Jump to ContentJump to Main Navigation
Light-Matter InteractionPhysics and Engineering at the Nanoscale$

John Weiner and Frederico Nunes

Print publication date: 2012

Print ISBN-13: 9780198567653

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780198567653.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 21 February 2017

(p.243) Appendix D Properties of Phasors

(p.243) Appendix D Properties of Phasors

Source:
Light-Matter Interaction
Publisher:
Oxford University Press

D.1 Introduction

In Section 2.10.1 we discussed the phasor form for electromagnetic propagating fields consisting of a single-frequency harmonic time dependence factor e iωt and the spatially dependent amplitude modulation factor e i(k· r+φ). The general form of the field is

(D.1)
A ( r , t ) = A 0 e i ( k · r + φ ) e i ω t
and we separated the field into a phasor factor, involving only the spatial dependence, and a harmonic time factor that was usually invariant.

In general, a “phasor” is any function with a sinusoidal modulation, and in this appendix we examine some of the useful properties of phasors and their application to circuit theory.

D.1.1 Phasor Addition

Since phasors can be represented as complex exponential, multiplication of two phasors Ae and Be is elementary.

(D.2)
A e i α · B e i β = A B e i ( α + β ) = C e i γ

The product of two phasors is another phasor with a phase angle equal to the algebraic sum of the two individual arguments of the exponential factors.

The addition of two phasors is less obvious. In many cases we are interested in real sin and cos functions, so let us examine the following sum of two real phasors, V 1 (t) = A sin ωt and V 2 (t)= B sin(ωt + φ).

(D.3)
V 3 ( t ) = V 1 ( t ) + V 2 ( t ) = A sin ω t + Bsin ( ω t + φ )

Intuitively we might suppose that the sum of two sin functions differing only in amplitude and relative phase would be another sin function. We posit therefore that the form of the sum is

(D.4)
V 3 ( t ) = C sin ( ω t + δ )
where C and δ are to be determined. Expanding V 3(t) in Eqs D.3, D.4 and setting coefficients of sin ωt and cos ωt equal, we find

(p.244)

Appendix D Properties of Phasors

Fig. D.1 Relations between two phasors A and B which differ in amplitude and phase. Phase of Aleads phase of B by φ. The interior angle γ is related to φ by γ = π − φ.

(D.5)
δ = tan 1 [ B sin φ A + B cos φ ]
(D.6)
C = [ A 2 + B 2 + 2 A B cos φ ] 1 / 2

The expression for C, the amplitude of the phasor sum, is highly reminiscent of the result we would expect for the length of a vector C that is the vector sum of A and B. In this case the length of the vector C would be determined by the law of cosines as indicated in Fig. D.1. The interior angle γ between the two component vectors is closely related to φ, the relative angle between the two phasors. In fact, it is easy to show that

(D.7)
γ = π φ
and substitution in Eq. D.6 results in
(D.8)
C = [ A 2 + B 2 2 AB cos γ ] 1 / 2
confirming that two phasors add with amplitude equivalent to the resultant “length” from the sum of two vectors.

D.2 Application of Phasors to Circuit Analysis

As an illustration of the usefulness of phasors to harmonic circuit finalysis, we consider a simple RLC circuit with

(D.9)
V ( t ) = R i ( t ) + L d i ( t ) d t + 1 C i ( t ) d t
where V, R, L, C have their usual meanings of voltage, resistance, inductance, and capacitance and i(t) is the time-dependent current running through all the lumped (p.245) circuit elements. We posit that the current source is harmonically oscillating at frequency ω and write i(t) as a phasor,
(D.10)
i ( t ) = i 0 e i ω t

Substituting the phasor form into Eq. D.9 we find

(D.11)
V ( t ) = R i 0 e i ω t + ω L i 0 e i ( ω t + π / 2 ) + 1 ω C i 0 e ( ω t π / 2 )
(D.12)
=V R +V L +V C

We see that the voltage drop across the resistor is in phase with the source, while the inductive reactance, ωL, shows a phase advance of π/ 2 and the capacitive reactance, /ωC, lags in phase by π/2. The resistive voltage drop is purely dissipative while the two reactances store energy originating at the source and return it to the circuit at different points along the harmonic cycle. Equation D.9 can be rewritten in terms of the impedance, Z (ω),

Appendix D Properties of Phasors

Fig. D.2 (a) Series RLC circuit with harmonic voltage source and a common current i(t) running through each circuit element. (b) Voltage phasor diagram showing the phase “lead” for the capacitive voltage term, the phase “lag” for the inductive voltage term and the net phase difference between the driving voltage and the resistive voltage drop. Panels (c) and (d) show similar diagrams for current phasors in RLC parallel circuit.

(p.246)

(D.13)
V ( t ) = Z ( ω ) i 0 e i ω t
(D.14)
Z ( ω ) = R + ( ω L ) e i ( π / 2 ) + ( 1 ω C ) e i ( π / 2 )
(D.15)
= R i ω L + ( i ω C )
(D.16)
= R + i X L ( ω ) + i X C ( ω )
where XL and XC are the inductive and capacitive reactances in phasor form. We can take this finalysis further by considering the RLC circuit in series and in parallel. Figure D.2 shows circuit and phasor diagrams for the two cases. In the series circuit the property common to all lumped elements is the current i(t) so the relevant phasor quantities are the voltages across the resistive, inductive, and capacitive elements. For the parallel circuit the common property is the voltage V (t) and the relevant phasor diagram is expressed in terms of the individual currents passing through each circuit element.