LEVEL 3 3015 COMMUNICATIONS, SIGNALS
AND SYSTEMS 2003
Peter H. Cole
March 13, 2003
Appendix H
SOME NOTES ON PHASOR
ANALYSIS
H.1
Objective
The objective of these notes is to clarify the basic concepts of phasor analysis of AC
circuits in the sinusoidal steady state.
H.2
A Simple Circuit
Our vehicle for this instruction will be the analysis of the simple circuit shown in the
Figure H.1.
i(t)
+
_
+ vL(t)
+
L
v(t)
R
vR(t)
_
_
Figure H.1: A simple circuit for analysis.
In analysing this circuit, we will assume the sinusoidal current with peak value Im and
phase and φ.
i(t) = Im cos(ωt + φ).
H.3
(H.1)
Analysis by Differential Equations
For the basic circuit elements R, L, and C, we can write differential equations relating
the terminal voltage and the element current, using reference directions for voltages and
currents as shown in Figure H.2.
231
232
APPENDIX H. SOME NOTES ON PHASOR ANALYSIS
i(t)
i(t)
i(t)
+
+
v(t)
v(t)
R
+
L
_
_
v(t) = Ldi(t)
dt
v(t) = R i(t)
v(t)
C
_
i(t) = C dv(t)
dt
Figure H.2: Sign conventions for circuit variables.
For a resistor R
v(t) = Ri(t) = RIm cos(ωt + φ).
(H.2)
For an inductor of self inductance L,
di(t)
= −ωLIm sin(ωt + φ).
dt
For a capacitor of capacitance C,
v(t) = L
1 8
1
v(t) =
Im sin(ωt + φ).
i(t)dt =
C
ωC
Thus for the circuit of Figure H.1 we have
di(t)
dt
= RIm cos(ωt + φ) − ωLIm sin(ωt + φ)
v(t) = Ri(t) + L
= Im R2 + (ωL)2 sin(ωt + φ + θ)
(H.3)
(H.4)
(H.5)
(H.6)
(H.7)
where
ωL
R
sin θ = and cos θ = .
R2 + (ωL)2
R2 + (ωL)2
H.4
(H.8)
Disadvantageous Features
Most people find this analysis burdensome, particularly in the last step where a sine wave
and cosine wave of excitation must be combined.
We will show in the below sections that analysis using complex phasors provides the
correct results with less trouble.
Although we achieve this result by introducing complex numbers to represent the
sinusoidal variables, all the voltages and currents within our circuit remain real numbers.
H.5. REPRESENTATION OF SINUSOIDS
H.5
233
Representation of Sinusoids
It is well known that the circular functions sin(ωt) and cos(ωt) bear simple relations to
the complex exponential ejωt . It is also well known that the complex exponential ejωt
has a particulary simple time derivative i.e. jωejωt , in which the functional form of the
variable does not change; it is merely multiplied by a constant.
This prompts us to investigate whether complex exponentials can be used to provide
a simpler version of AC circuit analysis than that just illustrated.
We first introdce the concept of phasor to represent a sine wave (or more correctly a
cosine wave) excitation.
In doing this, we first observe that, in a situation in which all variables are sinusoidal
and at a common angular frequency ω, the sinusoidal current i(t) = Im cos(ωt + φ) is
completely determined by its magnitude Im and its phase φ.
We may therefore represent that magnitude and phase information by a complex
number of magnitude Im and argument φ, i.e. the complex number Im ejφ as shown in the
Argand diagram on Figure H.3 at the point P1 .
imag
Rotation
P2
Im
P1
O
Instantaneous value at time t
Im
wt
Rotating arm
Phasor (fixed)
f
real
Instantaneous value at time t = 0
Figure H.3: Phasors, rotating arms and projections.
We notice that the point P1 does not move. Moreover, if we take the projection of the
point P1 on the horizontal axis, we obtain the value Im cos φ, which is the value of the
sinusoidal excitation at the time t = 0.
If we add an additional time-varying phase angle ωt to the phase angle φ of the phasor
leading to the point P1 , we obtaine the rotating arm in which the point P2 describes a
circle centred on the origin and of radius Im . At any time, the projection of the point P2 on
the horizontal axis gives the instantaneous value of the sinusoidal quantity Im cos(ωt + φ)
at time t. We notice that the horizontal projection is just the real part of the complex
number represented by the rotating arm.
234
APPENDIX H. SOME NOTES ON PHASOR ANALYSIS
H.6
Analysis Leading to Algebraic Equations
Let us now investigate the way in which taking our variables as the real parts of complex
exponentials may simplify the work involved in circuit analysis when sinusoidal signals
are present. We begin with
i(t) = {Im ej(ωt+φ) }.
(H.9)
Then
di(t)
d {Im ej(ωt+φ) }
=
dt
dt
dIm cos(ωt + φ)
=
dt
= −ω sin(ωt + φ)
=
{jωej(ωt+φ) }.
(H.10)
(H.11)
(H.12)
(H.13)
So we see that we may achieve the object of differentiation, which is initially to be
done externally to the operation of taking the real part, by performing the differentiation
first on the complex function, and then the taking the real part of the result. This is
actually a quite general result which applies to complex functions of a real variable when
we differentiat with respect to that real variable. What is, however, interesting for us
here is firstly that the differentiation operation has the simple effect of multiplying the
complex function by jω, and secondly that there is no other change in the functional form
of the complex function.
H.7
Application to Circuit Analysis
So in our analysis of the simple circuit given in Figure H.1 with the sinusoidal excitation
i(t) = {Im ej(ωt+φ) }
(H.14)
we may derive the input voltage by the steps
v(t) =
=
H.8
{RIm ej(ωt+φ) + jωLej(ωt+φ) }
{(R + jωL)Im ej(ωt+φ) }.
(H.15)
(H.16)
The Physical and Mathematical Systems
These steps lead to the following interpretation, in which it is useful to distinguish between
a physical system and a mathematical system.
H.8. THE PHYSICAL AND MATHEMATICAL SYSTEMS
H.8.1
235
The physical system
In the physical system, we have an excitation current and the resulting voltage which are
both real and sinusoidal, but are expressed as the real part of complex exponentials, as
below.
i(t) = {Im ej(ωt+φ } ; v(t) = {(R + jωL)Im ej(ωt+φ) }.
(H.17)
It is very evident that the voltage and current are real, as they must be.
H.8.2
The mathematical system
In the mathematical system we might treat our input current as the complex function
Im ej(ωt+φ)
(H.18)
and treat resistors, inductors, and capacitors as having complex impedances of R,
jωL and 1/(jωC) resepctively, and calculate voltages, (and in more complicated circuits
maybe other currents), by the same rules of circuit theory as we have been used to in DC
circuits, but where the resistances are replaced by the complex impedances defined in the
equations above. To find the answer to the physical problem, we take of the real parts of
the complex functions. Thus
i(t) = {Im ej(ωt+φ) }
(H.19)
and
v(t) =
=
H.8.3
{V e(jωt) }
{(R + jωL)Im ejφ ejωt }.
(H.20)
(H.21)
Deletion of the time function
We notice that the time function ejωt is a common factor of all of the above operations.
Thus we can delete it from from the function Im ej(ωt+φ) at the beginning of the calculation,
to obtain the phasor Im ejφ . Then we operate on that phasor by our complex impedances,
(in our case we multiply by our complex impedance (R +jωL), to obtain our input phasor
V = (R + jωL)Im ejφ . We then re-attach the common factor ejωt , and take the real part
of the result, to get the real solution to the physical problem.
v(t) =
=
{V ejωt }
{(R + jωL)Im ejφ ejωt }.
(H.22)
(H.23)