Overview
Second Order
Circuits II: Initial
Conditions
•  Recap class before break
o 2nd order circuit behavior options
o Analysis objective and method
•  Second order circuit solutions
o Step response
o Initial conditions
EGR 220, Chapter 8.2
March 22, 2016
2
High-Level Solution, Complete Only Once
1)  Develop 2nd order differential equation (KVL, KCL)
2)  Assume form of solution
o  i(t) = Aest or v(t) = Aest
•  What are the options for the
values s1 and s2?
3)  Substitute assumed solution into the 2nd order
diff. e. q., and obtain…
4)  Characteristic equation (s2 + as + b)=0
s1,2 = −α ± α 2 − ω02
o  This has two roots, so our solution must in fact be
some form of A1es t + A2es t
1
Form of Natural Response
•  What is the relationship
of α and ωo for each type
of response?
2
5)  Solve for ‘s1,2’ the roots of the characteristic
equation
6)  Identify type of damping and then…
7)  See handout for detailed process
3
4
1
Natural Responses (h.o.)
Oscillations!?
•  Overdamped α > ω0
o  The roots, ‘s,’ are negative, unequal and real
•  Where do the oscillations come from?
x(t) = A1e s1t + A2 e s 2 t
•  Critically damped α = ω0
2
s1,2 = −
" R%
R
1
± $ ' −
# 2L & LC
2L
=
− α ± α 2 − ω 02
o  The roots, ‘s,’ are equal and real
ۥ
x(t) = A1e s1t + A2 e s 2 t
x(t) = (A1 + A2 t)e −αt
Underdamped α < ω0
o  The roots, ‘s,’ are complex
x(t) = e −αt ( A1 cosω d t + A2 sinω d t)
€
•  Complete Response? … include X∞(t)
€
5
Solution Methodology To Use
1.  Find initial and final conditions
€
6
Initial & Final Values (problem 8.2)
o  Find: iR(0+), iL(0+) and iC(0+),
o  and: diR(0+)/dt, diL(0+)/dt and diC(0+)/dt,
o  and: iR(∞), iL(∞) and iC(∞)
o  Concepts of t = 0– and t = 0+; t = ∞
2.  Find α and ωo (instead of τ) & s1 and s2
3.  Identify type of damping (form for the
natural response)
4.  Construct complete response expression
5.  Solve for constants, using initial conditions
o  dvc(0+)/dt = ic(0+)/C or
o  diL(0+)/dt = vL(0+)/L
6.  Write out, and be able to graph, final
solution
7
8
2
Initial Values 1
Initial Values 2
1. Brainstorm and write down
everything we know about the
circuit
2. Begin solving: For just before time t = 0,
which we define as time t = 0–, find
ð iL(0–) and
ð vc(0–)
(Which also tell us what?)
9
Initial Values 3
10
Initial Values 4
4. Now find the first derivatives:
What do we know about vL(t)
in general, and what can we
deduce about vL(0+) ?
Use this to find diL(0+)/dt
3. For time t = 0+, find
ð iR(0+) (use KVL) and
ð ic(0+) (use KCL)
11
12
3
Initial Values 5
Initial Values 6
5. Finding diR(0+)/dt
ð Use ‘d(KVL)/dt’ to find an
expression for diR(0+)/dt
ð Use ic = C(dvc/dt) to solve
for diR(0+)/dt
6. Finally, find dic(0+)/dt
ð Use ‘d(KCL)/dt’ to find an
expression for dic(0+)/dt
13
14
Response of an RLC Circuit
Final Values
•  Find v(t) for t > 0
7. Find the final values, at t = ∞
ð iL(∞) =
ð iR(∞) =
ð iC(∞) =
15
16
4
Response of an RLC Circuit
Terminology Summary
•  Find v(t) for t > 0
•  Know how to find:
R
L
o  Characteristic equation s 2 + s +
1
LC
(series)
o  Natural frequencies s1,2 = −α ± α 2 − ω02
o  Resonant frequency ω0 ≡
o  Damping factor
α≡
1
LC
R
(series)
2L
≡
1
(parallel)
2RC
o  Damping frequency, ωd, for underdamped
systems
17
18
Summary
•  Second order circuits
o Finding initial and final conditions
•  Complete Response: Source-free,
forced and step responses
o Find natural frequencies, from the
“characteristic equation”
o Find constant coefficients, using the initial
conditions and complete response
expression
19
5