1. Circuit Analysis using Complex Exponentials
Physics Electricity and Magnetism Lecture 15E - AC Circuits & Resonance II No Y&F reference for slides on Complex analysis
Circuit Analysis using Complex Exponentials
Imaginary and Complex Numbers
Complex Exponential Phasors and Rotations
Phasors as Solutions of Steady State Oscillator Equations
Phasor representation applied to current versus voltage in R. L. C.
Series LCR Circuit using Complex Phasors
Parallel LCR Circuit using Complex Phasors
Transient Solution of Damped Oscillator using Complex Phasors

2. Summary: AC Series LCR CircuitZ Im Em F
wDt
VL-VC VR
XL-XC R
sketch shows
XL > XC L C E vR vC vL
VL = ImXL
+90º (p/2)
Lags VL by 90º
XL=wdL L
Inductor
VC = ImXC
-90º (-p/2)
Leads VC by 90º
XC=1/wdC C
Capacitor
VR = ImR
0º (0 rad)
In phase with VR R
Resistor
Amplitude Relation
Phase Angle
Phase of Current
Resistance or Reactance
Symbol
Circuit Element

3. Complex Exponentials in Circuit Analysis
is “pure imaginary”
Roots of a quadratic equation az2 + bz +c = 0
are complex if b2<4ac:
Complex number in rectangular form:
REVERSE SIGN OF IMAGINARY PART
Re{z} and Im{z} are both real numbers
Complex conjugate:
Addition:
EE’s use j instead of i (i is for current)

4. Representation using the Complex Plane
Imaginary axis (y)
Real axis (x) x y
|z| q
Polar form:
“Argand diagram”
Magnitude2:
Argument:
Picture also displays complex-valued functions

5. Complex Exponentials: Euler’s Formula
Taylor’s Series definition of exponential function:
Evaluate for u = iq:
use i2n+1 = i(-1)2n
Recognize series definitions of sine and cosine:
Hence:
Above is a complex number of magnitude 1, argument q:
Special Properties:

6. Rules for Complex Quantities
Euler’s Formula for magnitude r:
Polar form of arbitrary complex z
Complex conjugate of z
Sum:
Product:
Magnitude of a product = product of individual magnitudes
Argument of a product = sum of individual arguments
Two complex entities are equal if and only if their amplitudes
(magnitudes) are equal and their arguments (phases) are equal
Periodicity 2p:
DeMoivre’s Theorem:
for integer n & k = 0,1….n-1

7. Phasors, Rotation Operators, Evolution Operators:Im Re q -f +f
e+if rotates complex number z by angle +f in complex plane
Time Evolution: Let f = w(t-t0)
evolves z from t0 to t for tf > t0
A(t) is called a “time domain phasor”
Re {A} = is the measurable instantaneous value of A(t)
Time dependent exponentials like eiwt are sometimes called evolution operators
Cancelling common factors of eiwt leaves a “frequency domain phasor” A (often
simply called a “phasor”):
Represent sinusoidally varying real quantity a(t) as a vector in the complex plane, rotating (counterclockwise) at angular frequency w.

8. Phasor Definitions, continued
A phasor represents a sinusoidal, steady state signal whose amplitude Amax, phase f, and frequency w are time invariant.
peak value of a(t), real
frequency domain phasor A, complex, rotated by f from real axis
time domain phasor, complex
instantaneous value,
time domain, real
Phasor Transform P:
forward - time to frequency domain
inverse - frequency to time domain
Advantage of phasor transform: For sinusoidal signals, differential equations (time domain) become algebraic equations (frequency domain) as common factors of eiwt cancel).
Process: Replace real variables in time dependent analysis problem with variables written using complex exponentials. Cancel common factors of eiwt and solve remaining algebraic problem. Then return to real solution in time domain.

9. Time domain phasors are alternative solutions to sines and cosines in differential equations representing oscillations.
Recall:
a can be complex
chain rule:
First and Second Derivatives:
set a = wt+f, amplitude Amax
Example: Simple Harmonic Oscillator
Trig Solutions:
Complex Solutions:

10. Apply complex voltage & current (time domain phasors):
Complex Exponential Representation applied to Passive Circuit Elements: Revisit AC Voltage vs. Current in L, C, and R
Apply complex voltage & current
(time domain phasors):
Note: now using j = sqrt(-1)
vR( t )
i(t)
magnitudes:
exponents:
Resistor:
Inductor:
vL(t) L
i(t)
magnitudes:
exponents:
i(t)
vC ( t ) C
Capacitor:
magnitudes:
exponents:

11. Summary: Complex Exponential Representation of AC Voltage vs
Summary: Complex Exponential Representation of AC Voltage vs. Current in L, C, and R
& current:
Applied voltage:
time domain
Phase angle factor ejF rotates E CCW from current phasor
Passive circuit elements:
Resistor:
Inductor:
Capacitor:
VR& IRm in phase
VL leads ILm by p/2
VC lags ICm by p/2
e0 =1
e+jp/2 = +j
e-jp/2 = -j
Phasor rotations of voltage drops relative to currents:
Voltage drop phasors relative to currents: time domain
Frequency domain (factor out time dependence)
Define: Complex impedance z = E(t)/i(t) and |z| = Vmax/Imax
Impedances of simple circuit elements (complex):
Magnitudes of Impedances |z| = [zz*]1/2 (real)

12. Complex Impedance z Applied voltage: & current: time domain
Definition: complex impedance z (or simply impedance) is the ratio of the (complex) voltage phasor to the (complex) current phasor (in time or frequency domain).
Sketch shows positive Im{z}.
is positive, implying that
Em leads Im.
Definition: Phase angle F measures rotation of the applied voltage referenced to the current in the branch. It is also the angle between z and the real axis.
The rotation operator is ejF. Z F
Im(z)
Re(z) Re Im Em
1/z occurs when analyzing parallel branches:

13. Impedances of Series or Parallel Collections of Circuit Elements
Impedances of individual passive circuit elements:
The kth sub-circuit (arbitrary complexity)
consists of R, L, and/or C basic elements: zk
Assertion: Follow series or parallel resistor addition rules to compute equivalent impedances (complex)
Series branch formula: zk z2 z1
Parallel sub-circuit formula: z1 z2 zk

14. Revisit series LCR circuit using complex phasors
AC voltage:
Current:
time domain phasors R L C E vR vC vL
Currents are the same everywhere in an essential branch,
same phase, same magnitude.
Kirchhoff Loop rule for series LRC circuit:
Substitute the time domain voltage phasors for vR, vL, vC.
Divide equation (1) by i(t) ( = Imejwt )
Magnitude of Z: Im Em F
wDt VL VC VR
Sketch shows F positive for
VL>VC  XL>XC
A t = 0 sketch would show
phasors in frequency domain

15. Revisit the series LCR circuit, Continuation
Impedance z (Equation 1) can also be found by summing the impedances of the 3 basic circuit elements in series LCR circuit (invoke Series Formula*) Z F
XL-XC R Re Im
z is not a phasor, as it is time independent.
Previous “phasor diagrams” showed z rotating with Em.
Sketch shows F >0 for XL > XC
 VL > VC. F is positive when Im{z} is positive
Phase angle F for the circuit:
Power factor:
* The equivalent (complex) impedance for circuit elements in series (arbitrarily many) is the sum of the individual (complex) impedances.
Resonance: As before, z is real for XL= XC (w2 = 1/LC). |z| is minimized.
Current amplitude Im is maximized at resonance
Phase angle in terms of admittance 1/z:

16. Parallel LCR circuit using complex phasorsvC vL R L C E vR iR iL iC i a b
All steady state voltages and currents oscillate at driving frequency wD
AC voltage:
Instantaneous Current:
Kirchhoff Loop Rule (time domain):
Instantaneous voltages across parallel branches have the same magnitude and phase:
2 essential nodes “a” & “b”
4 essential branches,
Not all independent
Common voltage phase, all branches:
Common voltage peak, all branches:
Use voltage as reference instead of current
Kichhoff Current Rule (node a or b):
Current Amplitudes in each parallel branch (reference now to voltage drop):

17. Parallel LCR circuit, Continuation #1
IRm Im
Em, VR, VL, VC F
wDt
ICm Re
ILm
ICm- ILm
Instantaneous currents in each branch lead, lag, or are in phase with the (reference) voltages:
in phase with E(t)
lags E(t) by p/2
leads E(t) by p/2
Current amplitude addition rule is Pythagorean
Sketch shows F < 0 (applied voltage lags current) for
ICm > ILm  XL>XC
Substitute currents into junction rule equation (1):
Cancel ejwt factor and multiply by e-jF (current phasor – frequency domain)

18. Parallel LCR circuit, continuation #2
Recall, impedance
“admittance”
Substitute, then divide (Eq. 2.1) above by Em
Note that:
Admittance 1/z in Equation 3 is also the sum of the reciprocal impedances of the 3 basic circuit elements in parallel LCR circuit (invoke Parallel Formula)
Find |1/z|: multiply 1/z by complex conjugate (1/z)* and take square root:
Represent z in terms of 1/z:

19. Parallel LCR circuit, Continuation #3
Sketch shows F negative for 1/XC > 1/XL  IC > IL  XL > XC
1/z F
1/XC-1/XL
1/R E Im
Phase angle F:
For XL > XC: 1/XL – 1/XC < Tan (F) and F are positive in Series LCR circuit (see above), Voltage Em leads current Im Tan(F) and F are negative in Parallel LCR circuit Current Im leads applied voltage Em Converse for XC > XL F = 0 at resonance (XL = XC) in both Series and Parallel circuits

20. Parallel LCR circuit, Continuation #4
Resonance: Minimizes as function of frequency
Minimum of 1/|z| when XL = XC,
i.e. when w2 = 1/LC
Same resonant frequency as series LCR, but current is minimized instead of maximized at resonance
At resonance, the current amplitudes ILm and ICm in the L & C branches are equal, but are 180o apart in phase. These cancel at all times at nodes a and b of the circuit.

21. Using complex exponentials to solve a differential equation
Revisit Damped Oscillator: After “step response” to switch at ‘a’ saturates, turn switch to ‘b’. Decaying natural oscillations start when damping is weak. a L C E + b R
R dissipates potential energy
not a steady state system
solutions not simple sinusoids
second order equation for Q(t) – see Lect. 13
Trial solution…
… but if w is real, solution oscillates forever
wx and wy assumed real
 Assume complex frequency
Derivatives:
Substitute into Equation (1):
Cancel common factors of Q:
“characteristic equation”
Phasor-like trial solution (2) turned differential equation into polynomial equation

22. Using complex exponentials to solve a differential equation, #2
Equation (3) becomes 2 separate equations for real and imaginary terms
Expand:
Re{Eq 3} 
Substitute wy:
Im{Eq 3} 
Shifted natural frequency wx is
real for underdamping
imaginary for overdamping
For real wx (underdamping):
damping
oscillation
For critical damping, wx = 0:
No oscillations, decay only:

23. Using complex exponentials to solve a differential equation, #3
For overdamping:
wx becomes imaginary. Equations 4.1 & 4.2 invalid
Return to Eq Assume frequency is
pure imaginary ( no oscillation)
Eq. 3.0 becomes:
quadratic, real coefficients
Solution:
two roots
both pure imaginary
Both roots lead to decay w/o oscillation
+ root: w+ implies damping faster than e-Rt/2L
- root: w- implies damping slower than e-Rt/2L but not growth
Most general solution: linear combination of Q+ and Q-, each of form of Eq. 2.0
Recall definition, hyperbolic cosine:
Correctly reduces to critically damped Eq. 8.0