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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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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

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