laplace solution transient circuits
TRANSCRIPT
Laplace Transform Solutions of Transient Circuits
1
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
LEARNING OUTCOMES
Laplace transformProperties of Laplace transformLaplace Circuit Element ModelsApplication of Laplace transform in transient analysis
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
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
3
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
4
Definition of Laplace Transform
• Definition of the unilateral (one-sided) Laplace transform
where s=+j is the complex frequency, and f(t)=0 for t<0
• The inverse Laplace transform requires a course in complex variable analysis
0
dtetfstf stL F
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
5
Laplace Transform Pairs
Most of the time we find Laplace transforms from tables like Table 1 rather than using the defining integral
(t) F(s)
δ(t) 1
u(t) {a constant} s
1
e–at as
1
t 2
1
s
t e–at 2
1
as
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
6
Some Laplace Transform Properties
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Laplace Circuit Solutions
• In this section 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
7
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
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
8
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
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
9
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Capacitor
• Begin with the time domain relation for the element
• 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)
10
dt
v(t)dCi(t)
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Capacitor (cont’d.)
• Rearranging the above expression for the capacitor
• 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
11
s
v(0)
Cs
I(s)V(s)
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
s-Domain Model for C
s
vsI
sCsV
CvssCVvssVCsI
dt
tdvCti
)0()(
1)(or
)0()()0()()(
)()(
capacitor, aFor
Time domain s domain s domain
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Inductor
• Begin with the time domain relation for the element
• 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)
13
dt
i(t)dLv(t)
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Inductor (cont’d.)
• Rearranging the above expression for the inductor
• 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
14
s
i(0)
Ls
V(s)I(s)
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
s-Domain Models for an inductor
s
isV
sLsI
LissLIissILsV
dt
tdiLtv
)0()(
1)(or
)0()()0()()(
)()(
inductor,an For
Time domain s domain s domain
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Circuit Element Models
• Steps in applying the Laplace transform:
– Transform the circuit from the time domain to the s domain
– Solve the circuit using circuit analysis technique (nodal/mesh analysis, source transformation, etc.)
– Take the inverse Laplace transform of the solution and thus obtain the solution in the time domain
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Summary
s
isV
sLsI
LissLIsV
)0()(
1)(
)0()()(
s
vsI
sCsV
CvssCVsI
)0()(
1)(
)0()()(
For inductor: For capacitor:
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
Summary
Element Z(s)
Resistor R
Inductor sL
Capacitor 1/sC
*Assuming zero initial conditions
• Impedance in the s domain– Z(s)=V(s)/I(s)
• Admittance in the s domain– Y(s)=1/Z(s)=V(s)/I(s)
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
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
19
CE00436-1 ELECTRICAL PRINCIPLES LAPLACE TRANSFORM SOLUTION OF TRANSIENT CIRCUIT
EEE 202
Any Q ?