptance .....................4               group velocity......................7             complex .................6, 10             underdamped..................... 21
beta ......................................2       GSM..................................20       quarter-wave section ........... 6             uniform plane waves ........... 9
bn parameter.......................15              H magnetic field .................8           quasi-static .......................... 9      unitary matrix.................... 15
capacitance ..........................4            HF .....................................20    radar .................................. 20    vector differential equation18
carrier ..................................7        high frequency.....................9          rat race............................... 15     velocity of propagation . 2, 10
CDMA...............................20              high frequency resistance ..11                reciprocity ......................... 15       vg group velocity ................ 7
cellular...............................20          hybrid ring.........................15        reflected wave amplitude .. 15                 VHF .................................. 20
characteristic admittance .....3                   hyperbolic functions..........19              reflection coefficient ..... 3, 10             VLF................................... 20
characteristic impedance .2, 3                     impedance ...........................6        relative permittivity........... 12            vp velocity of propagation .. 2
circulator ...........................16               intrinsic ......................10        resistance                                     wave analogies .................. 10
communications frequencies                             waves.........................10              high frequency ............11              wave equation ............... 2, 17
  .........................................20      incident wave amplitude....15                 resistivity........................... 12      wave impedance................ 10
complex permittivity ...12, 21                     internally matched .............14            scattering matrix.......... 14, 16             wave input impedance....... 11
complex propagation                                intrinsic impedance ...........10             scattering parameter ... 14, 15,               wave number............... 10, 21
 constant........................6, 10             isotropic.............................21        16                                           wavelength .......................... 6
conductivity.......................12              j 18                                          self-matched ...................... 14         Wheeler's equation............ 14
conductor loss factor .........13                  J current density.................8           separation of variables....... 17              X-ray................................. 20
copper cladding .................13                k wave number .................10             series stub............................ 5      Y admittance....................... 4
cosmic rays........................20              k of a dielectric ..................12        sheet resistance.................. 11          y0 characteristic admittance 3
coupling factor...................16               lambda.................................6      SHF ................................... 20     Z0 characteristic impedance 3
D electric flux density ........8                  Laplacian ...........................19       shunt stub ............................ 5      α attenuation constant........ 6
dB ......................................16        LF......................................20    signs .................................... 2   αc conductor loss factor ... 13
dBm...................................18           light ...................................20   Sij scattering parameter..... 14               αd dielectric loss factor .... 13
del......................................18        linear .................................21    single-stub tuning................ 5           β phase constant................. 6
dielectric............................21           loss tangent..........................9       skin depth ............................ 7      δ loss tangent ..................... 9
dielectric constant..............12                    complex .....................12           SLF ................................... 20     δ skin depth........................ 7
dielectric loss factor...........13                lossless network ................15           Smith chart ...................... 4, 5        ε permittivity.................... 12
dielectric relaxation                              low frequency......................8          space derivative................. 18           εc complex permittivity.... 12
 frequency...........................8             magnetic permeability.......11                spectrum............................ 20        εr relative permittivity...... 12
directional coupler.............16                                                               square root of j .................. 18
directivity ..........................16
                                                   Maxwell's equations ............8                                                            γ complex propagation
                                                   MF.....................................20     stripline conductor............. 13              constant ............................. 6
div......................................18        microstrip conductors........13               stub length........................... 5
divergence .........................18                                                                                                          η intrinsic wave impedance
                                                   mode number.....................17            susceptance ......................... 4          ........................................ 10
E electric field ....................8             modulated wave ..................7            tan δ..................................... 9
effective permittivity .........13                                                                                                              λ wavelength...................... 6
                                                   nabla operator....................18          Taylor series...................... 19
EHF ...................................20                                                                                                       λ/4....................................... 6
                                                   network theory ..................14           TE waves........................... 17
electric conductivity ..........12                                                                                                              µ permeability.................. 11
                                                   normalize.............................4       telegrapher's equations ........ 2
electric permittivity ...........12                observation port...........14, 16             TEM assumptions ............... 9              ρ reflection coefficient ....... 3
electromagnetic spectrum..20                       omega-beta graph ................7            TEM waves ......................... 9          ρν volume charge density... 8
ELF....................................20          overdamped .......................21          thermal speed .................... 12          σ conductivity .................. 12
empirical............................21            parallel plate capacitance.....4              time domain ........................ 8         τ transmission coefficient... 3
envelope ..............................7           PCS ...................................20     time of flight ....................... 3       ∇ del ................................ 18
evanescent ...................17, 21               permeability ......................11         time variable...................... 21         ∇ divergence .................... 18
excitation port..............14, 16                permittivity........................12        time-harmonic ..................... 8          ∇ gradient ........................ 18
Faraday's law.......................8                  complex .....................12           TM waves ......................... 17          ∇2 Laplacian .................... 19
Fourier series .......................3                                                          transmission coefficient ...... 3              ∇2 Laplacian .................... 17
                                                       effective .....................13
frequency domain ................8                                                               transmission lines................ 2
frequency spectrum ...........20                       relative .......................12
                                                   phase constant .......2, 6, 8, 10             transverse .......................... 21




                            Tom Penick          tom@tomzap.com               www.teicontrols.com/notes          MicrowaveEngineering.pdf 1/30/2003 Page 1 of 21
                                                          TRANSMISSION LINES
         TELEGRAPHER'S EQUATIONS                                                                +/- WATCHING SIGNS
          ∂V      ∂I     ∂I      ∂V                                              By convention z is the variable used to describe
     (1)     = −L    (2)    = −C                                                 position along a transmission line with the origin z=0
          ∂z      ∂t     ∂z      ∂t                                              set at the load so that all other points along the line
By taking the partial derivative with respect to z of equation                   are described by negative position values.
1 and partial with respect to t of equation 2, we can get:
                                                                                          RS
            ∂ 2V     ∂ 2V                         ∂2I      ∂2I
     (i)         = LC 2                (ii)            = LC 2
            ∂z 2     ∂t                           ∂z 2     ∂t                        + VS
                                                                                     -        z=-l                               z =0    RL
                                                                                                                l

             SOLVING THE EQUATIONS                                               Ohm's law for right- and left-traveling disturbances:
To solve the equations (i) and (ii) above, we guess that                                      V+ = I + Z 0          V− = − I − Z 0
F ( u ) = F ( z ± vt ) is a solution to the equations. It is found
that the unknown constant v is the wave propagation
velocity.
                                                                                  vp VELOCITY OF PROPAGATION [cm/s]
            Vtotal = V+ ( z − vt ) + V− ( v + vt )       where:                  The velocity of propagation is the speed at which a
                                                                                 wave moves down a transmission line. The velocity
z is the position along the transmission line, where the load
                                                                                 approaches the speed of light but may not exceed the
    is at z=0 and the source is at z=-l, with l the length of the
    line.
                                                                                 speed of light since this is the maximum speed at
v is the velocity of propagation 1/ LC or ω / β , the speed                      which information can be transmitted. But vp may
                                                                                 exceed the speed of light mathematically in some
   at which the waveform moves down the line; see p 2
                                                                                 calculations.
t is time
                                                                                                     1    1   ω
                                                                                            vp =        =   =           where:
                                                                                                     LC   εµ β
          THE COMPLEX WAVE EQUATION
The general solutions of equations (i) and (ii) above                            L = inductance per unit length [H/cm]
yield the complex wave equations for voltage and                                 C = capacitance per unit length [F/cm]
current. These are applicable when the excitation is                             ε = permittivity of the material [F/cm]
sinusoidal and the circuit is under steady state                                 µ = permeability of the material [H/cm]
conditions.                                                                      ω = frequency [radians/second]
                                                                                 β = phase constant
            V ( z ) = V + e − jβz + V − e + jβz
                                                                                 Phase Velocity The velocity of propagation of a TEM
            I ( z ) = I + e − jβz + I − e + jβz                                  wave may also be referred to as the phase velocity.
                                                                                 The phase velocity of a TEM wave in conducting
                       V + e − jβz + V − e + jβz
            I ( z) =                                 where:
                                                                                 material may be described by:
                                  Z0                                                                     ω    2πδ     1
                                                                                            v p = ωδ =     =c     =c                 where:
e − jβz   e + jβz represent wave propagation in the +z
           and                                                                                           k     λ0    ε r eff
     and –z directions respectively,
                                                                                 δ = skin depth [m]
β = ω LC = ω / v is the phase constant,                                          c = speed of light 2.998 × 108 m/s
Z 0 = L / C is the characteristic impedance of the line.                         λ0 = wavelength in the material [m]
          These equations represent the voltage and current
          phasors.




                   Tom Penick         tom@tomzap.com          www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 2 of 21
 Z0 CHARACTERISTIC IMPEDANCE [Ω]                                                  τ TRANSMISSION COEFFICIENT
The characteristic impedance is the resistance                           The transmission coefficient is the ratio of total
initially seen when a signal is applied to the line. It is               voltage to the forward-traveling voltage, a value
a physical characteristic resulting from the materials                   ranging from 0 to 2.
and geometry of the line.
                                                                                                     Vtotal
                                 L                                                              τ≡          = 1+ ρ
                                    V    V                                                            V+
    Lossless line:     Z0 ≡        = + =− −
                                 C  I+   I−
                              R + j ωL                                               TOF TIME OF FLIGHT [s]
    Lossy line:     Z0 ≡               = Z 0 e jφ z
                              G + j ωC                                   The time of flight is how long it takes a signal to
                                                                         travel the length of the transmission line
L = inductance per unit length [H/cm]
C = capacitance per unit length [F/cm]                                                        l
V+ = the forward-traveling (left to right) voltage [V]                               TOF ≡      = l LC = LTOT CTOT
I+ = the forward-traveling (left to right) current [I]
                                                                                              v
V- = the reverse-traveling (right to left) voltage [V]                   l = length of the transmission line [cm]
I- = the reverse-traveling (right to left) current [I]                   v = the velocity of propagation 1/ LC , the speed at which
R = the line resistance per unit length [Ω/cm]                              the waveform moves down the line
G = the conductance per unit length [Ω-1/cm]                             L = inductance per unit length [H/cm]
φ = phase angle of the complex impedance [radians]                       C = capacitance per unit length [F/cm]
                                                                         LTOT = total inductance [H]
                                                                         CTOT = total capacitance [F]
y0 CHARACTERISTIC ADMITTANCE [Ω−1]
The characteristic admittance is the reciprocal of                                      DERIVED EQUATIONS
the characteristic impedance.
                                                                         V+ = z0 I + = (VTOT + ITOT z0 ) / 2
                         C  I    I
                    y0 ≡   = + =− −                                      V− = − z0 I − = (VTOT − I TOT z0 ) / 2
                         L  V+   V−
                                                                          I + = y0V+ = ( ITOT + VTOT y0 ) / 2
                                                                          I − = − y0V− = ( ITOT − VTOT y0 ) / 2
        ρ REFLECTION COEFFICIENT
The reflection coefficient is the ratio of reflected
voltage to the forward-traveling voltage, a value
ranging from –1 to +1 which, when multiplied by the
                                                                                        Cn FOURIER SERIES
wave voltage, determines the amount of voltage                           The function x(t) must be periodic in order to employ
reflected at one end of the transmission line.                           the Fourier series. The following is the exponential
                                                                         Fourier series, which involves simpler calculations
                              V−   I
                        ρ≡       =− −                                    than other forms but is not as easy to visualize as the
                              V+   I+                                    trigonometric forms.
                                                                                                1 t1 +T
                                                                                                  ∫ t1 x ( t ) e 0 dt
A reflection coefficient is present at each end of the                                                          − jnω t
transmission line:
                                                                                         Cn =
                                                                                                T
                    RS − z0                     RL − z0
        ρsource =                     ρload =                            Cn = amplitude                  n = the harmonic (an integer)
                    RS + z0                     RL + z0                  T = period [s]                  ω0 = frequency 2π/T [radians]
                                                                         t = time [s]
                                                                         The function x(t) may be delayed in time. All this does in a
                                                                         Fourier series is to shift the phase. If you know the Cns for
                                                                                                                   -jnω0α
                                                                         x(t), then the Cns for x(t-α) are just Cne      . (Here, Cns is
                                                                         just the plural of Cn.)




                Tom Penick       tom@tomzap.com       www.teicontrols.com/notes    MicrowaveEngineering.pdf 1/30/2003 Page 3 of 21
                    C CAPACITANCE [F]                                                                    SMITH CHART
                                                      dVcap                        First normalize the load impedance by dividing by the
             i dτ + v(0 )
        1 t
        C ∫0
v(t ) =                                  I cap = C                                 characteristic impedance, and find this point on the chart.
                                                       dt
v(t ) = v f + (v0 − v f )e − t / τ                      (          )
                                                                                   When working in terms of reactance X, an inductive load
                                         i (t ) = i f + i0 − i f e − t / τ         will be located on the top half of the chart, a capacitive load
                                                                                   on the bottom half. It's the other way around when working
P(t ) = i0 R e −2t / τ
               2
                                                                                   in terms of susceptance B [Siemens].
v(t) = voltage across the capacitor, at time t [V]                                 Draw a straight line from the center of the chart through the
                                                                                   normalized load impedance point to the edge of the chart.
vf = final voltage across the capacitor, steady-state voltage
     [V]                                                                           Anchor a compass at the center of the chart and draw a
v0 = initial voltage across the capacitor [V]                                      circle through the normalized load impedance point. Points
                                                                                   along this circle represent the normalized impedance at
t = time [s]
                                                                                   various points along the transmission line. Clockwise
τ = the time constant, RC [seconds]                                                movement along the circle represents movement from the
C = capacitance [F]                                                                load toward the source with one full revolution representing
Natural log:        ln x = b ⇔ e b = x                                             1/2 wavelength as marked on the outer circle. The two
                                                                                   points where the circle intersects the horizontal axis are the
                                                                                   voltage maxima (right) and the voltage minima (left).
                                                                                   The point opposite the impedance (180° around the circle) is
   C PARALLEL PLATE CAPACITANCE                                                    the admittance Y [Siemens]. The reason admittance (or
               εA                                 εA εwl εw                        susceptibility) is useful is because admittances in parallel
    C=                       Cper unit length =      =    =                        are simply added. (Admittance is the reciprocal of
                h                                 lh   lh   h                      impedance; susceptance is the reciprocal of reactance.)
ε = permittivity of the material [F/cm]
A = area of one of the capacitor plates [cm2]                                                                           z = distance from load
                                                                                            Γ( z ) = ΓL e j 2βz             [m]
h = plate separation [cm]
w = plate width [cm]                                                                     e j 2βz = 1∠2β z               j = −1
l = plate length [cm]
                                                                                                  Z( z ) − 1            ρ = magnitude of the
C = capacitance [F]                                                                     G( z ) =                            reflection coefficient
                                                                                                  Z( z ) + 1            β = phase constant
                                                                                          Γ −1               Z          Γ = reflection coefficient
          CAPACITOR-TERMINATED LINE                                                  ZL = L            Z= L             Z = normalized
          RS                                                                              ΓL + 1             Z0
                                                                                                                            impedance [Ω]

   + VS                                                                CL
   -
Where the incident voltage is                     (
                                     V+ = V0 1 − e − t / τ0   ),
                           2τ1 − t / τ1    2 τ 0 − t / τ0 
Vcap = V+ + V− = V0  2 +         e      −         e       
                         τ0 − τ1          τ0 − τ1         
V0 = final voltage across the capacitor [V]
t = time [s]
τ0 = time constant of the incident wave, RC [s]
τ1 = time constant effect due to the load, Z0CL [s]
C = capacitance [F]




                      Tom Penick     tom@tomzap.com             www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 4 of 21
             SINGLE-STUB TUNING                                                                                      FINDING A STUB LENGTH
The basic idea is to connect a line stub in parallel                                  Example: Find the lengths of open and shorted shunt stubs
(shunt) or series a distance d from the load so that the                              to match an admittance of 1-j0.5. The admittance of an
imaginary part of the load impedance will be canceled.                                open shunt (zero length) is Y=0; this point is located at the
                                                                                      left end of the Smith Chart x-axis. We proceed clockwise
                                                         d                            around the Smith chart, i.e. away from the end of the stub,
Shunt-stub: Select d
                                                                                      to the +j0.5 arc (the value needed to match –j0.5). The
so that the admittance             Y0                    Y0            YL
                                                                                      difference in the starting point and the end point on the
Y looking toward the
                                                                                      wavelength scale is the length of the stub in wavelengths.
load from a distance d    Open
                           or                                                         The length of a shorted-type stub is found in the same
is of the form Y0 + jB.   short     Y0                                                manner but with the starting point at Y=∞.
Then the stub
susceptance is chosen                    l                                               Open stub of                   j .5
                                                                                         length .074 λ
as –jB, resulting in a                                                                   matches an                                              Towa r d g ene r at or
                                                                                         admittance                                              .11 .12 .13 .14 .15
matched condition.                                                                       of 1-j.5                                    .09
                                                                                                                                            .1                                .16
                                                                                                                                                                                    .17




                                                                                                                                                          1.0
                                                                                                                            .08
                                                                                                                                                                                          .18
                                                              d                                                       .07                                                                                    Admittance
                                                                                                                                                                                                             (short)




                                                                                                                               0.




                                                                                                                                                                                2.0


                                                                                                                                                                                                .19
                                                                                                               .06
Series-stub: Select d                                                                                                                                                                                        Y= ∞




                                                                                                                                 5
                                                                                                         .05




                                                                                                                                                                                                      .2
so that the admittance             Z0                         Z0       ZL
                                                                                         .074 λ




                                                                                                                                                                                                           .21
                                                                                                    .04
Z looking toward the




                                                                                                                                                                                                             .22
                                                                                                    03
load from a distance d                                                                                                                                                                            5.0




                                                                                                 2 .
is of the form Z0 + jX.




                                                                                                                                                                                                              .23
                                                                                           .01 .0
                                   Z0
Then the stub




                                                                                                                                                                                                                  .24 .25 .26 .27
                                             l




                                                                                                                                                          1.0
                                                                                                                                           0.5
                                                                                                               0.1
susceptance is chosen




                                                                                                                                                                          2




                                                                                                                                                                                      5
                                                                                           0
as -jX, resulting in a                           Open
                                                                                                                       Admittance



                                                                                                   .49
                                                  or
matched condition.                               short                                                                 (open)
                                                                                                                       Y=0

                                                                                               .48
                                                                                                                                                                                                  5.
                                                                                                                                                                                                    0




                                                                                                                                                                                                                                  .28
                                                                                           .47
                                                                                        .324 λ




                                                                                                                                                                                                            .29
                                                                                                     .46
                                                                                                          .45




                                                                                                                                                                                                       .3
                                                                                                                                                                                    2.0



                                                                                                                                                                                                 .31
                                                                                                                            0.5
                                                                                                                .44
                                                                                                                                                                                          .32
                                                                                                                      .43
                                                                                                                                                                                    .33




                                                                                                                                                          1.0
                                                                                                                            .42                                               .34
                                                                                      Shorted stub of                                .41         .35
                                                                                      length .324 λ
                                                                                                                                            .4       .36 .37 .38 .39
                                                                                      matches an
                                                                                      admittance
                                                                                      of 1-j.5

                                                                                      In this example, all values were in units of admittance. If we
                                                                                      were interested in finding a stub length for a series stub
                                                                                      problem, the units would be in impedance. The problem
                                                                                      would be worked in exactly the same way. Of course in
                                                                                      impedance, an open shunt (zero length) would have the
                                                                                      value Z=∞, representing a point at the right end of the x-axis.




               Tom Penick         tom@tomzap.com                   www.teicontrols.com/notes              MicrowaveEngineering.pdf 1/30/2003 Page 5 of 21
              LINE IMPEDANCE [Ω]                                               γ COMPLEX PROPAGATION CONSTANT
The impedance seen at the source end of a lossless                            The propagation constant for lossy lines, taking into
transmission line:                                                            account the resistance along the line as well as the
                      1+ ρ     Z + jZ 0 tan ( β l )                           resistive path between the conductors.
         Z in = Z 0        = Z0 L
                      1− ρ     Z 0 + jZ L tan ( β l )                              γ = α + jβ = ZY =        ( R + jωL )( G + jωC )
For a lossy transmission line:                                                                    L     R
                            Z L + Z 0 tanh ( γl )
               Z in = Z 0
                            Z 0 + Z L tanh ( γl )                                                               G     C

Line impedance is periodic with spatial period λ/2.
                                                                              α=    RG attenuation constant, the real part of the
Z0 = L / C , the characteristic impedance of the line. [Ω]                        complex propagation constant, describes the loss
ρ = the reflection coefficient                                                β = 2π/λ, phase constant, the complex part of the complex
ZL = the load impedance [Ω]                                                       propagation constant
β = 2π/λ, phase constant                                                      Z = series impedance (complex, inductive) per unit length
γ = α+jβ, complex propagation constant                                            [Ω/cm]
                                                                              Y = shunt admittance (complex, capacitive) per unit length
                                                                                  [Ω-1/cm]
             λ WAVELENGTH [cm]                                                R = the resistance per unit length along the transmission
                                                                                  line [Ω/cm]
The physical distance that a traveling wave moves                             G = the conductance between conductors per unit length
during one period of its periodic cycle.                                          [Ω-1/cm]
                                                                              L = inductance per unit length [H/cm]
                            2π 2π v p
                      λ=      =   =                                           C = capacitance per unit length [F/cm]
                            β   k   f
β = ω LC = 2π/λ, phase constant
k = ω µε = 2π/λ, wave number
vp = velocity of propagation [m/s] see p 2.
f = frequency [Hz]


       λ/4 QUARTER-WAVE SECTION
A quarter-wave section of transmission line has the
effect of inverting the normalized impedance of the
load.
                          λ /4

       Zin                 Z0                              Z0
                                                    RL =
                                                           2

To find Zin, we can normalize the load (by dividing by the
characteristic impedance), invert the result, and
"unnormalize" this value by multiplying by the characteristic
impedance.
                                       Z0        1
In this case, the normalized load is      ÷ Z0 =
                                       2         2
                                                −1
                                           1
so the normalized input impedance is         =2
                                           2
and the actual input impedance is      Z in = 2 Z 0



                Tom Penick       tom@tomzap.com            www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 6 of 21
                 MODULATED WAVE                                                                     OMEGA - BETA GRAPH
Suppose we have a disturbance composed of two                                      This representation is commonly used for modulated
frequencies:                                                                       waves.
               sin ( ω0 − δω) t − ( β0 − δβ ) z 
                                                                                              ω
                                                                                                                    β
                                                                                                                 ω=
         and   sin ( ω0 + δω) t − ( β0 + δβ ) z 
                                                
                                                                                                                    LC
                                                                                                                      dω
                                                                                                            slope is     , group velocity
where ω0 is the average frequency and β0 is the average                                      ωc                       dβ
phase.                                                                                                  slope is phase velocity
                                                                                                        for a particular ω, β.
Using the identity 2cos  A − B  sin  A + B  = sin A + sin B
                                           
                         2   2                                                                                                    β
The combination (sum) of these two waves is

            2 cos ( δωt − δβz ) sin ( ω0t − β0 z )                                                  δ SKIN DEPTH [cm]
              144244 14     3        4244     3
                     envelope                carrier                               The depth into a material at which a wave is
The envelope moves at the group velocity, see p 7.                                 attenuated by 1/e (about 36.8%) of its original
δ = "the difference in"…                                                           intensity. This isn't the same δ that appears in the
ω0 = carrier frequency [radians/second]                                            loss tangent, tan δ.
ω = modulating frequency [radians/second]                                                           1    2
β0 = carrier frequency phase constant                                                       δ=        =               where:
β = phase constant                                                                                  α   ωµσ
So the sum of two waves                                                            α=    RG attenuation constant, the real part of the
will be a modulated wave                                                               complex propagation constant, describes loss
having a carrier frequency                                                         µ = permeability of the material, dielectric constant [H/cm]
equal to the average                                                               ω = frequency [radians/second]
frequency of the two waves,
and an envelope with a                                                             σ = (sigma) conductivity [Siemens/meter] see p12.
frequency equal to half the                                                                     Skin Depths of Selected Materials
difference between the two
                                                                                                      60 Hz         1 MHz                   1 GHz
original wave frequencies.
                                                                                     silver            8.27 mm           0.064 mm     0.0020 mm
                                                                                     copper            8.53 mm           0.066 mm     0.0021 mm
                                                                                     gold             10.14 mm           0.079 mm     0.0025 mm
                                                                                     aluminum         10.92 mm           0.084 mm     0.0027 mm
          vg GROUP VELOCITY [cm/s]                                                   iron              0.65 mm           0.005 mm     0.00016 mm
The velocity at which the envelope of a modulated
wave moves.

                 δω               ω
                                              2
                            1
          vg =      =          1 − c2                  where:
                 δβ        LCP    ω
L = inductance per unit length [H/cm]
CP = capacitance per unit length [F/cm]
ε = permittivity of the material [F/cm]
µ = permeability of the material, dielectric constant [H/cm]
ωc = carrier frequency [radians/second]
ω = modulating frequency [radians/second]
β = phase constant
Also, since β may be given as a function of ω, remember
                                        −1
                               dβ 
                         vg =     
                               dω 




                 Tom Penick      tom@tomzap.com                 www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 7 of 21
          MAXWELL'S EQUATIONS                                            ELECTROMAGNETIC WAVES
Maxwell's equations govern the principles of
guiding and propagation of electromagnetic                            MODELING MAXWELL'S EQUATIONS
energy and provide the foundations of all                          This is a model of a wave, analogous to a
electromagnetic phenomena and their                                transmission line model.
applications. The time-harmonic expressions can                                   L = µ [H/m]
be used only when the wave is sinusoidal.
                                                                                                     G= σ               C= ε
             STANDARD FORM         TIME-HARMONIC
               (Time Domain)        (Frequency Domain)                                               [Ω-1/m]            [F/m]
                         v
                  v    ∂B                 v      v                 L = inductance per unit length [H/cm]
Faraday's
              ∇× E = -                ∇ × E = -jωB                 µ = permeability of the material, dielectric constant [H/cm]
  Law
                       ∂t                                          G = the conductance per unit length [Ω-1/cm]
                            v                                      σ = (sigma) conductivity [Siemens/meter]
                v     v ∂D              v     v v
Ampere's
            ∇× H = J +              ∇ × H = jωD + J                C = capacitance per unit length [F/cm]
 Law*
                          ∂t                                       ε = permittivity of the material [F/cm]
                    v                       v
 Gauss'
  Law
               ∇ ⋅ D = ρv               ∇ ⋅ D = ρv                       propagation constant:           γ=     ( jωµ )( jωε + σ )
                    v                        v
no name
  law
                ∇⋅ B =0                 ∇⋅ B=0
                                                                                       LOW FREQUENCY
E = electric field [V/m]
                                                                   At low frequencies, more materials behave as
B = magnetic flux density [W/m2 or T] B = µ0H
                                                                   conductors. A wave is considered low frequency
t = time [s]
                                                                   when
D = electric flux density [C/m2] D = ε0E
ρ = volume charge density [C/m3]                                               σ             σ
                                                                         ω=                          is the dielectric relaxation frequency
H = magnetic field intensity [A/m]                                             ε             ε
J = current density [A/m2]
                                                                         1
*Maxwell added the ∂ D term to Ampere's Law.                        η=      (1 + j )         intrinsic wave impedance, see p 12.
                    ∂t                                                   σδ
                                                                   What happens to the complex propagation constant at low
                                                                   frequency? From the wave model above, gamma is
                                                                                                                              jωε
                                                                             γ=    ( jωµ )( jωε + σ ) =          jωµσ 1 +
                                                                                                                               σ
                                                                   Since both ω and ε/σ are small
                                                                                             1      ε
                                                                                  γ=   jωµσ  1 + j ω  =             jωµσ (1)
                                                                                                2   σ
                                                                   Since          1     1
                                                                             j=      +j
                                                                                   2     2
                                                                                        1     1   ωµσ    ωµσ
                                                                             γ = ωµσ       +j   =     +j
                                                                                        2      2   2      2
                                                                   So that, with γ = α + jβ

                                                                                       ωµσ                     ωµσ
                                                                    we get    α=                 ,     β=             or   γ=
                                                                                                                                1
                                                                                                                                  (1 + j )
                                                                                        2                       2               δ
                                                                   α = attenuation constant, the real part of the complex
                                                                       propagation constant, describes the loss
                                                                   β = phase constant, the complex part of the complex
                                                                       propagation constant
                                                                   σ = (sigma) conductivity [Siemens/cm]
                                                                   δ = skin depth [cm]
                                                                   So the wave is attenuating at the same rate that it is
                                                                   propagating.


              Tom Penick   tom@tomzap.com       www.teicontrols.com/notes    MicrowaveEngineering.pdf 1/30/2003 Page 8 of 21
                     HIGH FREQUENCY                                                                   TEM WAVES
At high frequencies, more materials behave as                                            Transverse Electromagnetic Waves
dielectrics, i.e. copper is a dielectric in the gamma
                                                                                Electromagnetic waves that have single, orthogonal
ray range. A wave is considered high frequency when
                                                                                vector electric and magnetic field components (e.g., Ex
            σ           σ                                                       and Hy ), both varying with a single coordinate of space
     ω?                     is the dielectric relaxation frequency              (e.g., z), are known as uniform plane waves or
            ε           ε                                                       transverse electromagnetic (TEM) waves. TEM
                                                                                calculations may be made using formulas from
                 µ
         η=            intrinsic wave impedance, see p 12.                      electrostatics; this is referred to as quasi-static
                 ε                                                              solution.
What happens to the complex propagation constant at high                                       Characteristics of TEM Waves
frequency?
                                                                                • The velocity of propagation (always in the z direction) is
                                                         σ                      v p = 1 / µε , which is the speed of light in the material
         γ=     ( jωµ )( jωε + σ ) =        jωµ jωε 1 +     
                                                        jωε 
                                                                                • There is no electric or magnetic field in the direction of
Since both 1/ω and σ/ε are small                                                  propagation. Since this means there is no voltage drop in
                  1 σ                       σ µ                                 the direction of propagation, it suggests that no current
       γ = jω µε 1 +                  γ=         + jω µε                         flows in that direction.
                  2 jωε 
                                             2 ε
                                                                              • The electric field is normal to the magnetic field
With   γ = α + jβ                                                               • The value of the electric field is η times that of the
                                                                                  magnetic field at any instant.
                               σ µ                                              • The direction of propagation is given by the direction of
              we get    α=              ,    β = ω µε ,
                               2 ε                                                E×H.
                                                                                • The energy stored in the electric field per unit volume at
                                                                                  any instant and any point is equal to the energy stored in
                                                                                  the magnetic field.
                 tan δ LOSS TANGENT
The loss tangent, a value between 0 and 1, is the loss
coefficient of a wave after it has traveled one                                                 TEM ASSUMPTIONS
wavelength. This is the way data is usually presented
in texts. This is not the same δ that is used for skin                          Some assumptions are made for TEM waves.
depth.
                                                                                 E =0
                                                                                  z                                Hz = 0
                                    σ
                            tan δ =                                              σ=0                              time dependence e
                                                                                                                                       j ωt
                                    ωε
Graphical representation of
                                       Imag. ( I )
loss tangent:
                                                ωε
For a dielectric,
                                                     δ
tan δ = 1 .
       1              π                                      Re ( I )
α≈       ( tan δ ) β = tan δ                             σ
       2              λ
ωε is proportional to the amount of current going through
the capacitance C.
σ is proportional to the amount current going through the
conductance G.




                     Tom Penick    tom@tomzap.com            www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 9 of 21
                      WAVE ANALOGIES                                                             k WAVE NUMBER [rad./cm]
Plane waves have many characteristics analogous to                                  The phase constant for the uniform plane wave; the
transmission line problems.                                                         change in phase per unit length. It can be considered
    Transmission Lines                            Plane Waves                       a constant for the medium at a particular frequency.

       Phase constant                            Wave number                                                   ω          2π
                                                                                                        k=       = ω µε =
                       ω 2π                                   ω 2π                                             v          λ
 β = ω LC =               =                k = ω µε =            =
                       vp   λ                                 vp   λ                k appears in the phasor forms of the uniform plane wave

Complex propagation const.                  Complex propagation                                      E x ( z ) = E1e − jkz + E 2 e jkz , etc.
                                                 constant                           k has also been used as in the "k of a dielectric" meaning εr.
 γ = α + jβ
  =    ( R + jωL )( G + jωC )             γ=      ( jωµ )( jωε + σ )
                                                                                     η (eta) INTRINSIC WAVE IMPEDANCE [Ω]
  Velocity of propagation                        Phase velocity
                  1   ω                          1  ω         2πδ                   The ratio of electric to magnetic field components.
        vp =        =                    vp =      = = ωδ = c                       Can be considered a constant of the medium. For
                  LC β                           µε k          λ
                                                                                    free space, η = 376.73Ω. The units of η are in ohms.
 Characteristic impedance                     Intrinsic impedance
                                                                                                 E x+    E y+                         E x−    E y−
             L V+                                   µ Ex +                                  η=        =−                   −η=             =−
        Z0 =  =                                  η=  =                                           H y+    H x+                         H y−    H x−
             C I+                                   ε H y+

              Voltage                             Electric Field                        at low frequencies                     at high frequencies

V ( z ) = V+ e− jβz + V− e jβz           Ex ( z ) = E+ e− jkz + E− e jkz                         1                                              µ
                                                                                            η=      (1 + j )                             η=
                                                                                                 σδ                                             ε
              Current                            Magnetic Field
                                                                                    When an electromagnetic                          H
         1                                          1
 I (z) =    V+ e − jβz − V− e jβz      H y ( z ) =  E+ e − jkz − E− e jkz       wave encounters a sheet of
         Z0                                       η                             conductive material it sees an                         E
                                                                                    impedance. K is the direction                               K
   Line input impedance                    Wave input impedance                     of the wave, H is the magnetic
                                                                                    component and E is the
              Z L + jZ 0 tan ( β l )                 η L + jη0 tan ( kl )
 Z in = Z 0                               ηin = η0                                  electrical field. E × H gives the
              Z 0 + jZ L tan ( βl )                  η0 + jηL tan ( kl )            direction of propagation K.

              Z L + Z 0 tanh ( γl )                  η L + η0 tanh ( γl )
 Z in = Z 0                               ηin = η0
              Z 0 + Z L tanh ( γl )                  η0 + ηL tanh ( γl )

    Reflection coefficient                   Reflection coefficient
                Z L − Z0                                ηL − η0
         ρ=                                       ρ=
                Z L + Z0                                ηL + η0




                   Tom Penick          tom@tomzap.com           www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 10 of 21
             SHEET RESISTANCE [Ω]                                                  ηin WAVE INPUT IMPEDANCE [Ω]
Consider a block of material with conductivity σ.                          The impedance seen by a wave in a medium.

                                      l                                    For example, the impedance of a metal sheet in a vacuum:

                                                                                                              metal                 vacuum
                     w


                                                                                                                η0                   ηL
                t
                                                                                                                l
                                                                                                ηin
                           l
It's resistance is   R=           Ω.
                                                                           Note that a transmission line model is used here because it
                          wt σ                                             is analogous to a wave traveling in a medium. The "load" is
                                                                           the element most remote in the direction of propagation.

                                                                                                                       η L + η0 tanh (γl )
If the length is equal to the width, this reduces to
                                                                                                          ηin = η0                               Ω.
                                                                                                                       η0 + η L tanh (γl )
                                                                           The input impedance is
                                  1
                          R=              Ω.
                                 tσ
                                                                           In this example, l is the thickness of a metal sheet. If the
And this is sheet resistance.                                              metal thickness is much greater than the skin depth, then
                                                                                              1          
                                                                            tanh (λl ) = tanh  (1 + j )l  = tanh [(big number )(1 + j )] ≈ 1
                                                                                              δ          
   HIGH FREQUENCY RESISTANCE [Ω]                                           If l is much less than the skin depth δ, then
                                                                                              1          
                                                                            tanh (λl ) = tanh  (1 + j )l  = tanh [(small number )(1 + j )]
When a conductor carries current at high frequency,
the electric field penetrates the outer surface only                                          δ          
about 1 skin depth so that current travels near the
                                                                            = (same small number )(1 + j ) = (1 + j )
                                                                                                                 l
surface of the conductor. Since the entire cross-
section is not utilized, this affects the resistance of the                                                     δ
conductor.
                                               w                               µ MAGNETIC PERMEABILITY [H/m]
Cross-section
of a conductor                                                             The relative increase or decrease in the resultant
showing current                   t                         δ              magnetic field inside a material compared with the
flow near the surface:                                                     magnetizing field in which the given material is
                                                                           located. The product of the permeability constant and
                                                                           the relative permeability of the material.
                      1           ωµ 0 1
         R≈                     =                                                       µ = µ 0µ r where µ = 4π×10-7 H/m
               σδ ( perimeter )   2σ 2w + 2t                                                              0

                                                                                   Relative Permeabilities of Selected Materials
σ = (sigma) conductivity (5.8×105 S/cm for copper)
     [Siemens/meter]                                                       Air                   1.00000037           Mercury        0.999968
ω = frequency [radians/second]                                             Aluminum              1.000021             Nickel         600
                                                                           Copper                0.9999833            Oxygen         1.000002
δ = skin depth [cm]
                                                                           Gold                  0.99996              Platinum       1.0003
µ0 = permeability of free space µ0 = 4π×10-9 [H/cm]                        Iron (99.96% pure)    280,000              Silver         0.9999736
w = width of the conductor [cm]                                            Iron (motor grade)    5000                 Titanium       1.00018
t = thickness of the conductor [cm]                                        Lead                  0.9999831            Tungsten       1.00008
                                                                           Manganese             1.001                Water          0.9999912




                Tom Penick      tom@tomzap.com         www.teicontrols.com/notes    MicrowaveEngineering.pdf 1/30/2003 Page 11 of 21
      ε ELECTRIC PERMITTIVITY [F/m]                                                 σ CONDUCTIVITY [S/m] or [1/(Ω·m)]
The property of a dielectric material that determines                           A measure of the ability of a material to conduct
how much electrostatic energy can be stored per unit                            electricity, the higher the value the better the material
of volume when unit voltage is applied, also called the                         conducts. The reciprocal is resistivity. Values for
dielectric constant. The product of the constant of                             common materials vary over about 24 orders of
permittivity and the relative permittivity of a material.                       magnitude. Conductivity may often be determined
                                                                                from skin depth or the loss tangent.
             ε = ε0 εr    where ε0 = 8.85×10-14 F/cm
                                                                                                               2
                                                                                                         nc qe l
                                                                                                    σ=           S/m         where
                                                                                                          me vth
           εc COMPLEX PERMITTIVITY
                                                                                nc = density of conduction electrons (for copper this is
                                                                                             28    -3
                                             ε′′                                    8.45×10 ) [m ]
             εc = ε′ − jε′′        where         = tan δc                       qe = electron charge? 1.602×10-23 [C]
                                             ε′
                                                                                l = vthtc the product of the thermal speed and the mean
In general, both ε′ and ε′′ depend on frequency in                                   free time between collisions of electrons, the average
complicated ways. ε′ will typically have a constant                                  distance an electron travels between collisions [m]
maximum value at low frequencies, tapering off at higher                        me = the effective electron mass? [kg]
frequencies with several peaks along the way. ε′′ will                          vth = thermal speed, usually much larger than the drift
typically have a peak at the frequency at which ε′ begins to                    velocity vd. [m/s]
decline in magnitude as well as at frequencies where ε′ has                        Conductivities of Selected Materials [1/(Ω·m)]
peaks, and will be zero at low frequencies and between
peaks.                                                                          Aluminum            3.82×107          Mercury                   1.04×106
                                                                                Carbon              7.14×104          Nicrome                   1.00×106
                                                                                Copper (annealed)   5.80×107          Nickel                    1.45×107
                                                                                Copper (in class)   6.80×107          Seawater                  4
           εr RELATIVE PERMITTIVITY                                             Fresh water         ~10-2             Silicon                   ~4.35×10-4
                                                                                Germanium           ~2.13             Silver                    6.17×107
The permittivity of a material is the relative permittivity                     Glass               ~10-12            Sodium                    2.17×107
multiplied by the permittivity of free space                                    Gold                4.10×107          Stainless steel           1.11×106
                                                                                Iron                1.03×107          Tin                       8.77×106
                              ε = ε r × ε0                                      Lead                4.57×10           Titanium                  2.09×106
                                                                                                                      Zinc                      1.67×107
In old terminology, εr is called the "k of a dielectric". Glass
(SiO2) at εr = 4.5 is considered the division between low k
and high k dielectrics.
     Relative Permittivities of Selected Materials
                                                                                                     P POWER [W]
Air (sea level)     1.0006           Polystyrene            2.6                 Power is the time rate of change of energy.
Ammonia             22               Polyethylene           2.25
                                                                                                                       % power = ρ ×100
Bakelite            5                Rubber                 2.2-4.1                                                                         2
                                                                                Power reflected at a discontinuity:
Glass               4.5-10           Silicon                11.9
Ice
Mica
                    3.2
                    5.4-6
                                     Soil, dry
                                     Styrofoam
                                                            2.5-3.5
                                                            1.03
                                                                                                                                        (
                                                                                Power transmitted at a discontinuity: % power = 1 − ρ 2 × 100     )
most metals         ~1               Teflon                 2.1
Plexiglass          3.4              Vacuum                 1
Porcelain           5.7              Water, distilled       81
Paper               2-4              Water, seawater        72-80
Oil                 2.1-2.3
NOTE: Relative permittivity data is given for materials at
low or static frequency conditions. The permittivity for
most materials varies with frequency. The relative
permittivities of most materials lie in the range of 1-25. At
high frequencies, the permittivity of a material can be quite
different (usually less), but will have resonant peaks.




                  Tom Penick      tom@tomzap.com            www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 12 of 21
              MICROSTRIP CONDUCTORS                                                                 STRIPLINE CONDUCTOR
How fast does a wave travel in a microstrip? The
                                                                                       Also called shielded microstrip. The effective relative
question is complicated by the fact that the dielectric
                                                                                       permittivity is used in calculations.
on one side of the strip may be different from the
dielectric on the other side and a wave may travel at
                                                                                                                 t
different speeds in different dielectrics. The solution is                                                                 w                 h2
to find an effective relative permittivity εr eff for the
combination.
                             t                                                                                                               h1
                                       w
                                                                                                                                      εr1h1 + εr 2 h2
                                                                                           assuming w ≥ 10 h ,            ε r eff =                   where
                                                   h                                                                                     h1 + h2
                                                                                       εr1 = the relative permittivity of the dielectric of thickness h1.
                       Some Microstrip Relations                                       εr2 = the relative permittivity of the dielectric of thickness h2.
Z 0 = Z 0 ε r eff                                C air Z 0 = ε0µ 0
   air                                                       air


                                                                                                        COPPER CLADDING
L = Z0            ε 0µ 0 = C total ( Z 0 )       L C air = ε 0µ 0
            air                              2


                                                                                       The thickness of copper on a circuit board is
                  L                                               L                    measured in ounces. 1-ounce cladding means that 1
Z0 =                                                        =
                                                      air
                                                 Z0                                    square foot of the copper weighs 1 ounce. 1-ounce
            C total                                             C air
                                                                                       copper is 0.0014" or 35.6 µm thick.
                                                              C total
γ = jβ = jω ε0µ 0 ε r eff                        ε r eff    = air
                                                              C
                1                  1                                                    αd DIELECTRIC LOSS FACTOR [dB/cm]
vp =                     =
            ε 0µ0 εr eff         L C total                                                                           β 0 ε r (ε r eff − 1)
                                                                                                    α d = 8.68                               tan δ
It's difficult to get more than 200Ω for Z0 in a microstrip.                                                         2 ε r eff (ε r − 1)
                       Microstrip Approximations
            εr + 1      εr − 1
ε r eff =          +                                                                    αc CONDUCTOR LOSS FACTOR [dB/cm]
              2      2 1 + 12h / w
      60          8h w                                  w                                           R
              ln      +     ,                          for ≤ 1                          α c = 8.68                                        ωµ 0
       ε r eff  w 4h                                                                                       ,                  1                  1
                                                        h                                                         R=                  =
                                                                                                     2Z 0                σδ(perimeter )   2σ (perimeter )
Z0 =                      120π                            w
                                                     , for > 1
      ε r eff w                     w                 h
               h + 1.393 + 0.667 ln  h + 1.444  
                                               
   8e A                                                                w
   e2 A − 2 ,                                                            <2
w                                                                      h
 =
h 2                         ε −1                       0.61  
                                                                       w
        B − 1 − ln ( 2 B − 1) + r    ln ( B − 1) + 0.39 −                >2
  π                           2ε r 
                                                                ,
                                                            εr  
                                                                    h

where A = Z 0        εr + 1 εr − 1          0.11  ,   377π
                           +         0.23 +       B=
                  60   2     εr + 1          εr      2Z0 εr




                      Tom Penick       tom@tomzap.com              www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 13 of 21
               WHEELER'S EQUATION                                                            NETWORK THEORY
Another approximation for microstrip calculations is
Wheeler's equation.                                                                    Sij SCATTERING PARAMETER
              
                                                                                                             Si j
                                        
                                          2                        
              14 + 8       14 +
                                   8
                                              1+
                                                  1
                                                                   
     42.4   4h     εr 4h       εr 4h                          
                                                                                         observation port              excitation port
                                                 εr
Z0 =     ln 1+        × +         ×  +π 2
                                                                   
     1+εr   w′  11    w′  11      w′       2                              A scattering parameter, represented by Sij, is a
                                                                
                                                              
                                                                               dimensionless value representing the fraction of wave
                                                                         amplitude transmitted from port j into port i, provided
                                                                               that all other ports are terminated with matched loads
                        4                                    1                 and only port j is receiving a signal. Under these
                     7+                                  1+
                        εr         Z0                    εr                same conditions, Sii is the reflection coefficient at port
               8h            exp  42.4 ε r + 1  − 1 + 0.81                 i.
where                 11                        
        w′ =
                                  Z0                                         To experimentally determine the scattering
                             exp       εr + 1  − 1
                                  42.4                                       parameters, attach an impedance-matched generator
                                                                               to one of the ports (excitation port), attach
                                                                               impedance-matched loads to the remaining ports, and
                                                                               observe the signal received at each of the ports
                                                                               (observation ports). The fractional amounts of signal
                                                                               amplitude received at each port i will make up one
                                                                               column j of the scattering matrix. Repeating the
                                                                               process for each column would require n2
                                                                               measurements to determine the scattering matrix for
                                                                               an n-port network.


                                                                                           Sij SCATTERING MATRIX
                                                                                                 S11    S12        L S1N 
                                                                                                S       S 22       L S2N 
                                                                                                 21                       
                                                                                                 M       M         O M 
                                                                                                                          
                                                                                                S N1    SN 2       L S NN 
                                                                               The scattering matrix is an n×n matrix composed of
                                                                               scattering parameters that describes an n-port
                                                                               network.
                                                                               The elements of the diagonal of the scattering matrix
                                                                               are reflection coefficients of each port. The elements
                                                                               of the off-diagonal are transmission coefficients, under
                                                                               the conditions outlined in "SCATTERING
                                                                               PARAMETER".
                                                                               If the network is internally matched or self-matched, then
                                                                                S11 = S 22 = L = S NN = 0 , that is, the diagonal is all zeros.
                                                                               The sum of the squares of each column of a scattering
                                                                               matrix is equal to one, provided the network is lossless.




                    Tom Penick      tom@tomzap.com         www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 14 of 21
    an, bn INCIDENT/REFLECTED WAVE                                                                  LOSSLESS NETWORK
                             AMPLITUDES                                             A network is lossless when
                                                                                                                  S S =/
                                                                                                                     †
The parameters an and bn describe the incident and
reflected waves respectively at each port n. These
parameters are used for power and scattering matrix                                 † means to take the complex conjugate and transpose the
                                                                                         matrix. If the network is reciprocal, then the transpose
calculations.                                                                            is the same as the original matrix.
                                                                                    / = a unitary matrix. A unitary matrix has the properties:
The amplitude of the wave incident to
                                                                  Vn+
port n is equal to the amplitude of the              an =                                     N                                  N
incident voltage at the port divided by
the square root of the port impedance.
                                                                   Z0n                       ∑S
                                                                                             k =1
                                                                                                    ki S ki = 1
                                                                                                         *
                                                                                                                             ∑S  k =1
                                                                                                                                        ki   S kj = 0
                                                                                                                                               *



Amplitude of the wave reflected at port                                −            In other words, a column of a unitary matrix multiplied by its
                                                                  V
n is equal to the amplitude of the                   bn =             n             complex conjugate equals one, and a column of a unitary
reflected voltage at the port divided by                          Z 0n              matrix multiplied by the complex conjugate of a different
the square root of the port impedance.                                              column equals zero.

The scattering parameter is equal to the wave
amplitude output at port i divided by the wave                             bi
amplitude input at port j provided the only                   Sij =                  RAT RACE OR HYBRID RING NETWORK
source is a matched source at port j and all                               aj                                                                3λ
other ports are connected to matched loads.                                         The rat race or hybrid                                   4
                                                                                    ring network is lossless,
The relationship between the S-parameters                                           reciprocal, and                      1
and the a- and b-parameters can be written in
matrix form where S is the scattering matrix                  b = Sa                internally matched.
                                                                                                                                 λ                    λ   4
and a and b are column vectors.                                                                                                  4                    4
                                                                                                                                             λ


                                                         (                      )
                                                                                                                             2               4
Power flow into any port is shown as              1 2
                                               P=   a −b
                                                         2
                                                                                                                                                  3
a function of a- and b-parameters.
                                                  2                                 The signal splits upon entering the network and half travels
The ratio of the input power at port                          2                     around each side. A signal entering at port 1 and exiting at
j to the output power at port I can          Pin j       aj                1        port 4 travels ¾ of a wavelength along each side, so the
be written as a function of a- and                   =        2
                                                                  =            2    signals are in phase and additive. From port 1 to port 3 the
b-parameters or the S-parameter.
                                            Pout i       bj            Sij          signal travels one wavelength along one side and ½
                                                                                    wavelength along the other, arriving a port 3 out of phase
                                                                                    and thus canceling. From port 1 to port 2 the paths are ¼
                                                                                    and 5/4 wavelengths respectively, thus they are in phase
                           RECIPROCITY                                              and additive.

A network is reciprocal when Sij = Sji in the scattering
matrix, i.e. the matrix is symmetric across the
diagonal. Also, Zij = Zji and Yij = Yji. Networks
constructed of “normal materials” exhibit reciprocity.
Reciprocity Theorem:
                  v         v            v    v
             ∫E
              S
                      a   × H b ⋅ ds = ∫ Eb × H a ⋅ ds
                                       S

Ea and Hb are fields from two different sources.




               Tom Penick          tom@tomzap.com             www.teicontrols.com/notes    MicrowaveEngineering.pdf 1/30/2003 Page 15 of 21
             DIRECTIONAL COUPLER                                                            MAXWELL'S EQUATIONS, TIME
                                             1                        2
                                                                                                 HARMONIC FORM
The directional coupler is a 4-
port network similar to the rat                                                                  ∇ × E = - jωµH           "curl on E"
race. It can be used to
measure reflected and                                                                            ∇ × H = - jωµE           "curl on H"
transmitted power to an                      3                        4            E =  E x ( x, y ) x + E y ( x, y ) y + Ez ( x, y ) z  e jωt −γz
                                                                                                     ˆ                ˆ                ˆ
antenna.
An input at one port is divided between two of the remaining                       H =  H x ( x, y ) x + H y ( x, y ) y + H z ( x, y ) z  e jωt −γz
                                                                                                     ˆ                ˆ                ˆ
ports. The coupling factor, measured in dB, describes the
division of signal strength at the two ports. For example if                      From the curl equations we can derive:
the coupler has a coupling factor of –10 dB, then a signal
                                                                                        ∂ Ez                                  ∂ Hz
input at port 1 would appear at port 4 attenuated by 10 dB                        (1)        + γE y = − jωµH x          (4)        + γH y = jωεEx
with the majority of the signal passing to port 2. In other                             ∂y                                     ∂y
words, 90% of the signal would appear at port 2 and 10% at
                                                                                  (2) −
                                                                                           ∂ Ez                        ∂ Hz
port 4. (-10 dB means "10 dB down" or 0.1 power, -6 dB                                          − γEx = − jωµH y (5) −      − γH x = jωεE y
means 0.25 power, and –3 dB means 0.5 power.) A                                            ∂x                           ∂x
reflection from port 2 would appear at port 3 attenuated by
the same amount. Meters attached to ports 3 and 4 could                                   ∂ Ey       ∂ Ex                     ∂ Hy       ∂ Hx
be used to measure reflected and transmitted power for a                          (3)            −        = − jωµH z    (6)          −        = jωεEz
system with a transmitter connected to port 1 and an
                                                                                          ∂x         ∂y                       ∂x          ∂y
antenna at port 2. The directivity of a coupler is a
measurement of how well the coupler transfers the signal to                       From the above equations we can obtain:
the appropriate output without reflection due to the coupler                                               1          ∂ Ez    ∂ Hz 
itself; the directivity approaches infinity for a perfect coupler.                (1) & (5) H x =                 jωε      −γ      
                                                                                                        γ + ω µε 
                                                                                                         2   2
                                                                                                                       ∂y       ∂x 
 directivity = 10 log ( p3 / p1 ) , where the source is at port 1
and the load is at port 2.                                                                                 1          ∂ Ez    ∂ Hz 
                                                                                  (2) & (4) H y =                 jωε      −γ      
The directional coupler is lossless          0           p   0 −q                                     γ + ω µε 
                                                                                                          2  2
                                                                                                                        ∂x      ∂y 
and reciprocal. The scattering               p           0   q  0
                                                                  
matrix looks like this. In a real
                                             0               0 p                                           1      ∂ Ez        ∂ Hz 
coupler, the off-diagonal zeros                           q                       (2) & (4) E x = −                 −γ    + jωµ      
                                                                                                        γ + ω µε 
                                                                                                              22
                                                                                                                        ∂x        ∂y 
would be near zero due to leakage.           −q          0   p 0

                                                                                                                1      ∂ Ez        ∂ Hz 
                                                                                  (1) & (5) E y = −                    −γ    + jωµ      
                      CIRCULATOR                                                                             γ + ω µε 
                                                                                                              2   2
                                                                                                                           ∂y        ∂x 
                                     1                                            This makes it look like if Ez and Hz are zero, then Hx, Hy, Ex,
The circulator is a 3-port                            l
                                                                                  and Ey are all zero. But since ∞ × 0 ≠ 0 , we could have
network that can be                                  β+               2           non-zero result for the TEM wave if
used to prevent
                                                                                  γ 2 = −ω2µε ⇒ γ = jω µε . This should look familiar.
reflection at the antenna                    β−
from returning to the                    l                        l
source.

                                                 3
Port 3 is terminated internally by a matched load. With a
source at 1 and a load at 2, any power reflected at the load
is absorbed by the load resistance at port 3. A 3-port
network cannot be both lossless and reciprocal, so the
circulator is not reciprocal.
Schematically, the                  The circulator is lossless
circulator may be depicted          but is not reciprocal. The
like this:                          scattering matrix looks like
                                    this:
                                                  0 0 1 
                                                  1 0 0 
                                                        
                                                  0 1 0 
                                                        




               Tom Penick      tom@tomzap.com                 www.teicontrols.com/notes     MicrowaveEngineering.pdf 1/30/2003 Page 16 of 21
                     WAVE EQUATIONS                                                          TM, TE WAVES IN PARALLEL PLATES
From Maxwell's equations and a vector identity on                                         TM, or transverse magnetic,
curl, we can get the following wave equations:                                            means that magnetic waves
                                                                                          are confined to the transverse                   x
           ∇ 2 E = -ω2µεE                  "del squared on E"                             plane. Similarly, TE                      d              y
           ∇ 2 H = ω2µεH                   "del squared on H"                             (transverse electric) means
                                                                                          that electrical waves are
The z part or "del squared on Ez" is:                                                     confined to the transverse            (z direction is into page)
                                                                                          plane.
                     γ 2 E z γ 2 Ez γ 2 Ez
          ∇ 2 Ez =          +      +       = −ω2µεEz                                      Transverse plane means the plane that is transverse to
                     ∂ x2 ∂ y 2 ∂ z 2                                                     (perpendicular to) the direction of propagation. The
                                                                                          direction of propagation is taken to be in the z direction, so
Using the separation of variables, we can let:                                            the transverse plane is the x-y plane. So for a TM wave,
                     E z = X ( x ) ⋅Y ( y ) ⋅ Z ( z )                                     there is no Hz component (magnetic component in the z
                                                                                          direction) but there is an Ez component.

                                                                                                              Ez = A sin ( k x x ) e −γz
We substitute this into the previous equation and divide by
X·Y·Z to get:
            1 d 2 X 1 d 2Y 1 d 2 Z                                                        A = amplitude [V]
                    +      +          1 µε
                                    = −ω23
                                       2                                                          mπ
            X 3 Y dy
            424 1 3 123 a constant
            1 dx
                  2      2        2
                      2
                             Z dz                                                          kx =        The magnetic field must be zero at the plate
               −kx
                2               2
                             −k y2             − kz                                               d
                                                                                              boundaries. This value provides that characteristic.
Since X, Y, and Z are independent variables, the only way                                     [cm-1]
the sum of these 3 expressions can equal a constant is if all                             x = position; perpendicular distance from one plate. [cm]
3 expressions are constants.                                                              d = plate separation [cm]
                        1 d 2Z          d 2Z                                              γ = propagation constant
                               = −k z ⇒      = − Zk z
                                     2                2
So we are letting            2             2                                              z = position along the direction of propagation [cm]
                        Z dz            dz                                                m = mode number; an integer greater than or equal to 1
A solution could be       Z = e −γz
                                                                                                              γ = −ω2µε + ( kx )
                                                                                                                                      2
            2 − γz          2 −γz
           γe        = −k z e              −k z = γ
                                                      2       2
so that                              and
                                                                                          Notice than when ( kx ) 2 ≥ ω2µε , the quantity under the
Solutions for X and Y are found
                                                                                          square root sign will be positive and γ will be purely real. In
      1    d2X
                  = −k x ⇒ X = A sin ( k x x ) + B cos ( k x x )                          this circumstance, the wave is said to be evanescent. The
                         2

      X     dx 2                                                                          wavelength goes to infinity; there is no oscillation or
           d 2Y                                                                           propagation. On the other hand, when ( kx )2 < ω2µε , γ is
                 = − k y ⇒ Y = C sin ( k y y ) + D cos ( k y y )
      1                 2

      Y    dy 2                                                                           purely imaginary.
                                                                                                                                           x
                                                                                          The magnitude of Ez is related
                                       k x + k y − γ = ω µε                                                                                           m =1
                                           2              2       2    2
giving us the general solution                                                            to its position between the
                                                                                          plates and the mode number                                  m =2
For a particular solution we need to specify initial conditions                                                                 d
                                                                                          m. Note that for m = 2 that
and boundary conditions. For some reason, initial
conditions are not an issue. The unknowns are kx, ky, A, B,                               d = λ.                                                       Ez
                                                                                                                                    -max       +max
C, D. The boundary conditions are
                                               ∂ H tan
                 Etan = 0                              =0
                                                ∂n
Etan = the electric field tangential to a conducting surface
Htan = the magnetic field tangential to a conducting surface
n = I don't know




                 Tom Penick          tom@tomzap.com                   www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 17 of 21
                                                                    GENERAL MATHEMATICAL
       COMPLEX TO POLAR NOTATION
j in polar notation:                                                             j                 ∇ NABLA, DEL OR GRAD OPERATOR
                                     π                         j
                                                                    Imag.                       Compare the ∇ operation to taking the time derivative.
                                 j
                    j=e              2                                                          Where ∂/∂t means to take the derivative with respect
                                                                                 Re             to time and introduces a s-1 component to the units of
So we can find the square root of j:                                                            the result, the ∇ operation means to take the
                π            π
                                                                                                derivative with respect to distance (in 3 dimensions)
            j            j               1     1                                                and introduces a m-1 component to the units of the
    j= e        2
                    =e       4
                                 =          +j                unit circle
                                          2     2                                               result. ∇ terms may be called space derivatives and
                                                                                                an equation which contains the ∇ operator may be
                                                                                                called a vector differential equation. In other words
   dBm DECIBELS RELATIVE TO 1 mW                                                                ∇A is how fast A changes as you move through
                                                                                                space.
The decibel expression for power. The logarithmic
                                                                                                in rectangular              ∂A    ∂A    ∂A
nature of decibel units translates the multiplication and                                                           ∇A = xˆ    +y
                                                                                                                                ˆ    +z
                                                                                                                                      ˆ
division associated with gains and losses into addition                                         coordinates:                ∂x    ∂y    ∂z
and subtraction.                                                                                in cylindrical                ∂A ˆ 1 ∂A       ∂A
                                                                                                coordinates:        ∇A = r  ˆ   +φ         +z
                                                                                                                                            ˆ
     0 dBm = 1 mW                                                                                                          ∂r    r ∂φ    ∂z
    20 dBm = 100 mW                                                                             in spherical               ∂A ˆ 1 ∂A ˆ 1 ∂A
   -20 dBm = 0.01 mW                                                                            coordinates:        ∇A = r
                                                                                                                         ˆ    +θ      +φ
                                                                                                                           ∂r    r ∂θ    r sin θ ∂φ
                    P ( dBm ) = 10 log  P ( mW ) 
                                                 
                             P ( mW ) = 10
                                                    P ( dBm ) /10                                                   ∇ GRADIENT
                                                                                                         v                "The gradient of the vector Φ" or
                                                                                                        ∇Φ = −E           "del Φ" is equal to the negative of
                                                                                                                          the electric field vector.
                     PHASOR NOTATION
                                                                                                ∇Φ is a vector giving the direction and magnitude of the
To express a derivative in phasor notation, replace                                             maximum spatial variation of the scalar function Φ at a point
 ∂                                                                                              in space.
    with jω . For example, the                                                                                  v     ∂Φ    ∂Φ    ∂Φ
∂t                                                                                                             ∇Φ = x
                                                                                                                    ˆ    +y
                                                                                                                          ˆ    +z
                                                                                                                                ˆ
                       ∂V       ∂I                                                                                    ∂x    ∂y    ∂z
Telegrapher's equation     = −L
                        ∂z      ∂t
         ∂V                                                                                                       ∇⋅ DIVERGENCE
becomes       = − LjωI .
          ∂z                                                                                    ∇⋅ is also a vector operator, combining the "del" or
                                                                                                "grad" operator with the dot product operator and is
                                                                                                read as "the divergence of". In this form of Gauss'
                                                                                                law, where D is a density per unit area, with the
                                                                                                operators applied, ∇⋅D becomes a density per unit
                                                                                                volume.

                                                                                                                          ∂ Dx ∂ Dy ∂ Dz
                                                                                                        div D = ∇ ⋅ D =       +    +     =ρ
                                                                                                                           ∂x   ∂y   ∂z
                                                                                                D = electric flux density vector D = εE [C/m2]
                                                                                                ρ = source charge density [C/m3]




                    Tom Penick               tom@tomzap.com                 www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 18 of 21
               ∇2 THE LAPLACIAN
∇2 is a combination of the divergence and del
operations, i.e. div(grad Φ) = ∇⋅∇ Φ = ∇2 Φ. It is read
as "the LaPlacian of" or "del squared".

                       ∂2 Φ ∂2 Φ ∂2 Φ
              ∇ 2F =       +     +
                       ∂ x2 ∂ y 2 ∂ z 2
Φ = electric potential [V]


GRAPHING TERMINOLOGY
  With x being the horizontal axis and y the vertical, we have
  a graph of y versus x or y as a function of x. The x-axis
  represents the independent variable and the y-axis
  represents the dependent variable, so that when a graph
  is used to illustrate data, the data of regular interval (often
  this is time) is plotted on the x-axis and the corresponding
  data is dependent on those values and is plotted on the y-
  axis.


           HYPERBOLIC FUNCTIONS
               j sin θ = sinh ( jθ )
                    j cos θ = cosh ( jθ )
                    j tan θ = tanh ( jθ )


                  TAYLOR SERIES
                           1
                  1+ x ≈ 1+ x , x = 1
                           2
           1
                ≈ 1 + x 2 + x4 + x6 + L , x < 1
         1− x 2

                    1
                         ≈ 1m x , x = 1
                 1± x




              Tom Penick      tom@tomzap.com        www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 19 of 21
                                        ELECTROMAGNETIC SPECTRUM
  FREQUENCY             WAVELENGTH           DESIGNATION                                    APPLICATIONS
                         (free space)

< 3 Hz                 > 100 Mm                                Geophysical prospecting
3-30 Hz                10-100 Mm        ELF                    Detection of buried metals
30-300 Hz              1-10 Mm          SLF                    Power transmission, submarine communications
0.3-3 kHz              0.1-1 Mm         ULF                    Telephone, audio
3-30 kHz               10-100 km        VLF                    Navigation, positioning, naval communications
30-300 kHz             1-10 km          LF                     Navigation, radio beacons
0.3-3 MHz              0.1-1 km         MF                     AM broadcasting
3-30 MHz               10-100 m         HF                     Short wave, citizens' band
30-300 MHz             1-10 m           VHF                    TV, FM, police
   54-72                                                         TV channels 2-4
   76-88                                                         TV channels 5-6
   88-108                                                        FM radio
   174-216                                                       TV channels 7-13
0.3-3 GHz              10-100 cm        UHF                    Radar, TV, GPS, cellular phone
   470-890 MHz                                                   TV channels 14-83
   915 MHz                                                       Microwave ovens (Europe)
   800-2500 MHz                              "money band"        PCS cellular phones, analog at 900 MHz, GSM/CDMA at 1900
   1-2                                                           L-band, GPS system
   2.45                                                          Microwave ovens (U.S.)
   2-4                                                           S-band
3-30 GHz               1-10 cm          SHF                    Radar, satellite communications
   4-8                                                           C-band
   8-12                                                          X-band (Police radar at 11 GHz)
   12-18                                                         Ku-band (dBS Primestar at 14 GHz)
   18-27                                                         K-band (Police radar at 22 GHz)
30-300 GHz             0.1-1 cm         EHF                    Radar, remote sensing
   27-40                                                         Ka-band (Police radar at 35 GHz)
   40-60                                                         U-band
   60-80                                                         V-band
   80-100                                                        W-band
0.3-1 THz              0.3-1 mm         Millimeter             Astromony, meteorology
                       3-300 µm
  12        14
10 -10 Hz                               Infrared               Heating, night vision, optical communications
             14
3.95×10 -              390-760 nm       Visible light          Vision, astronomy, optical communications
   7.7×1014 Hz            625-760                                 Red
                          600-625                                 Orange
                          577-600                                 Yellow
                          492-577                                 Green
                          455-492                                 Blue
                          390-455                                 Violet
1015-1018 Hz           0.3-300 nm       Ultraviolet            Sterilization
  16        21
10 -10 Hz                               X-rays                 Medical diagnosis
                                        γ-rays
  18        22
10 -10 Hz                                                      Cancer therapy, astrophysics
       22
> 10 Hz                                 Cosmic rays            Astrophysics




                  Tom Penick    tom@tomzap.com     www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 20 of 21
                                                               GLOSSARY
anisotropic materials materials in which the electric
  polarization vector is not in the same direction as the electric
  field. The values of ε, µ, and σ are dependent on the field
  direction. Examples are crystal structures and ionized
  gases.
complex permittivity ε The imaginary part accounts for heat
  loss in the medium due to damping of the vibrating dipole
  moments.
dielectric An insulator. When the presence of an applied field
  displaces electrons within a molecule away from their
  average positions, the material is said to be polarized.
  When we consider the polarizations of insulators, we refer to
  them as dielectrics.
empirical A result based on observation or experience rather
  than theory, e.g. empirical data, empirical formulas. Capable
  of being verified or disproved by observation or experiment,
  e.g. empirical laws.
evanescent wave A wave for which β=0. α will be negative.
   That is, γ is purely real. The wave has infinite wavelength—
   there is no oscillation.
isotropic materials materials in which the electric polarization
   vector is in the same direction as the electric field. The
   material responds in the same way for all directions of an
   electric field vector, i.e. the values of ε, µ, and σ are constant
   regardless of the field direction.
linear materials materials which respond proportionally to
   increased field levels. The value of µ is not related to H and
   the value of ε is not related to E. Glass is linear, iron is non-
   linear.
overdamped system in the case of a transmission line, this
   means that when the source voltage is applied the line
   voltage rises to the final voltage without exceeding it.
time variable materials materials whose response to an
   electric field changes over time, e.g. when a sound wave
   passes through them.
transverse plane perpendicular, e.g. the x-y plane is
   transverse to z.
underdamped system in the case of a transmission line, this
   means that after the source voltage is applied the line
   voltage periodically exceeds the final voltage.
wave number k The phase constant for the uniform plane
   wave. k may be considered a constant of the medium at a
   particular frequency.




                 Tom Penick      tom@tomzap.com        www.teicontrols.com/notes   MicrowaveEngineering.pdf 1/30/2003 Page 21 of 21

				
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Description: admittance ...........................4 AM ....................................20 Ampere's law.......................8 a n parameter.......................15 anisotropic.........................21 attenuation constant.............6 B magnetic flux density.....8 B susceptance .....................4 beta......................................2 b n parameter.......................15 capacitance ..........................4 carrier ..................................7 CDMA...............................20 cellular...............................20 characteristic admittance .....3 characteristic impedance .2, 3 circulator ...........................16 communications frequencies .........................................20 complex permittivity ...12, 21 complex propagation constant........................6, 10 conductivity.......................12 conductor loss factor .........13 copper cladding .................13 cosmic rays........................20 coupling factor...................16 D electric flux density ........8 dB......................................16 dBm...................................18 del......................................18 dielectric............................21 dielectric constant..............12 dielectric loss factor...........13 dielectric relaxation frequency...........................8 directional coupler.............16 directivity ..........................16 div......................................18 divergence .........................18 E electric field ....................8 effective permittivity.........13 EHF ...................................20 electric conductivity ..........12 electric permittivity ...........12 electromagnetic spectrum..20 ELF....................................20 empirical............................21 envelope ..............................7 evanescent ...................17, 21 excitation port..............14, 16 Faraday's law.......................8 Fourie