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The Phasor Transform and Impedance
1. Objective
We will see how the phasor transform can simplify the voltage-current relationship for
inductors and capacitors, eliminating the need for derivatives and integrals. In fact, the
voltage-current relationship for resistors, inductors and capacitors in the phasor domain
looks just like Ohm’s law, where voltage equals current time a scaling constant. We call
the scaling constant impedance. It serves the same role as resistance, but in the phasor
domain. It is a constant like resistance, but turns out to be complex valued and varies
with the frequency of the signal involved.
2. What is the phasor transform?
Sinusoids are special signals. Note that the integral and derivative of a sinusoid is a
sinusoid. Thus, the voltage-current relationships for inductors and capacitors, which are
characterized by integrals and derivatives, tell us that a sinusoidal current produces a
sinusoidal voltage. The only difference between the sinusoidal voltage across and current
through these devices is possibly the amplitude and phase. The frequency of the current
will be the same as the frequency of the voltage. Thus, if we only consider sinusoidal
signals, all we need to keep track of is magnitude and phase of the voltages and currents.
This is where the phasor transform come in!
A complex number, unlike a real number, contains both a magnitude and a phase.
Thus, a complex number is a very convenient way to describe the various sinusoids in our
circuits. Here is how the phasor transform works. We begin by considering a sinusoid in
the so-called time domain (where our voltage and current equations are functions of time)
x(t ) = X m cos(ω t + φ ) .
(1)
In the phasor domain, this signal is expressed as a single complex number X with a
magnitude, X m , equal to the sinusoid amplitude and a phase, φ , equal to the sinusoid’s
phase. Any complex number can be written in two distinct ways, rectangular form and
polar form. While these forms looks very different, Euler proved that they are equivalent
with the famous theorem that bears his name. Consider a complex number Z written in
rectangular form and polar form respectively,
Z = X + jY = re jθ .
(2)
Here X is the real part of Z , Y is the purely imaginary part of Z . The variable r is the
magnitude of Z , θ is the phase, j = −1 , and e = 2.71828182846... . The two forms of a
complex number can be found from one another using the following relationships:
r = X 2 +Y2 ,
(3)
θ = tan −1 (Y / X ) ,
(4)
R. C. Hardie Department of Electrical and Computer Engineering University of Dayton
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and
X = r cos θ
(5)
Y = r sin θ .
(6)
Note that the polar form is often written in shorthand notation like so
Z = re jθ r ∠θ .
(7)
This hides the rather bizarre e j part, and highlights the fact that the magnitude and phase
are the important pieces of information carried by a complex number.
A TI-85 (and up) will do the conversions between the two forms (rectangular and
polar) in (2). To enter the complex number 3 + j 2 = 3.6e j.59 into your TI, use (3,2) for
rectangular form or (3.6∠.59) for polar entry. To convert the form, enter the number in
the form you know and now set the mode to RectC or PolarC to specify the form you
want. With the correct mode set, press enter and your previously entered number will
appear in new format.
The phasor domain representation for the sinusoid in (1) is given by
X = X m e jφ = X m cos φ + jX m sin φ .
(8)
We simply let the magnitude of the cosine equal the magnitude of the phasor (the
complex number) and the phase of the cosine becomes the phase term in the phasor.
Look at what we have done. We have replaced a function of time with a single
number (albeit, a complex number) that contains the only information we need to keep
track of, magnitude and phase. Performing a phasor transform means writing Equation
(8) from Equation (1). Performing an inverse phasor transform means writing Equation
(1) from Equation (8). Of course, in performing an inverse phasor transform, we must
know what the original frequency, ω , is. Here is an example. The voltage produce by a
standard AC outlet is described by
v(t ) = 169.7 cos(120π t + 60°) ,
(9)
where we have arbitrarily set the phase to be 60 degrees. The phasor transform for this is
V = 169.7e j 60° = 169.7∠60° .
(10)
Be sure to mark the angular units (radians or degrees) the same in both your time domain
and phasor domain representations. Specify degrees with the ° symbol, otherwise,
radians will be assumed.
Note that adding sinusoidal signals of the same frequency in the time domain is
equivalent to adding their phasor transforms in the phasor domain. This “trick” can be
helpful in visualizing the sum of two or more sinusoids of the same frequency.
R. C. Hardie Department of Electrical and Computer Engineering University of Dayton
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3. What is impedance?
For an inductor, capacitor, resistor, or combination, a sinusoidal current produces a
sinusoidal voltage of the same frequency (although, as we said, the magnitude and phase
may change). Thus, we have
and
i (t ) = I m cos(ω t + φi )
(11)
v(t ) = Vm cos(ω t + φv ) .
(12)
I = I m e jφi
(13)
V = Vm e jφv .
(14)
In the phasor domain, we have
and
Now to the amazing part… Note that the phasor voltage, V , can be written in
terms of phasor current, I , multiplied by a complex number (which scales the magnitude
as needed and adds to the phase as needed). This gives
V = IZ
(15)
for some complex number Z = Z m e jφz , which we call the impedance. Again, the concept
of impedance applies to R’s, L’s and C’s and combinations thereof. Writing the complex
numbers in (15) in polar form yields
Vm e jφv = I m e jφi Z m e jφz = I m Z m e j (φi +φz ) .
(16)
Thus, the magnitude of the sinusoidal voltage across some impedance is the magnitude of
the current times the impedance magnitude. The phase of the voltage is the phase of the
current plus the phase of the impedance.
Solving for impedance, assuming we know the voltage and current, we get
Z = Z m e jφz =
V Vm e jφv Vm j (φv −φi )
=
=
e
.
I I m e jφi I m
(17)
Thus, impedance can be determined by determining the relationship between voltage and
current magnitudes and phases, for a particular element (or combination of elements). In
particular, the impedance magnitude is given by
Zm =
Vm
,
Im
and the impedance phase is given by
R. C. Hardie Department of Electrical and Computer Engineering University of Dayton
(18)
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φ z = φv − φi .
(19)
You will find that impedance, for elements other than resistors, varies with the
frequency of the voltage or current source exciting the element. That is, the same
physical component will have different impedances when used with different source
frequencies.
Here is the real beauty. When the source or sources in your circuit are
sinusoidal, you can express all the voltages and currents in the circuit as complex
numbers (expressing them as phasors in the phasor domain). You can express all the
components (R, L & C’s) by their impedance (complex numbers). Now, because all the
voltage-current relationships in this phasor domain are given by (15), which is just like
Ohm’s law, the circuit can be analyzed like a DC resistive circuit. One solves for the
phasor domain voltage or current that is of interest, and finally, performs an inverse
phasor transform to get the time domain voltage or current which is generally the more
intuitive form of the solution (the one you are accustomed to). No more differential
equations!!!
So what’s the catch? Well… one, we have to deal with complex numbers. And
two, the source in the circuit has to be sinusoidal. However, Fourier analysis tells us that
virtually any waveform can be written in terms of sinusoids. Thus, the concept of
impedance allows us to solve any RLC circuit with any input source without differential
equations.
It can be shown that the theoretical impedance for an ideal resistor is simply its
resistance, R. The theoretical impedance for an ideal inductor is
Z = jω L = ω Le j 90° = ω L∠90° .
(20)
The impedance for an ideal capacitor is give by
Z=
1
−j
1 − j 90°
1
=
=
e
=
∠ − 90° .
ωC
jω C ω C ω C
(21)
Note that ω is frequency in radians/second and ω = 2π f , where f is frequency in
cycles/second (Hz).
R. C. Hardie Department of Electrical and Computer Engineering University of Dayton