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Exam 1 Crib Sheet
Ohm’s Law – Linear relationship
between voltage and current in a
resistor
V = I R
V – Voltage, Volts [V]
I – Current, Amps [A]
R – Resistance, Ohms [Ω]
Power
P = V I
P – Power, Watts [W]
Using the above polarities (which may ot be correct)
For P > 0, the component consumes power
For P < 0, the component produces power
V
I
Node – a connection between two or
more components
Loop – a closed path through which
current can flow
KCL – Kirchoff’s Current Law
N
n n 1
I 0
The sum of the currents leaving a node is zero
(signs determined by polarity).
I1 - I2 + I3 = 0
I1 I
I
KVL – Kirchoff’s Voltage Law N
n n 1
V 0
The sum of the voltages around any closed loop
is zero (signs determined by polarity).
V1 + V2 - V3 = 0
V
V
V
Superposition – For each independent source, turn off all other independent sources and find the
contribution from that source. Sum the contribution from each source to get the parameter of interest.
Source transformation
Rs
Vs (^) Is=Vs/Rs Rs
Resistors in series – REQ R 1 R 2
R1 R
Resistors in parallel -
1 1 1
R EQ
R R
R1 R
Exam 1 Crib Sheet
Example includes a Current Controlled Voltage Source (CCVS) as a dependent source and I1 as an
independent source.
R2 (^) i
I
0
A
2000Ix R3 C
i
Ix
R
B
R i
V A VA VB
R R
VB VB VA V C
I
R R R
C B x
V V I
B x
V
I
R
Node Analysis Mesh Analysis
i 1^ R^1^^ ^ i 1^ R^3^^ ^ i 1^ ^ i 2^ R^2 ^0
2000 I x i 3 R 4 i 2 i 1 R 2 0
3 2 1 i i I
i 1 (^) i 2 (^) Ix
Thevenin voltage ( VTH ) – Open circuit the load, find the voltage across the load nodes
Norton current ( IN )– Short circuit the load, find the current through that short circuit
Thevenin resistance ( RTH ) – Turn off all independent sources, replace the load with a test voltage
(Vtest), find the current (Itest) through the test voltage, RTH = Vtest/Itest.
VTH = IN RTH (Ohm’s Law relationship)
Comparator
If V1 < V2 , Vout = V
saturation
If V1 > V2 , Vout = V
V
Vout
U
OUT
V
Inverting amplifier circuit
nn 1
R
Vout Vi R
U
OPAMP
R
R
0
Vin Vout
Non-inverting amplifier circuit
1 nn 1
R
Vout Vi R
U
OUT
R
R
0
Vout
Vin
Summing amplifier circuit
Rf Rf Vout V V R R
U
R
V
V
0
R2 Rf
Vout
Exam 2 Crib Sheet
First order circuits
Differential equation:
dy y f t dt
, with solution y t yh t y p t
f t represents a source function or n
th derivative of the source function, with appropriate
coefficients
yh t represents the homogeneous/transient part of the solution
For first order circuits, the homogeneous solution always takes the form
t
yh t Ae
y p t represents the particular/forced part of the solution.
The particular solution is always the same type of function as the source. τ is the time constant
For RC circuits, RC
For RL circuits, L R
Second order circuits
Differential equation:
2 2 2 (^2) o
d y dy y f t dt dt
, with solution y t yh t yp t
s-domain
2 2
s Y s 2 sY s oY s F s
yh t represents the homogeneous/transient part of the solution
The form of the homogeneous solution depends on the damping
y p t represents the particular/forced part of the solution.
The particular solution is always the same type of function as the source.
f t represents a source function or n
th derivative of the source function
F s represents the Laplace transform of the function f(t)
Overdamped
(α > ωo )
1 2 1 2
t t
yh t A e A e
2 2 1 , 2 o
y (^) 0 A 1 (^) A 2 yp 0
1 1 2 2
p
dy dy
A A
dt dt
Critically
Damped (α = ωo )
(^1 )
t t
yh t A e A te
^ ^ from the differential equation
y (^) 0 A 1 yp 0
1 2
dy 0 dyp 0
A A
dt dt
Underdamped
(α < ωo )
1 cos^ 2 sin
t h
y t e A t A t
from the differential equation
2 2 o
y (^) 0 A 1 yp 0
1 2
p
dy dy
A A
dt dt
RLC series circuit
R
L
o
LC
RLC parallel circuit
2 RC
o
LC
Exam 2 Crib Sheet
Partial Fraction Expansion
Simple Real Poles:
Real, Equal Poles – Double Pole:
n
1 n n
1 n1 n 2 1 n n
2 n2 n s p
n p t p t p t 1 n1 n
Real, Equal Poles Double Pole:
A A A
Expand F(s) .. [ ] s p s p (s p )
A (s p ) F(s) ; Cover-Up Rule
Usually Find A from evaluating F(0) or F(1)
f(t) (A e .... A e A te )
^
t 0
Simple Poles Repeated Poles
Complex Conjugate Poles
1
1
1
1 p t t 1
In General:
A A A Expand F(s) .... s p s j s j
Find A and A A / from Cover-Up Rule
t 0
Simple Poles
f(t) A e .... 2 A e co
Complex Pole
s )
s
( t
Exam 3 Crib Sheet
Complex Numbers
Rectangular form :
A A R jAI
Polar form:
A A
Rectangular to polar
2 2 A A R AI
1 tan I A R
A
A
^
Polar to rectangular
AR A cos A
AI A sin A
Euler’s Law: cos^ ^ ^ sin^
j
e j
Mathematics with complex number
Addition/Subtraction – Rectangular form
A B AR BR j AI BI
A B AR BR j AI BI
Complex conjugate
A A R jAI
A A R jAI
Multiplication/Dvision – Rectangular form
AB A B A B
A B
A A
B B
Complex conjugate
A A A
A A A
AC Steady State signals
Time domain signals
F t Ao sin t
Ao – amplitude ω – radial frequency, 2πf
ϴ – phase
Phasor signals
F A o
Ao – amplitude
ϴ – phase
(Rectangular form)
sin (^)
j t (^) j F t Ao t Ao e A eo Ao
(Phasor form)
Impedances – Laplace domain (zero initial conditions)
R
Z R L
Z sL
1
C
Z
sC
Impedances – AC steady state
R
Z R
0
R
Z R
L
Z j L
90
L
Z L
1
C Z
j C
1
90
C
Z
C
Exam 3 Crib Sheet
Impedance, Z [Ω], properties have the same characteristics as resistance
In series add, Z (^) EQ Z 1 (^) Z 2 In parallel, inverse relationship,
1 1 2
1 2 1 2
EQ
Z Z
Z
Z Z Z Z
(^) (^)
Admittance, Y [mho], properties have characteristics that are the ‘inverse’ of impedance
In parallel, add, YEQ Y 1 (^) Y 2 In series, inverse relationship,
1 1 2
1 2 1 2
EQ
Y Y
Y
Y Y Y Y
(^) (^)
AC Steady State Power
S P jQ
S – Complex power
P – Real power, [W] Q – Reactive power, [VAR]
|S| – Total power, [VA]
Using Ohm’s Law relationships for impedances (Z)
Complex Power
2 *^2
o o o o
V
S V I I Z
Z
Total Power
(^2 )
V o (^) VRMS S Z Z
where 2
o RMS
V
V
Capacitive reactance is negative (Q < 0)
Inductive reactance is positive (Q > 0)
Power produced by the source(s) is equal to the sum of the
power produced/stored for each impedance in the circuit
Power factor – a metric over how efficient power
consumption/production appears to be
0 < power factor < 1
Power factor = cos S
P
S
Ideal Transformers
Np : number of windings on the primary
Ns : number of windings on the secondary
Primary: source side of the transformer
Secondary: load side of the transformer
The winding ratio,
Ns N Np
Voltage relationship, Vs NVp
Current relationship,
Ip Is N
Np:Ns
Primary
Vs
Is
Vp Secondary
Ip
Bode Plots Crib Sheet
Bode Plots
Decade – a change in frequency by one order of magnitude, for example
100 rad/s → 1000 rad/s
10
4 Hz → 10
5 Hz
dB – decibel
dB = 20 log |F(jω)| Note the argument of the logarithm
is a magnitude expression
A change of 20dB corresponds to a of |F(jω)|
by one order of magnitude
Bode plot magnitude approximations
n H s s Slope +20dB/decade
1
n
H s
s
Slope -20dB/decade
H (^) s K ‘Flat’, dB = 20log|K|
Sketching Bode plot magnitudes (real poles and zeros)
Crossing an n-pole: Slope changes by -20*n dB/decade
Crossing an n-zero: Slope changes by +20*n dB/decade
‘n’ indicates the number of poles or zeros
‘Crossing’ rules apply when going from a lower
frequency to a higher frequency
Sketching Bode plot phases (real poles and zeros)
Crossing an n-pole: Phase changes by * 2
n
Crossing an n-zero: Phase changes by *
2
n
Phase changes are ‘spread out’ over two decades, one decade on either side of the pole or
zero
Corrections for Bode plot magnitudes (real poles and zeros)
At an n-pole: The ‘real’ dB valule is -3n dB ‘below’ the asymtote
At an n-zero: The ‘real’ dB valule is +3n dB ‘above’ the asymtote
The asymptote is the straight line approximation
of the Bode plots
‘Far away’ from poles and zeros, the
asymptotes are an accurate representation of the Bode plot
Bode Plots Crib Sheet
Second Order Circuits
Damping ratio,
o
, a metric of the damping
α is the attenuation constant
ωo is the resonant frequency
δ > 1, overdamped
δ = 1, critically damped
δ < 1, underdamped
Lowpass/Highpass filters
Overdamped and critically damped cases, the Bode plots follow the procedure on the previous page
Underdamped cases, use the critically damped approximation, add a ‘correction’ of
20 log
at the resonant
frequency, ωo
Bandpass filters
Overdamped, the Bode plots follow the procedure on the previous page
Critically damped and underdamped cases
At the resonant frequency, the magnitude Bode plot is 0dB
The vertex where the stopbands meet is 20 log 2
Note: The above discussion is for second order circuits only. If there is a gain stage, the Bode plot moves ‘up’ or
‘down’ and the dB value of the gain determines the reference for adding corrections/stopband vertices
Cascaded Filters – Magnitude Bode Plots
H(s) = H 1 (s)H 2 (s)H 3 (s) (three stages) → dB = 20log|H 1 (jω)H 2 ( jω)H 3 ( jω)| = 20log|H 1 (jω)| + 20log|H 2 (jω)| + 20log|H 3 (jω)|
angle = H 1 j H 2 j H 3 j H 1 j H 2 j H 3 j
Bode Plots Crib Sheet
Second order filters
Filter name pole/zero ID
2 poles Low pass filter
2 zeros at zero High pass filter
2 poles
1 zero at zero Bandpass filter
2 poles
Bandstop filter
Schematic s( ) H s( )
o
2
s
2 2 s o
2
s
2
s
2 2 s o
2
2 s
s
2 2 s o
2
s
2 o
2
s
2 2 s o
2
