1. ECE 342 1. Networks & Systems Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois

2. What is Capacitance? 3 1 2 Voltage = 6V Voltage=0 Voltage build up
No Charge
No Current 1 2
Voltage build up
Charge build up
Transient Current 3
Voltage = 6V
Charge=Q
No Current

3. What is Capacitance? 4 6 5 Voltage=0 Voltage decaying Voltage=6V
Charge=Q
No Current 4
Voltage decaying
Charge decaying
Transient Current 5
Voltage=0
No Charge
No Current 6

4. Capacitance
Relation: Q = Cv
Q: charge stored by capacitor
C: capacitance
v: voltage across capacitor
i: current into capacitor

5. What is Inductance? Current in wire produces magnetic field
Flux is magnetic field integrated over area

6. Inductance

7. Inductance
Relation: Y = Li
Y: flux stored by inductor
L: inductance
i: current through inductor
v: voltage across inductor

8. Low-Pass Circuit
Need to solve for vo(t)

9. Low-Pass Circuit
Need to solve for vo(t)

10. Reactive Circuit
Need to solve for vx(t)

11. Laplace Transforms
The Laplace transform F(s) of a function f(t) is defined as:
To a mathematician, this is very meaningful; however circuit engineers seldom use that integral
The Laplace transform provides a conversion from the time domain into a new domain, the s domain

12. s = jw Laplace Transforms
The Laplace transform of the derivative of a function f(t) is given by
Differentiation in the time domain becomes a multiplication in the s domain. That is why Laplace transforms are useful to circuit engineers
s = jw

13. Low-Pass Circuit
Time
domain s
domain
Time domain
s domain

14. Low-Pass Circuit - Solution
Pole exists at
Assume that Vi(s)=Vd/s
Inversion

15. Reactive Circuit

16. Reactive Circuit - Solution
Exercise: Use inversion to solve for vx(t)

17. Table of Laplace Transforms
Function
Transform

18. Capacitance – s Domain The capacitor is a reactive element
Assume time-harmonic behavior of voltage and current variables jw = s
The capacitor is a reactive element

19. Inductance – s Domain The inductor is a reactive element
Assume time-harmonic behavior of voltage and current variables jw = s
The inductor is a reactive element

20. Thevenin Equivalent Principle Application
Any linear two-terminal network consisting of current or voltage sources and impedances can be replaced by an equivalent circuit containing a single voltage source in series with a single impedance.
Application
To find the Thevenin equivalent voltage at a pair of terminals, the load is first removed leaving an open circuit. The open circuit voltage across this terminal pair is the Thevenin equivalent voltage.
The equivalent resistance is found by replacing each independent voltage source with a short circuit (zeroing the voltage source), replacing each independent current source with an open circuit (zeroing the current source) and calculating the resistance between the terminals of interest. Dependent sources are not replaced and can have an effect on the value of the equivalent resistance.

21. Norton Equivalent Principle Application
Any linear two-terminal network consisting of current or voltage sources and impedances can be replaced by an equivalent circuit containing a single current source in parallel with a single impedance.
Application
To find the Norton equivalent current at a pair of terminals, the load is first removed and replaced with a short circuit. The short-circuit current through that branch is the Norton equivalent current.
The equivalent resistance is found by replacing each independent voltage source with a short circuit (zeroing the voltage source), replacing each independent current source with an open circuit (zeroing the current source) and calculating the resistance between the terminals of interest. Dependent sources are not replaced and can have an effect on the value of the equivalent resistance.

22. Thevenin Equivalent - Example
Circuit for calculating impedance
Calculating Thevenin voltage
KCL law at node A’ gives 1+Iy = Ix
Also KVL gives 10=2Iy+3Ix
Combining these equations gives Ix = 2.4 mA
From which we calculate
Vx = Vth=3(2.4)=7.2 V

23. Thevenin Example
Find voltage across R4
From which

24. Thevenin Equivalent Circuit for calculating impedance
(current source is replaced with open circuit)

25. Circuit with Thevenin Equivalent
ZTH
VTH

26. Need to find the current i1(t) in resistor R3
Thevenin Example
Need to find the current i1(t) in resistor R3

27. Thevenin voltage calculation
Thevenin Example
Thevenin voltage calculation

28. Calculating the impedance
Thevenin Example
Calculating the impedance

29. Thevenin Example

30. Expanding using partial fraction The time-domain current i1(t) is:
Thevenin Example
Expanding using partial fraction
With K1 = 3.33, K2 = -5, K3 = 1.67
The time-domain current i1(t) is: