ECE 222
Electric Circuit Analysis II
Chapter 5
Duality in
Electrical Engineering
Herbert G. Mayer, PSU
Status 5/1/2016
For use at CCUT Spring 2016
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Syllabus
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Definition
Duality Examples
L & C Duality
Series, Parallel Circuit
Bibliography
1
Definition
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In EE, a dual relationship exists between certain
pairs of electric devices and units, e.g. voltage and
current
Duality manifests itself by the ability to
interchange dual units in an expression, yielding
two dual, different, yet valid expressions
A dual expression is formed by interchanging the
two and thus creating its corresponding, dual rule
Ultimate reason behind this is the duality of
electrical and magnetic phenomena in nature
Example: v(t) = L di / dt  i(t) = C dv / dt
2
Duality Examples
Voltage

Capacitance

Resistance

Parallel

CVS Short Circuit 
KCL

Impedance

Thévenin Theorem 
Reactance

Current
Inductance
Conductance
Serial
CCS Open Circuit
KVL
Admittance
Norton Theorem
Susceptance
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Duality Examples
Resistor & Conductor:
v=iR  i=vG
Inductor Voltage & Capacitor Current – differential form:
iC = C dvC / dt  vL = L diL / dt
Capacitor & Inductor – integral form:
vC(t) = V0 + 1/C iC(t) dt  iL(t) = I0 + 1/L vL (t) dt
Voltage Division & Current Division
vR1 = v * R1 / ( R1 + R2 )  iG1 = i * G1 / ( G1 + G2 )
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Duality Examples
Worst case for CCS is
open terminals
Worst case for CVS is
short-circuited terminals
Instantaneous change of
current is not possible in
an inductor
Instantaneous change of
voltage is not possible in a
capacitor
Instantaneous change of
voltage at the terminals of
an inductor is quite
possible
Instantaneous change of
current (displacement
current) in a capacitor is
quite possible
Inductor current is out of
phase (runs behind) with
the voltage by + π/2
Capacitor current
(displacement current) is
out of phase (runs ahead)
with the voltage by - π/2
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L & C Duality
n Connected in Series
Leq
Leq
= L1 + L2 + . . Ln
= Σ Li with i = 1..n
1/Ceq = 1/C1 + 1/C2 . . + 1/Cn
1/Ceq = Σ 1/Ci with i = 1..n
n Connected in Parallel
1/Leq = 1/L1 + 1/L2 + . . 1/Ln
1/Leq = Σ 1/Li with i = 1..n
Ceq
Ceq
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= C1 + C2 . . + Cn
= Σ Ci with i = 1..n
L & C Duality
Time Constant τ in L
Time Constant τ in C
τ = L/R
Units of R
= [V A-1]
Units of L = [H] = [V s A-1]
Unit of τ
= [s]
τ = C*R
Units of R
= [V A-1]
Units of C = [F] = [A s V-1]
Unit of τ
= [s]
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Series & Parallel Circuit
KCL in Parallel RLC Circuit
+
1/R dv/dt + v/L + C d2v/dt2 = 0
d2v / dt2 + 1 / (RC) dv / dt + v / (LC) = 0
v’’ + v’ / RC + v / LC = 0
+
iC
V0
iL
I0
C
iR
L
R
-
Natural Response of Parallel RLC Circuit
L
R
R di/dt + i/C + L d2i/dt2 + 0 = 0
I0
V0
C
i
d2i / dt2 + R / L * di / dt + i / (LC) = 0
+
Natural response in Series RLC circuit
v
-
KCL in Series RLC Circuit
-
i’’ + i’ R / L + i / LC = 0
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a
b
Bibliography
1.
2.
Wiki on duality:
https://en.wikipedia.org/wiki/Duality_(electrical_circ
uits)
Electric Circuits, 10nd edition, Nilsson and Riedel,
Pearsons Publishers, © 2015 ISBN-13: 978-0-13376003-3
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