Laplace Transform Solutions of Transient Circuits

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					          Laplace Transform
         Solutions of Transient
                Circuits

                Dr. Holbert
               March 5, 2008


Lect13             EEE 202        1
              Introduction
• In a circuit with energy storage elements,
  voltages and currents are the solutions to
  linear, constant coefficient differential
  equations
• Real engineers almost never solve the
  differential equations directly
• It is important to have a qualitative
  understanding of the solutions

Lect13              EEE 202                    2
         Laplace Circuit Solutions
• In this chapter we will use previously
  established techniques (e.g., KCL, KVL,
  nodal and loop analyses, superposition,
  source transformation, Thevenin) in the
  Laplace domain to analyze circuits
• The primary use of Laplace transforms
  here is the transient analysis of circuits


Lect13               EEE 202                   3
 Laplace Circuit Element Models
• Here we develop s-domain models of circuit
  elements
• DC voltage and current sources basically remain
  unchanged except that we need to remember
  that a dc source is really a constant, which is
  transformed to a 1/s function in the Laplace
  domain




Lect13                EEE 202                   4
                     Resistor
• We start with a simple (and trivial) case, that of the
  resistor, R
• Begin with the time domain relation for the element
                   v(t) = R i(t)
• Now Laplace transform the above expression
                   V(s) = R I(s)
• Hence a resistor, R, in the time domain is simply
  that same resistor, R, in the s-domain



Lect13                    EEE 202                          5
                    Capacitor
• Begin with the time domain relation for the element
                              d v(t)
                     i(t)  C
                                dt
• Now Laplace transform the above expression
                    I(s) = s C V(s) – C v(0)
• Interpretation: a charged capacitor (a capacitor with
  non-zero initial conditions at t=0) is equivalent to an
  uncharged capacitor at t=0 in parallel with an
  impulsive current source with strength C·v(0)

Lect13                    EEE 202                           6
           Capacitor (cont’d.)
• Rearranging the above expression for the capacitor
                        I(s) v(0)
                 V(s)      
                        sC     s
• Interpretation: a charged capacitor can be replaced
  by an uncharged capacitor in series with a step-
  function voltage source whose height is v(0)
• Circuit representations of the Laplace transformation
  of the capacitor appear on the next page



Lect13                   EEE 202                       7
                   Capacitor (cont’d.)
                                          iC(t)
         Time                         +
         Domain                     vC(t)                 C
                                      –


               IC(s)
                                                  IC(s)
           +
                           1/sC                   +              Cv(0)
                                                          1/sC
         VC(s)                               VC(s)
                       +   v(0)
                       –    s                 –
           –
                 Laplace (Frequency) Domain Equivalents
Lect13                            EEE 202                                8
                   Inductor
• Begin with the time domain relation for the
  element                  d i(t)
                     v(t)  L
                                 dt
• Now Laplace transform the above expression
                   V(s) = s L I(s) – L i(0)
• Interpretation: an energized inductor (an
  inductor with non-zero initial conditions) is
  equivalent to an unenergized inductor at t=0 in
  series with an impulsive voltage source with
  strength L·i(0)
Lect13                 EEE 202                      9
            Inductor (cont’d.)
• Rearranging the above expression for the
  inductor           V(s) i(0)
                  I(s)         
                           sL        s
• Interpretation: an energized inductor at t=0 is
  equivalent to an unenergized inductor at t=0 in
  parallel with a step-function current source with
  height i(0)
• Circuit representations of the Laplace
  transformation of the inductor appear on the
  next page
Lect13                     EEE 202                    10
                       Inductor (cont’d.)

          Time                        +
          Domain                     vL(t)              L
                                               iL(0)
                                      –

               IL(s)                            IL(s)
           +                                   +
                            sL
         VL(s)                                          sL   i(0)
                                              VL(s)            s
                        –
                            Li(0)              –
           –            +

               Laplace (Frequency) Domain Equivalents
Lect13                              EEE 202                         11
          Analysis Techniques
• In this section we apply our tried and
  tested analysis tools and techniques to
  perform transient circuit analyses
     – KVL, KCL, Ohm’s Law
     – Voltage and Current division
     – Loop/mesh and Nodal analyses
     – Superposition
     – Source Transformation
     – Thevenin’s and Norton’s Theorems
Lect13                 EEE 202              12
           Transient Analysis
• Sometimes we not only must Laplace transform
  the circuit, but we must also find the initial
  conditions

         Element     DC Steady-State
         Capacitor   I = 0; open circuit
         Inductor    V = 0; short circuit



Lect13                 EEE 202                     13
           Class Examples
• Drill Problems P6-4, P6-5




Lect13              EEE 202   14