1. Reading Assignment: Chapter 8 in Electric Circuits, 9th Ed. by Nilsson
Chapter EGR 272 – Circuit Theory II 1
Reading Assignment: Chapter 8 in Electric Circuits, 9th Ed. by Nilsson
2nd-order circuits have 2 independent energy storage elements (inductors and/or capacitors)
Analysis of a 2nd-order circuit yields a 2nd-order differential equation (DE)
A 2nd-order differential equation has the form:
Solution of a 2nd-order differential equation requires two initial conditions: x(0) and x’(0)
All higher order circuits (3rd, 4th, etc) have the same types of responses as seen in 1st-order and 2nd-order circuits

2. Series RLC and Parallel RLC Circuits
Chapter EGR 272 – Circuit Theory II 2
Series RLC and Parallel RLC Circuits
Since 2nd-order circuits have two energy-storage types, the circuits can have the following forms:
1) Two capacitors
2) Two inductors
3) One capacitor and one inductor
A) Series RLC circuit *
B) Parallel RLC circuit *
C) Others
* The textbook focuses on these two types of 2nd-order circuits
Series RLC circuit
Parallel RLC circuit

3. Form of the solution to differential equations
Chapter EGR 272 – Circuit Theory II 3
Form of the solution to differential equations
As seen with 1st-order circuits in Chapter 7, the general solution to a differential equation has two parts:
where xh or xn is due to the initial conditions in the circuit
and xp or xf is due to the forcing functions (independent voltage and current sources for t > 0).
The forced response
The forced response is due to the independent sources in the circuit for t > 0. Since the natural response will die out once the circuit reaches steady-state, the forced response can be found by analyzing the circuit at t = . In particular,
x(t) = xh + xp = homogeneous solution + particular solution
or x(t) = xn + xf = natural solution + forced solution
xf = x()

4. A 2nd-order differential equation has the form:
Chapter EGR 272 – Circuit Theory II 4
The natural response
A 2nd-order differential equation has the form:
where x(t) is a voltage v(t) or a current i(t).
To find the natural response, set the forcing function f(t) (the right-hand side of the DE) to zero.
Substituting the general form of the solution Aest yields the characteristic equation:
s2 + a1s + ao = 0
Finding the roots of this quadratic (called the characteristic roots or natural frequencies) yields:

5. Roots are real and distinct [ (a1)2 > 4ao ] Solution has the form:
Chapter EGR 272 – Circuit Theory II 5
Characteristic Roots
The roots of the characteristic equation may be real and distinct, repeated, or complex. Thus, the natural response to a 2nd-order circuit has 3 possible forms:
1) Overdamped response
Roots are real and distinct [ (a1)2 > 4ao ]
Solution has the form:
Sketch the form of the solution.
Discuss the concept of the dominant root.

6. 2) Critically damped response
Chapter EGR 272 – Circuit Theory II 6
2) Critically damped response
Roots are repeated [ (a1)2 = 4ao ] so s1 = s2 = s = -a1/2
Solution has the form:
Sketch the form of the solution.

7. 3) Underdamped response
Chapter EGR 272 – Circuit Theory II 7
3) Underdamped response
Roots are complex [ (a1)2 < 4ao ] so s1 , s2 =   j
Show that the solution has the form:
Sketch the form of the solution.
Discuss the concept of the exponential envelope.
Sketch xn if A1 = 0, A2 = 10,  =-1, and  = .
Sketch xn if A1 = 0, A2 = 10,  =-10, and  = 100.

8. Discuss the possible responses for v(t)
Chapter EGR 272 – Circuit Theory II 8
Illustration: The transient response to a 2nd-order circuit must follow one of the forms indicated above (overdamped, critically damped, or underdamped). Consider the circuit shown below. v(t) is 0V for t < 0 and the steady-state value of v(t) is 10V. How does it get from 0 to 10V?
Discuss the possible responses for v(t)
Define the terms damping, rise time, ringing, and % overshoot

9. A cruise-control circuit The output of a logic gate
Chapter EGR 272 – Circuit Theory II 9
Examples: When is each of the 3 types of responses desired? Discuss the following cases:
An elevator
A cruise-control circuit
The output of a logic gate
The start up voltage waveform for a DC power supply

10. s2 + 2s + wo2 = 0 Series and Parallel RLC Circuits
Chapter EGR 272 – Circuit Theory II 10
Series and Parallel RLC Circuits
Two common second-order circuits are now considered:
series RLC circuits
parallel RLC circuits.
Relationships for these circuits can be easily developed such that the characteristic equation can be determined directly from component values without writing a differential equation for each example.
A general 2nd-order differential equation has the form:
A general 2nd-order characteristic equation has the form:
where
 = damping coefficient
wo = resonant frequency
s2 + 2s + wo2 = 0

11. Series RLC Circuit - develop expressions for  and wo
Chapter EGR 272 – Circuit Theory II 11
Series RLC Circuit - develop expressions for  and wo

12. Parallel RLC Circuit - develop expressions for  and wo
Chapter EGR 272 – Circuit Theory II 12
Parallel RLC Circuit - develop expressions for  and wo

13. Chapter 8 EGR 272 – Circuit Theory II13
Procedure for analyzing 2nd-order circuits (series RLC and parallel RLC)
1. Find the characteristic equation and the natural response
Is the circuit a series RLC or parallel RLC? (for t > 0 with independent sources killed)
Find  and wo2 and use these values in the characteristic equation: s2 + 2s + wo2.
Find the roots of the characteristic equation (characteristic roots or natural frequencies).
Determine the form of the natural response based on the type of characteristic roots:
Overdamped: Real, distinct roots (s1 and s2):
Underdamped: Complex roots (s1,s2 = j):
Critically damped: Repeated roots (s=s1=s2):
Find the forced response - Analyze the circuit at t =  to find xf = x().
Find the initial conditions, x(0) and x’(0).
A) Find x(0) by analyzing the circuit at t = 0- (find all capacitor voltages and inductor currents)
Analyze the circuit at t = 0+ (using vC(0) and iL (0) from step 3B) and find:
4. Find the complete response
Find the total response, x(t) = xn + xf .
Use the two initial conditions to solve for the two unknowns in the total response.

14. Example: Determine v(t) in the circuit shown below for t > 0 if :
Chapter EGR 272 – Circuit Theory II 14
Example: Determine v(t) in the circuit shown below for t > 0 if :
A) R = 7
B) R = 2
C) R =

15. Example: B) Continued with R = 2
Chapter EGR 272 – Circuit Theory II 15
Example: B) Continued with R = 2

16. Example: C) Continued with R =
Chapter EGR 272 – Circuit Theory II 16
Example: C) Continued with R =

17. Example: Determine v(t) in the circuit shown below for t > 0.
Chapter EGR 272 – Circuit Theory II 17
Example: Determine v(t) in the circuit shown below for t > 0.

18. Example: Determine v(t) in the circuit shown below for t > 0.
Chapter EGR 272 – Circuit Theory II 18
Example: Determine v(t) in the circuit shown below for t > 0.
Note: In determining if a circuit is a series RLC or parallel RLC circuit, consider the circuit for t > 0 with all independent sources killed.

19. Example: Determine i(t) in the circuit shown below for t > 0.
Chapter EGR 272 – Circuit Theory II 19
Example: Determine i(t) in the circuit shown below for t > 0.
20H