1. 0 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

2. 1 Syllabus Definition Duality Examples L & C Duality Series, Parallel Circuit Bibliography

3. 2 Definition 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

4. 3 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

5. 4 Duality Examples Resistor & Conductor: v = i R  i = v G Inductor Voltage & Capacitor Current – differential form: i C = C dv C / dt  v L = L di L / dt Capacitor & Inductor – integral form: v C (t) = V 0 + 1/C i C (t) dt  i L (t) = I 0 + 1/L v L (t) dt Voltage Division & Current Division v R1 = v * R 1 / ( R 1 + R 2 )  i G1 = i * G 1 / ( G 1 + G 2 )

6. 5 Duality Examples Worst case for CCS is open terminals Instantaneous change of current is not possible in an inductor Instantaneous change of voltage at the terminals of an inductor is quite possible Inductor current is out of phase (runs behind) with the voltage by + π/2 Worst case for CVS is short-circuited terminals Instantaneous change of voltage is not possible in a capacitor Instantaneous change of current (displacement current) in a capacitor is quite possible Capacitor current (displacement current) is out of phase (runs ahead) with the voltage by - π/2

7. 6 L & C Duality L eq = L 1 + L 2 +.. L n L eq = Σ L i with i = 1..n 1/C eq = 1/C 1 + 1/C 2.. + 1/C n 1/C eq = Σ 1/C i with i = 1..n n Connected in Series n Connected in Parallel 1/L eq = 1/L 1 + 1/L 2 +.. 1/L n 1/L eq = Σ 1/L i with i = 1..n C eq = C 1 + C 2.. + C n C eq = Σ C i with i = 1..n

8. 7 L & C Duality τ= L / R Units of R= [V A -1 ] Units of L = [H]= [V s A -1 ] Unit of τ= [s] Time Constant τ in L Time Constant τ in C τ=C * R Units of R = [V A -1 ] Units of C = [F]= [A s V -1 ] Unit of τ= [s]

9. 8 Series & Parallel Circuit KCL in Parallel RLC Circuit R di/dt + i/C + L d 2 i/dt 2 + 0 = 0 d 2 i / dt 2 + R / L * di / dt + i / (LC) = 0 i’’ + i’ R / L + i / LC = 0 1/R dv/dt + v/L + C d 2 v/dt 2 = 0 d 2 v / dt 2 + 1 / (RC) dv / dt + v / (LC) = 0 v’’ + v’ / RC + v / LC = 0 KCL in Series RLC Circuit

10. 9 Bibliography  Wiki on duality: https://en.wikipedia.org/wiki/Duality_(electrical_circ uits)  Electric Circuits, 10 nd edition, Nilsson and Riedel, Pearsons Publishers, © 2015 ISBN-13: 978-0-13- 376003-3