Exam #2 Review
Dr. Holbert
March 26, 2008
LectR2
EEE 202
1
Don’t Forget the Essentials
•
•
•
•
Verify voltage polarity and current direction
Obey the passive sign convention
The Fundamentals: Ohm’s Law; KCL; KVL
Series/Parallel Impedance combinations
Z series  Z1  Z 2    Z N   Z j
1
1
1
1
1




Z par Z1 Z 2
ZM
Zi
LectR2
EEE 202
2
Circuit Analysis Techniques
• All these circuit analysis techniques have
wide applicability: DC, AC, and Transient
• Voltage and Current Division
• Nodal and Loop/Mesh Analyses
• Source Transformation
• Superposition
• Thevenin’s and Norton’s Theorems
LectR2
EEE 202
3
Transient Circuit Analysis
•
•
•
•
•
First and second order circuit responses
Differential equation approach
Laplace transform approach
Inspection (step-by-step) method
Bottom line: Using appropriate techniques
can you find v(t) and/or i(t) in transient
RLC circuits?
LectR2
EEE 202
4
RLC Characteristics
Element
Resistor
V/I Relation
Capacitor
d vC (t )
iC (t )  C
dt
I = 0; open circuit
Inductor
d iL (t )
vL (t )  L
dt
V = 0; short circuit
vR (t )  R iR (t )
DC Steady-State
V=IR
ELI and the ICE man
LectR2
EEE 202
5
Circuit ODE Solutions
• Determine the circuit differential equation(s)
• Find the forced (particular) and natural
(complementary) solutions
• First-order vs. second-order circuits
– First-order: find time constant (=RC; =L/R)
– Second-order: Compute the natural frequency, 0,
and the damping ratio,  (or compute the roots, s1,2, of
the characteristic equation)
• Transient and steady-state waveforms
LectR2
EEE 202
6
Damping Summary
Damping
Poles (s1, s2)
Ratio
ζ>1
Real and unequal
ζ=1
Real and equal
Damping
Overdamped
Critically damped
0 < ζ < 1 Complex conjugate
Underdamped
pair set
ζ=0
Purely imaginary pair Undamped
LectR2
EEE 202
7
Laplacian Domain
• Determining the Laplace transform from
– The defining integral
– Transform pairs in conjunction with properties
u(t) ↔ 1/s
e-at ↔ 1/(s+a)
– Circuit element representations in s domain
• Finding the transfer function
• Performing the inverse Laplace transform
to find the time-domain response
– Three possible cases based on poles
LectR2
EEE 202
8
Laplacian of Circuit Elements
Using Ohm’s Law,
impedance (Z) can
be defined via:
V=IZ
IC(s)
+
VC(s)
–
LectR2
Circuit
Element
Resistor
Capacitor
Inductor
IL(s)
+
1/sC
+
–
Impedance
(ohms)
R
1 / (sC)
sL
VL(s)
–
v(0)
s
EEE 202
sL
i(0)
s
9
Transfer Function
• The transfer function, H(s), is the ratio of some
output variable (y) to some input variable (x)
Y(s) Output
H(s) 

X(s) Input
• The transfer function is shown in block diagram
form as (where h(t) is the impulse response)
X(s) ↔ x(t)
Input
LectR2
System
H(s) ↔ h(t)
EEE 202
Y(s) ↔ y(t)
Output
10
Some Terminology & Quantities
Our vocabulary has
expanded with
several new terms,
including:
• Phasor & impedance
• Impulse (delta) and
step functions
• Transfer function
• Impulse response
• Poles and zeros
LectR2
• Initial and final value
theorems
• Linearity and time
invariance
• Convolution integral
• Period, frequency,
and amplitude
• Characteristic
equation
• Over/under damped
EEE 202
11