1. Laplace Circuit Analysis
Preamble
In this course we have modeled circuit elements (resistors, inductors, and capacitors) as linear and time invariant. We have used linear differential equations to determine the transients. We have modified this analysis for AC steady state by using jw with inductors and capacitors to form impedance. We coupled this with phasors to solve AC steady state circuits.
In the case of Laplace transforms we take the Laplace transform of the voltages and currents that describe the circuit elements (resistors, capacitors, inductors). This converts the circuit models to functions of s as illustrated here. We then use all the methods of analysis that we learned for DC circuits such as nodal, mesh, Thevenin, Norton, to solve the Laplace circuits which we say are in the complex frequency domain. Everything is algebraic and it makes the problem(s) easier to handle than if we carried out everything in the time domain with integral, differential equations. Once we have found the particular variable we seek, such a voltage or current, we then take the inverse Laplace to go to the time domain.
Electrical and Computer Engineering Department
University of Tennessee
Knoxville, Tennessee
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2. Laplace Circuit Analysis
Circuit Element Modeling

3. Laplace Circuit Analysis
Circuit Element Modeling
Resistance
Time Domain
Complex Frequency Domain

4. Laplace Circuit Analysis
Circuit Element Modeling
Inductor

5. Laplace Circuit Analysis
Capacitor

6. Laplace Circuit Analysis
Linear Transformer

7. Laplace Circuit Analysis
Time domain to complex frequency domain
Oh man!
What a mess.

8. Laplace Circuit Analysis
Circuit Application:
Given the circuit below. Assume zero IC’s. Use Laplace to find vc(t).
The time domain circuit:
Laplace circuit

9. Laplace Circuit Analysis
Circuit Application:

10. Laplace Circuit Analysis
Circuit Application:
Given the circuit below. Assume vc(0) = - 4 V. Use Laplace to find vc(t).
The time domain circuit:
Laplace circuit:

11. Laplace Circuit Analysis
Circuit Application:
Check the boundary conditions
vc(0) = - 4 V
vc(oo) = 2 V

12. Laplace Circuit Analysis
Circuit Application:
Find i0(t) using Laplace

13. Laplace Circuit Analysis
Circuit Application:
Find i0(t) using Laplace
Mesh 1

14. Laplace Circuit Analysis
Circuit Application:
Find i0(t) using Laplace
Mesh 2

15. Laplace Circuit Analysis
Circuit Application:
Find i0(t) using Laplace
Add these 2
equations

16. Laplace Circuit Analysis
Circuit Application:
Find i0(t) using Laplace
Is final value of
i2(t) reasonable?

17. That’s all Folks !