EGR260-Ch8 (Second Order)

Chapter 8 EGR 260 – Circuit Analysis 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:
d2x dx
2
 a1  a o x(t)  f(t)
dt dt
• 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
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
t=0 i(t)
t=0
R L i(t)
+ + +
Vs Is
_ v (t) R L C v (t)
C
_
_
Series RLC circuit Parallel RLC circuit

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:
x(t) = xh + xp = homogeneous solution + particular solution
or x(t) = xn + xf = natural solution + forced solution
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,
xf = x()
The natural response

A 2nd-order differential equation has the form:
d2x dx
2
 a 1  a o x(t)  f(t)
dt dt
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. d 2 x dx
2
 a 1  a o x(t)  0
dt dt
Substituting the general form of the solution Aest yields the characteristic
equation:
s2 + a1 s + ao = 0
Finding the roots of this quadratic (called the characteristic roots or natural
frequencies) yields:
-a1   a1 
2
- 4a o
s1 , s2 
2
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:
x n  A1e s1t
 A 2e s 2t
Sketch the form of the solution.

Discuss the concept of the dominant root.
2) Critically damped response

Roots are repeated [ (a1)2 = 4ao ] so s1 = s2 = s = -a1/2
Solution has the form:
xn   A1t  A2  e st

3) Underdamped response
Roots are complex [ (a1)2 < 4ao ] so s1 , s2 =   j
Show that the solution has the form:
xn  e t
 A1 cos(  t)  A 2 sin(  t) 
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.
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
t=0
R L
+ +
10 V _
C v (t)
_
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
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:
d2x dx
  o x(t)  f(t)
2
2
2α ω
dt dt
A general 2nd-order characteristic equation has the form:
s2 + 2s + wo2 = 0
where
 = damping coefficient
wo = resonant frequency
Series RLC Circuit - develop expressions for  and wo

t=0 i(t)
R L
+ +
Vs _
C v (t)
_
Parallel RLC Circuit - develop expressions for  and wo
t=0
i(t)
+
Is
R L C v (t)
_
Procedure for analyzing 2nd-order circuits (series RLC and parallel RLC)
1. Find the characteristic equation and the natural response
A) Is the circuit a series RLC or parallel RLC? (for t > 0 with independent sources killed)
B) Find  and wo2 and use these values in the characteristic equation: s2 + 2s + wo2.
R 1 1
 (series RLC)  (parallel RLC) wo 
2
(both series and parallel RLC)
2L 2RC LC
C) Find the roots of the characteristic equation (characteristic roots or natural frequencies).
D) Determine the form of the natural response based on the type of characteristic roots:
a) Overdamped: Real, distinct roots (s1 and s2): x n  A1es1t  A 2es2t
 1
b) Underdamped: Complex roots (s1,s2 = j): x  e t A cos(  t)  A sin(  t)
n 2 
c) Critically damped: Repeated roots (s=s1=s2):
xn   A1t  A2  e
st
2. Find the forced response - Analyze the circuit at t =  to find xf = x().

3. 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)
B) Analyze the circuit at t = 0+ (using vC(0) and iL (0) from step 3B) and find:
 
dvC 0  i C 0 

  or
diL 0   

v L 0  
dt C dt L
4. Find the complete response
A) Find the total response, x(t) = xn + xf .
B) Use the two initial conditions to solve for the two unknowns in the total response.
Example: Determine v(t) in the circuit shown below for t > 0 if :

t=0 i(t)
A) R = 7
B) R = 2 R 1H
+
+
C) R = 2 10 10 V _
0.1 F v (t)
_
Example: B) Continued with R = 2

Example: C) Continued with R = 2 10

Example: Determine v(t) in the circuit shown below for t > 0.

t=0
3 i(t)
+
8A 1 5H 0.05 F v (t)
_
Example: Determine v(t) in the circuit shown below for t > 0.
i(t) Note: In determining if a circuit is

6 + a series RLC or parallel RLC
4u(t) A 0.1 F v (t) circuit, consider the circuit for t > 0
_ with all independent sources killed.
1H
Example: Determine i(t) in the circuit shown below for t > 0.

10 i(t) 3
t=0
+ + +
20 V _ 1 _ 4V
4H 20H F v(t)
20
_

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Chapter 8 EGR 260 – Circuit Analysis 1

Reading Assignment: Chapter 8 in Electric Circuits, 9th Ed. by Nilsson

Series RLC and Parallel RLC Circuits

Series RLC circuit Parallel RLC circuit

Form of the solution to differential equations

where xh or xn is due to the initial conditions in the circuit

The forced response

The natural response

Sketch the form of the solution.

2) Critically damped response

Sketch the form of the solution.

Illustration: The transient response to a 2nd-order circuit must follow one of

Examples: When is each of the 3 types of responses desired? Discuss the

Series and Parallel RLC Circuits

Series RLC Circuit - develop expressions for  and wo

Parallel RLC Circuit - develop expressions for  and wo

2. Find the forced response - Analyze the circuit at t =  to find xf = x().

Example: Determine v(t) in the circuit shown below for t > 0 if :

Example: B) Continued with R = 2

Example: C) Continued with R = 2 10

Example: Determine v(t) in the circuit shown below for t > 0.

Example: Determine v(t) in the circuit shown below for t > 0.

i(t) Note: In determining if a circuit is

Example: Determine i(t) in the circuit shown below for t > 0.

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