Basic Quantities.ppt

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					          EEE 498/598
Overview of Electrical Engineering

                   Lecture 12:
          Overview Of Circuit Theory;
    Lumped Circuit Elements; Topology Of
       Circuits; Resistors; KCL and KVL;
     Resistors in Series and Parallel; Energy
     Storage Elements; First-Order Circuits

                                                1
        Lecture 12 Objectives
 To commence our study of circuit theory.
 To develop an understanding of the
  concepts of Lumped circuit elements;
  topology of circuits; resistors; KCL and
  KVL; resistors in series and parallel;
  energy storage elements; and first-order
  circuits.

                                  Lecture 12
    Overview of Circuit Theory
 Electrical circuit elements are idealized
  models of physical devices that are defined
  by relationships between their terminal
  voltages and currents. Circuit elements can
  have two or more terminals.
 An electrical circuit is a connection of
  circuit elements into one or more closed
  loops.
                                   Lecture 12
      Overview of Circuit Theory
   A lumped circuit is one where all the terminal
    voltages and currents are functions of time only.
    Lumped circuit elements include resistors,
    capacitors, inductors, independent and
    dependent sources.
   An distributed circuit is one where the terminal
    voltages and currents are functions of position
    as well as time. Transmission lines are
    distributed circuit elements.

                                          Lecture 12
    Overview of Circuit Theory
 Basic quantities are voltage, current, and
  power.
 The sign convention is important in
  computing power supplied by or absorbed
  by a circuit element.
 Circuit elements can be active or passive;
  active elements are sources.
                                   Lecture 12
        Overview of Circuit Theory
   Current is moving positive electrical charge.
   Measured in Amperes (A) = 1 Coulomb/s
   Current is represented by I or i.
   In general, current can be an arbitrary function
    of time.
     Constant current is called direct current (DC).
     Current that can be represented as a sinusoidal
      function of time (or in some contexts a sum of
      sinusoids) is called alternating current (AC).

                                              Lecture 12
    Overview of Circuit Theory
 Voltage is electromotive force provided by
  a source or a potential difference between
  two points in a circuit.
 Measured in Volts (V): 1 J of energy is
  needed to move 1 C of charge through a 1
  V potential difference.
 Voltage is represented by V or v.

                                  Lecture 12
    Overview of Circuit Theory
 The lower case symbols v and i are usually
  used to denote voltages and currents that
  are functions of time.
 The upper case symbols V and I are usually
  used to denote voltages and currents that
  are DC or AC steady-state voltages and
  currents.

                                  Lecture 12
       Overview of Circuit Theory
   Current has an assumed direction of flow; currents in
    the direction of assumed current flow have positive
    values; currents in the opposite direction have negative
    values.
   Voltage has an assumed polarity; volt drops in with the
    assumed polarity have positive values; volt drops of the
    opposite polarity have negative values.
   In circuit analysis the assumed polarity of voltages are
    often defined by the direction of assumed current flow.


                                               Lecture 12
      Overview of Circuit Theory
   Power is the rate at which energy is being
    absorbed or supplied.
   Power is computed as the product of voltage
    and current:
              pt   vt it  or P  VI
   Sign convention: positive power means that
    energy is being absorbed; negative power means
    that power is being supplied.
                                             Lecture 12
  Overview of Circuit Theory

                              • If p(t) > 0, then the circuit
                              element is absorbing power
                 i(t)
                              from the rest of the circuit.
Rest of                       • If p(t) < 0, then the circuit
            +
circuit                       element is supplying power
          v(t)                to the rest of the circuit.

            -
                        Circuit element under
                        consideration

                                            Lecture 12
    Overview of Circuit Theory
 If power is positive into a circuit element,
  it means that the circuit element is
  absorbing power.
 If power is negative into a circuit element,
  it means that the circuit element is
  supplying power. Only active elements
  (sources) can supply power to the rest of a
  circuit.
                                    Lecture 12
     Active and Passive Elements
   Active elements can generate energy.
       Examples of active elements are independent and
        dependent sources.
   Passive elements cannot generate energy.
       Examples of passive elements are resistors,
        capacitors, and inductors.
   In a particular circuit, there can be active
    elements that absorb power – for example, a
    battery being charged.

                                                Lecture 12
    Independent and Dependent
             Sources
 An independent source (voltage or
current) may be DC (constant) or time-
varying; its value does not depend on other
voltages or currents in the circuit.
 A dependent source has a value that
depends on another voltage or current in the
circuit.

                                   Lecture 12
    Independent Sources


vs t           is t 


Voltage Source   Current Source




                             Lecture 12
Dependent Sources


          v=f(vx)             v=f(ix)
     +                   +
      -                   -



   Voltage             Current
  Controlled          Controlled
Voltage Source      Voltage Source
   (VCVS)              (CCVS)
                               Lecture 12
Dependent Sources


          I=f(Vx)             I=f(Ix)




   Voltage             Current
 Controlled          Controlled
Current Source      Current Source
   (VCCS)              (CCCS)
                               Lecture 12
    Passive Lumped Circuit Elements
   Resistors
                 R




   Capacitors
                 C




   Inductors
                 L

                             Lecture 12
        Topology of Circuits
 A lumped circuit is composed of lumped
  elements (sources, resistors, capacitors,
  inductors) and conductors (wires).
 All the elements are assumed to be
  lumped, i.e., the entire circuit is of
  negligible dimensions.
 All conductors are perfect.

                                   Lecture 12
         Topology of Circuits
 A schematic diagram is an electrical
  representation of a circuit.
 The location of a circuit element in a
  schematic may have no relationship to its
  physical location.
 We can rearrange the schematic and have
  the same circuit as long as the connections
  between elements remain the same.
                                    Lecture 12
          Topology of Circuits
   Example: Schematic of a circuit:
                               “Ground”: a
                               reference point
                               where the voltage
                               (or potential) is
                               assumed to be zero.




                                       Lecture 12
            Topology of Circuits
   Only circuit elements that are in closed loops
    (i.e., where a current path exists) contribute to
    the functionality of a circuit.
                                         This circuit
                                         element can be
                                         removed without
                                         affecting
                                         functionality. This
                                         circuit behaves
                                         identically to the
                                         previous one.

                                            Lecture 12
          Topology of Circuits
  A node is an equipotential point in a circuit. It
is a topological concept – in other words, even if
the circuit elements change values, the node
remains an equipotential point.
 To find a node, start at a point in the circuit.
From this point, everywhere you can travel by
moving only along perfect conductors is part of a
single node.

                                          Lecture 12
         Topology of Circuits
 A loop is any closed path through a circuit in
which no node is encountered more than once.
 To find a loop, start at a node in the circuit.
From this node, travel along a path back to the
same node ensuring that you do not encounter any
node more than once.
 A mesh is a loop that has no other loops inside
of it.

                                      Lecture 12
         Topology of Circuits
  If we know the voltage at every node of a
circuit relative to a reference node (ground),
then we know everything about the circuit –
i.e., we can determine any other voltage or
current in the circuit.
 The same is true if we know every mesh
current.

                                     Lecture 12
      Topology of Circuits

N1   N2    N3        N4   • In this example there
                          are 5 nodes and 2
                          meshes.
                          • In addition to the
      M1        M2
                          meshes, there is one
                          additional loop
            N0            (following the outer
                          perimeter of the circuit).




                                    Lecture 12
                  Resistors
 A resistor is a circuit element that
  dissipates electrical energy (usually as heat).
 Real-world devices that are modeled by
  resistors: incandescent light bulb, heating
  elements (stoves, heaters, etc.), long wires
 Parasitic resistances: many resistors on
  circuit diagrams model unwanted
  resistances in transistors, motors, etc.
                                      Lecture 12
                       Resistors
                       i(t)

                                         vt   Ri t 
           The                +
          Rest of        R        v(t)
           the
          Circuit             -


   Resistance is measured in Ohms (W)
   The relationship between terminal voltage and current
    is governed by Ohm’s law
   Ohm’s law tells us that the volt drop in the direction of
    assumed current flow is Ri
                                                Lecture 12
              KCL and KVL
 Kirchhoff’s Current Law (KCL) and Kirchhoff’s
Voltage Law (KVL) are the fundamental laws of
circuit analysis.
 KCL is the basis of nodal analysis – in which
the unknowns are the voltages at each of the
nodes of the circuit.
 KVL is the basis of mesh analysis – in which
the unknowns are the currents flowing in each of
the meshes of the circuit.
                                      Lecture 12
                  KCL and KVL
   KCL                                  i1(t)     i5(t)
     The sum of all currents
                                 i2(t)                 i4(t)
      entering a node is zero,
      or                                       i3(t)

     The sum of currents
                                   n
      entering node is equal
      to sum of currents          i (t )  0
                                  j 1
                                           j
      leaving node.


                                                 Lecture 12
                 KCL and KVL
   KVL
     The sum of voltages          n
      around any loop in a      v j 1
                                          j   (t )  0
      circuit is zero.


           -       + v2(t) -   +
        v1(t)                  v3(t)
           +                   -

                                              Lecture 12
                 KCL and KVL
   In KVL:
     A voltage encountered + to - is positive.
     A voltage encountered - to + is negative.

   Arrows are sometimes used to represent voltage
    differences; they point from low to high voltage.

                   +
                   v(t)   ≡             v(t)
                    -
                                               Lecture 12
           Resistors in Series
 A single loop circuit is one which has only
  a single loop.
 The same current flows through each
  element of the circuit - the elements are in
  series.



                                     Lecture 12
          Resistors in Series
Two elements are in series if the current that
flows through one must also flow through
the other.

                               Series
           R1      R2




                                    Lecture 12
              Resistors in Series
Consider two resistors in series with a
voltage v(t) across them:
                                Voltage division:
         i(t)
                                                       R1
         +          +
                                    v1 (t )  v(t )
               R1       v1(t)                       R1  R2
                    -
        v(t)                                          R2
                    +
                                    v2 (t )  v(t )
               R2       v2(t)                       R1  R2
          -         -
                                                    Lecture 12
           Resistors in Series
 If we wish to replace the two series
resistors with a single equivalent resistor
whose voltage-current relationship is the
same, the equivalent resistor has a value
given by
                 Req  R1  R2

                                     Lecture 12
          Resistors in Series
 For N resistors in series, the equivalent
resistor has a value given by

            R1
                   R2                  Req
            R3



         Req  R1  R2  R3    RN
                                       Lecture 12
           Resistors in Parallel
   When the terminals of two or more circuit
    elements are connected to the same two
    nodes, the circuit elements are said to be in
    parallel.




                                       Lecture 12
            Resistors in Parallel
Consider two resistors in parallel with a
voltage v(t) across them:
                          Current division:
       i(t)
                                                    R2
        +
                i1(t)   i2(t)    i1 (t )  i (t )
                                                  R1  R2
      v(t)       R1      R2                         R1
                                i2 (t )  i (t )
                                                 R1  R2
        -
                                           Lecture 12
         Resistors in Parallel
 If we wish to replace the two parallel
resistors with a single equivalent resistor
whose voltage-current relationship is the
same, the equivalent resistor has a value
given by
                         R1 R2
                  Req 
                        R1  R2

                                     Lecture 12
         Resistors in Parallel
 For N resistors in parallel, the equivalent
resistor has a value given by

          R1   R2   R3                 Req



                   1
      Req 
            1   1  1     1
                  
            R1 R2 R3    RN
                                     Lecture 12
       Energy Storage Elements
 Capacitors store energy in an electric field.
 Inductors store energy in a magnetic field.

 Capacitors and inductors are passive
  elements:
     Can  store energy supplied by circuit
     Can return stored energy to circuit

     Cannot supply more energy to circuit than is
      stored.
                                        Lecture 12
     Energy Storage Elements
 Voltages and currents in a circuit without
  energy storage elements are solutions to
  algebraic equations.
 Voltages and currents in a circuit with
  energy storage elements are solutions to
  linear, constant coefficient differential
  equations.

                                    Lecture 12
       Energy Storage Elements
   Electrical engineers (and their software tools)
    usually do not solve the differential equations
    directly.
   Instead, they use:
      LaPlace transforms
      AC steady-state analysis
   These techniques covert the solution of
    differential equations into algebraic problems.

                                         Lecture 12
      Energy Storage Elements
   Energy storage elements model electrical
    loads:
      Capacitors model computers and other
        electronics (power supplies).
      Inductors model motors.
   Capacitors and inductors are used to build
    filters and amplifiers with desired frequency
    responses.
   Capacitors are used in A/D converters to
    hold a sampled signal until it can be
    converted into bits.
                                         Lecture 12
                     Capacitors
   Capacitance occurs when two conductors are separated
    by a dielectric (insulator).
   Charge on the two conductors creates an electric field
    that stores energy.
   The voltage difference between the two conductors is
    proportional to the charge.
                           qt   C vt 
   The proportionality constant C is called capacitance.
   Capacitance is measured in Farads (F).
                                               Lecture 12
             Capacitors
 The
             +                             dv(t )
 rest i(t)
   of              v(t)
                                i (t )  C
  the
                                            dt
             -
circuit
                            t
                        1
                 v(t )   i ( x)dx
                        C 
                                       t
                                  1
                 v(t )  v(t 0 )   i ( x)dx
                                  C t0
                                           Lecture 12
               Capacitors
 The voltage across a capacitor cannot
  change instantaneously.
 The energy stored in the capacitors is given
  by
                      1 2
             wC (t )  Cv (t )
                      2


                                    Lecture 12
                      Inductors
   Inductance occurs when current flows through a (real)
    conductor.
   The current flowing through the conductor sets up a
    magnetic field that is proportional to the current.
   The voltage difference across the conductor is
    proportional to the rate of change of the magnetic flux.
   The proportionality constant is called the inductance,
    denoted L.
   Inductance is measured in Henrys (H).

                                                Lecture 12
                 Inductors
 The             +
                                            di(t )
 rest i(t)                        v(t )  L
   of        L       v(t)                    dt
  the
                 -
circuit
                                  t
                               1
                       i (t )   v( x)dx
                               L 
                                           t
                                           1
                        i (t )  i (t 0 )   v( x)dx
                                           L t0
                                               Lecture 12
                Inductors
 The current through an inductor cannot
  change instantaneously.
 The energy stored in the inductor is given
  by
                      1 2
             wL (t )  Li (t )
                      2


                                   Lecture 12
     Analysis of Circuits Containing
       Energy Storage Elements
   Need to determine:
     The  order of the circuit.
     Forced (particular) and natural
      (complementary/homogeneous) responses.
     Transient and steady state responses.
     1st order circuits - the time constant.
     2nd order circuits - the natural frequency
      and the damping ratio.

                                       Lecture 12
      Analysis of Circuits Containing
        Energy Storage Elements
   The number and configuration of the energy
    storage elements determines the order of the
    circuit.
            n  # of energy storage elements
   Every voltage and current is the solution to a
    differential equation.
   In a circuit of order n, these differential
    equations are linear constant coefficient and
    have order n.
                                          Lecture 12
      Analysis of Circuits Containing
        Energy Storage Elements
   Any voltage or current in an nth order
    circuit is the solution to a differential
    equation of the form
        d n v(t )        d n1v(t )
             n
                   an1      n 1
                                     ...  a0v(t )  f (t )
          dt               dt
    as well as initial conditions derived from
    the capacitor voltages and inductor
    currents at t = 0-.
                                                     Lecture 12
    Analysis of Circuits Containing
      Energy Storage Elements
   The solution to any differential equation consists
    of two parts:
                    v(t) = vp(t) + vc(t)
   Particular (forced) solution is vp(t)
      Response particular to the source

   Complementary/homogeneous (natural)
    solution is vc(t)
      Response common to all sources
                                          Lecture 12
    Analysis of Circuits Containing
      Energy Storage Elements
 The particular solution vp(t) is typically a
  weighted sum of f(t) and its first n
  derivatives.
 If f(t) is constant, then vp(t) is constant.

 If f(t) is sinusoidal, then vp(t) is sinusoidal.




                                        Lecture 12
     Analysis of Circuits Containing
       Energy Storage Elements
   The complementary solution is the
    solution to
          n              n 1
        d v(t )         d v(t )
            n
                 an 1     n 1
                                  ...  a0v(t )  0
         dt              dt

   The complementary solution has the form
                                 n
                    vc (t )   K i e sit
                                i 1

                                               Lecture 12
       Analysis of Circuits Containing
         Energy Storage Elements
 s1through sn are the roots of the
  characteristic equation
                        n 1
            s  an1s
             n
                                ...  a1s  a0  0




                                                      Lecture 12
    Analysis of Circuits Containing
      Energy Storage Elements
 If si is a real root, it corresponds to a
  decaying exponential term Ki e s t , si  0  i



 If si is a complex root, there is another
  complex root that is its complex conjugate,
  and together they correspond to an
  exponentially decaying sinusoidal term
           e  it  Ai cos  d t  Bi sin  d t 

                                                     Lecture 12
    Analysis of Circuits Containing
      Energy Storage Elements
 The steady state (SS) response of a circuit
  is the waveform after a long time has
  passed.
    DC SS if response approaches a
     constant.
    AC SS if response approaches a sinusoid.
 The transient response is the circuit
  response minus the steady state response.
                                   Lecture 12
    Analysis of Circuits Containing
      Energy Storage Elements
 Transients usually are associated with the
  complementary solution.
 The actual form of transients usually
  depends on initial capacitor voltages and
  inductor currents.
 Steady state responses usually are
  associated with the particular solution.
                                    Lecture 12
           First-Order Circuits

 Any circuit with a single energy storage
  element, an arbitrary number of sources,
  and an arbitrary number of resistors is a
  circuit of 1st order.
 Any voltage or current in such a circuit is
  the solution to a 1st order differential
  equation.

                                     Lecture 12
           First-Order Circuits

   Examples of 1st order circuits:
     Computer   RAM
        A dynamic RAM stores ones as charge on a
         capacitor.
        The charge leaks out through transistors
         modeled by large resistances.
        The charge must be periodically refreshed.

                                        Lecture 12
               First-Order Circuits
   Examples of 1st order circuits (Cont’d):
     The RC low-pass filter for an envelope detector in a
      superheterodyne AM receiver.
     Sample-and-hold circuit:
        The capacitor is charged to the voltage of a
         waveform to be sampled.
        The capacitor holds this voltage until an A/D
         converter can convert it to bits.
     The windings in an electric motor or generator can
      be modeled as an RL 1st order circuit.
                                              Lecture 12
              First-Order Circuits
                                 vR(t) -
                             +

                                  R        +
                         +
               vS(t)                  C        vC(t)
                         -                 -



   1st Order Circuit:
     One capacitor and one resistor
     The source and resistor may be equivalent to a
      circuit with many resistors and sources.
                                                       Lecture 12
            First-Order Circuits
                              vR(t)
                          +           -
                               R          +
                      +
              vS(t)                   C       vC(t)
                      -                   -
                              i(t)



   Let’s derive the (1st order) differential
    equation for the mesh current i(t).

                                                      Lecture 12
      First-Order Circuits
KVL around the loop:
           vS t   vR t   vC t 
We have
           vR t   Ri t 
                                   t
           vC t   vC 0   ix dx
                             1
                             C0


                                         Lecture 12
        First-Order Circuits
The KVL equation becomes:
                          t
      Ri t   vC 0   ix dx  vS t 
                        1
                        C0
Differentiating both sides w.r.t. t, we have
            dit  1        dvS t 
          R        it  
             dt    C          dt
or
        dit  1           1 dvS t 
                  it  
         dt     RC         R dt
                                        Lecture 12
            First-Order Circuits
                                   iL(t)
                           iR(t)           +

               iS(t)       R           L   v(t)

                                           -



   1st Order Circuit:
     One inductor and one resistor
     The source and resistor may be equivalent to a
      circuit with many resistors and sources.
                                                  Lecture 12
            First-Order Circuits
                                iL(t)
                        iR(t)           +

              iS(t)      R          L   v(t)

                                        -




   Let’s derive the (1st order) differential
    equation for the node voltage v(t).

                                               Lecture 12
       First-Order Circuits
KCL at the top node:
            iS t   iR t   iL t 
We have
                     vt 
           iR t  
                      R
                                  t
           iL t   iL 0    v x dx
                              1
                              L0
                                          Lecture 12
        First-Order Circuits
The KVL equation becomes:
       vt 
                         t
              iL 0   vx dx  iS t 
                       1
        R              L0
Differentiating both sides w.r.t. t, we have
         1 dvt  1        diS t 
                  vt  
         R dt     L          dt
or
         dvt  R          diS t 
                vt   R
          dt    L            dt
                                        Lecture 12
             First-Order Circuits
   For all 1st order circuits, the diff. eq. can be
    written as       dvt  1
                            vt   f t 
                       dt    
   The complementary solution is given by

                      vC t   Ke
                                     t
                                          



    where K is evaluated from the initial conditions.

                                              Lecture 12
            First-Order Circuits
   The time constant of the complementary
    response is .
     For an RC circuit,  = RC
     For an RL circuit,  = L/R

    is the amount of time necessary for an
    exponential to decay to 36.7% of its initial
    value.

                                       Lecture 12
            First-Order Circuits
   The particular solution vp(t) is usually a
    weighted sum of f(t) and its first derivative.
     If f(t) is constant, then vp(t) is constant.
     If f(t) is sinusoidal, then vp(t) is sinusoidal.




                                              Lecture 12

				
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