This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.
Phasors may be used to analyze the behavior of electrical and mechanical systems that have reached a kind of equilibrium called sinusoidal steady state. In the sinusoidal steady state, every voltage and current (or force and velocity) in a system is sinusoidal with angular frequency ωω. However, the amplitudes and phases of these sinusoidal voltages and currents are all different. For example, the voltage across a resistor might lead the voltage across a capacitor by 90∘90∘ ( π2π2 radians) and lag the voltage across an inductor by 90∘(π2radians)90∘(π2radians).
In order to make our application of phasors to electrical systems concrete, we consider the series RLC circuit illustrated in Figure 1. The arrow labeled i(t)i(t) denotes a current that flows in response to the voltage applied,and the + and - on the voltage source indicate that the polarity of the applied voltage is positive on the top and negative on the bottom. Our convention is that current flows from positive to negative, in this case clockwise in the circuit.
We will assume that the voltage source is an audio oscillator that pro-
duces the voltage
V(t)=Acos(ωt+φ).V(t)=Acos(ωt+φ).
(1)We represent this voltage as the complex signal
V
(
t
)
↔
A
e
j
φ
e
j
ω
t
V
(
t
)
↔
A
e
j
φ
e
j
ω
t
(2)and give it the phasor representation
V(t)↔V;V=Aejφ.V(t)↔V;V=Aejφ.
(3)We then describe the voltage source by the phasor V and remember that we can always compute the actual voltage by multiplying by ejωtejωt and taking the real part:
V
(
t
)
=
Re
{
V
e
j
ω
t
}
.
V
(
t
)
=
Re
{
V
e
j
ω
t
}
.
(4)
Show that Re [Vejωt]=Acos(ωt+φ) Re [Vejωt]=Acos(ωt+φ) when V=Aejφ.V=Aejφ.
Circuit Laws. In your circuits classes you will study the Kirchhoff laws that govern the low frequency behavior of circuits built from resistors (R)(R), inductors (L)(L), and capacitors (C)(C). In your study you will learn that the voltage dropped across a resistor is related to the current that flows through it by the equation
V
R
(
t
)
=
R
i
(
t
)
.
V
R
(
t
)
=
R
i
(
t
)
.
(5)You will learn that the voltage dropped across an inductor is proportional to
the derivative of the current that flows through it, and the voltage dropped
across a capacitor is proportional to the integral of the current that flows
through it:
V
L
(
t
)
=
L
d
i
d
t
(
t
)
V
C
(
t
)
=
1
C
∫
i
(
t
)
d
t
.
V
L
(
t
)
=
L
d
i
d
t
(
t
)
V
C
(
t
)
=
1
C
∫
i
(
t
)
d
t
.
(6)Phasors and Complex Impedance. Now suppose that the current in the preceding equations is sinusoidal, of the form
i
(
t
)
=
B
cos
(
ω
t
+
θ
)
.
i
(
t
)
=
B
cos
(
ω
t
+
θ
)
.
(7)We may rewrite i(t)i(t) as
i(t)= Re {Iejωt}i(t)= Re {Iejωt}
(8)where I
I is the phasor representation of i(t)i(t).
Find the phasor
I
I in terms of
B
B and
θ
θ in Equation 8.
The voltage dropped across the resistor is
V
R
(
t
)
=
R
i
(
t
)
=
R
Re
{
I
e
j
ω
t
}
=
Re
{
R
I
e
j
ω
t
}
.
V
R
(
t
)
=
R
i
(
t
)
=
R
Re
{
I
e
j
ω
t
}
=
Re
{
R
I
e
j
ω
t
}
.
(9)Thus the phasor representation for VR(t)VR(t) is
VR(t)↔VR;VR=RI.VR(t)↔VR;VR=RI.
(10)We call R the impedance of the resistor because R is the scale constant that
relates the “phasor voltage VR' to the “phasor current I.”
The voltage dropped across the inductor is
V
L
(
t
)
=
L
d
i
d
t
(
t
)
=
L
d
d
t
Re
{
I
e
j
ω
t
}
.
V
L
(
t
)
=
L
d
i
d
t
(
t
)
=
L
d
d
t
Re
{
I
e
j
ω
t
}
.
(11)The derivative may be moved through the Re [
] Re [
] operator (see Exercise 3)
to produce the result
V
L
(
t
)
=
L
Re
{
j
ω
I
e
j
ω
t
}
=
Re
{
j
ω
L
I
e
j
ω
t
}
.
V
L
(
t
)
=
L
Re
{
j
ω
I
e
j
ω
t
}
=
Re
{
j
ω
L
I
e
j
ω
t
}
.
(12)Thus the phasor representation of
V
L
(
t
)
V
L
(t)
VL(t)↔VL;VL=jωLI.VL(t)↔VL;VL=jωLI.
(13)We call jωLjωL the impedance of the inductor because jωLjωL is the complex scale
constant that relates “phasor voltage
V
L
V
L
' to “phasor current
I
I.”
Prove that the operators ddtddt and Re [] Re [] commute:
d
d
t
Re
{
e
j
ω
t
}
=
Re
{
d
d
t
e
j
ω
t
}
.
d
d
t
Re
{
e
j
ω
t
}
=
Re
{
d
d
t
e
j
ω
t
}
.
(14) The voltage dropped across the capacitor is
V
C
(
t
)
=
1
C
∫
i
(
t
)
d
t
=
1
C
∫
Re
{
I
e
j
ω
t
}
d
t
.
V
C
(
t
)
=
1
C
∫
i
(
t
)
d
t
=
1
C
∫
Re
{
I
e
j
ω
t
}
d
t
.
(15)The integral may be moved through the Re [
] Re [
] operator to produce the result
V
C
(
t
)
=
1
C
Re
{
I
j
ω
e
j
ω
t
}
=
Re
{
I
j
ω
C
e
j
ω
t
}
.
V
C
(
t
)
=
1
C
Re
{
I
j
ω
e
j
ω
t
}
=
Re
{
I
j
ω
C
e
j
ω
t
}
.
(16)Thus the phasor representation of
V
C
(
t
)
V
C
(t)
is
VC(t)↔VC;VC=IjωCVC(t)↔VC;VC=IjωC
(17)We call 1jωC1jωC the impedance of the capacitor because 1jωC1jωC is the complex scale
constant that relates “phasor voltage
V
C
V
C
" to “phasor current
I
I.”
Kirchhoff's Voltage Law. Kirchhoff's voltage law says that the voltage dropped in the series combination of
RR, L
L, and
C
C illustrated in Figure 1 equals the voltage generated by the source (this is one of two fundamental conservation laws in circuit theory, the other being a conservation law for current):
V
(
t
)
=
V
R
(
t
)
+
V
L
(
t
)
+
V
C
(
t
)
.
V
(
t
)
=
V
R
(
t
)
+
V
L
(
t
)
+
V
C
(
t
)
.
(18)If we replace all of these voltages by their complex representations, we have
Re
{
V
e
j
ω
t
}
=
Re
{
(
V
R
+
V
L
+
V
C
)
e
j
ω
t
}
.
Re{V
e
j
ω
t
}=Re{(
V
R
+
V
L
+
V
C
)
e
j
ω
t
}.
(19)An obvious solution is
V
=
V
R
+
V
L
+
V
C
=
(
R
+
j
ω
L
+
1
j
ω
C
)
I
V
=
V
R
+
V
L
+
V
C
=
(
R
+
j
ω
L
+
1
j
ω
C
)
I
(20)where I is the phasor representation for the current that flows in the circuit.
This solution is illustrated in Figure 2, where the phasor voltages RI,jωLIRI,jωLI,
and 1jωCI1jωCI are forced to add up to the phasor voltage
V
V.
Redraw Figure 2 for R=ωL=1ωC=1R=ωL=1ωC=1.
Impedance. We call the complex number R+jωL+1jωCR+jωL+1jωC the complex
impedance for the series RLC network because it is the complex number that
relates the phasor voltage
V
V to the phasor current
I
I:
V
=
Z
I
Z
=
R
+
j
ω
L
+
1
j
ω
C
.
V
=
Z
I
Z
=
R
+
j
ω
L
+
1
j
ω
C
.
(21)The complex number
Z
Z depends on the numerical values of resistance (R)(R), inductance (L)(L), and capacitance (C)(C), but it also depends on the angular frequency (ω)(ω) used for the sinusoidal source. This impedance may be manipulated as follows to put it into an illuminating form:
Z
=
R
+
j
(
ω
L
-
1
ω
C
)
=
R
+
j
L
C
(
ω
L
C
-
1
ω
L
C
)
.
Z
=
R
+
j
(
ω
L
-
1
ω
C
)
=
R
+
j
L
C
(
ω
L
C
-
1
ω
L
C
)
.
(22)The parameter
ω
0
=
1
L
C
ω
0
=
1
L
C
is a parameter that you will learn to call an "undamped natural frequency" in your more advanced circuits courses. With it, we may write the impedance as
Z
=
R
+
j
ω
0
L
(
ω
ω
0
-
ω
0
ω
)
.
Z
=
R
+
j
ω
0
L
(
ω
ω
0
-
ω
0
ω
)
.
(23)The frequency
ω
ω
0
ω
ω
0
is a normalized frequency that we denote by
ν
ν. Then the impedence, as a function of normalized frequency, is
Z
(
ν
)
=
R
+
j
ω
0
L
(
ν
-
1
ν
)
.
Z
(
ν
)
=
R
+
j
ω
0
L
(
ν
-
1
ν
)
.
(24)When the normalized frequency equals one
(
ν
=
1
)
(ν=1), then the impedance is entirely real and
Z
=
R
Z=R. The circuit looks like it is a single resistor.
|
Z
(
ν
)
|
=
R
[
1
+
(
ω
0
L
R
)
2
(
ν
-
1
ν
)
2
]
1
/
2
.
arg Z(ν)=tan-1ω0LR(ν-1ν).
|
Z
(
ν
)
|
=
R
[
1
+
(
ω
0
L
R
)
2
(
ν
-
1
ν
)
2
]
1
/
2
.
arg Z(ν)=tan-1ω0LR(ν-1ν).
(25)The impedance obeys the following symmetries around ν=1ν=1:
Z
(
ν
)
=
Z
*
(
1
ν
)
|
Z
(
ν
)
|
=
|
Z
(
1
ν
)
|
a
r
g
Z
(
ν
)
=
-
a
r
g
Z
(
1
ν
)
.
Z
(
ν
)
=
Z
*
(
1
ν
)
|
Z
(
ν
)
|
=
|
Z
(
1
ν
)
|
a
r
g
Z
(
ν
)
=
-
a
r
g
Z
(
1
ν
)
.
(26)In the next paragraph we show how this impedance function influences the current that flows in the circuit.
Resonance. The phasor representation for the current that flows the current that flows in the series RLC circuit is
I
=
V
Z
(
ν
)
=
1
|
Z
(
ν
)
|
e
-
j
arg
Z
(
ν
)
V
I
=
V
Z
(
ν
)
=
1
|
Z
(
ν
)
|
e
-
j
arg
Z
(
ν
)
V
(27)The function
H
(
ν
)
=
1
Z
(
ν
)
H(ν)=
1
Z
(
ν
)
displays a "resonance phenomenon." that is,
|
H
(
ν
)
|
|H(ν)|
peaks at
ν
=
1
ν=1 and decreases to zero and
ν
=
0
ν=0
and
ν
=
∞
ν=∞:
|
H
(
ν
)
|
=
0
,
ν
=
0
1
R
ν
=
1
0
,
ν
=
∞
.
|
H
(
ν
)
|
=
0
,
ν
=
0
1
R
ν
=
1
0
,
ν
=
∞
.
(28)When
|
H
(
ν
)
|
=
0
|H(ν)|=0, no current flows.
The function |H(ν)||H(ν)| is plotted against the normalized frequency ν=ν=ωω0ωω0 in Figure 3.14. The resonance peak occurs at ν=1ν=1, where |H(ν)|=|H(ν)|=1R1R meaning that the circuit looks purely resistive. Resonance phenomena
underlie the frequency selectivity of all electrical and mechanical networks.
(MATLAB) Write a MATLAB program to compute and plot |H(ν)||H(ν)| and
argH(ν)argH(ν) versus
ν
ν for
ν
ν ranging from 0.1 to 10 in steps of 0.1. Carry out your computations for ω0LR=10,1,0.1ω0LR=10,1,0.1, and 0.01, and overplot
your results.
Circle Criterion and Power Factor. Our study of the impedance
Z(ν)Z(ν) and the function H(ν)=1Z(ν)H(ν)=1Z(ν) brings insight into the resonance of an
RLC circuit and illustrates the ffequency selectivity of the circuit. But there
is more that we can do to illuminate the behavior of the circuit.
V
=
R
I
+
j
(
ω
L
-
1
ω
C
)
I
.
V
=
R
I
+
j
(
ω
L
-
1
ω
C
)
I
.
(29)This equation shows how voltage is divided between resistor voltage RI and inductor-capacitor voltage
j
(
ω
L
-
1
ω
C
)
I
.
j(ωL-
1
ω
C
)I.
V
=
R
I
+
j
ω
0
L
(
ω
ω
0
-
ω
0
ω
)
I
V
=
R
I
+
j
ω
0
L
(
ω
ω
0
-
ω
0
ω
)
I
(30)or
V
=
R
I
+
j
ω
0
L
R
(
ν
-
1
ν
)
R
I
.
V
=
R
I
+
j
ω
0
L
R
(
ν
-
1
ν
)
R
I
.
(31)In order to simplify our notation, we can write this equation as
V
=
V
R
+
j
k
(
ν
)
V
R
V
=
V
R
+
j
k
(
ν
)
V
R
(32)where
V
R
V
R
is the phasor voltage RIRI and k(ν)k(ν) is the real variable
k
(
ν
)
=
ω
0
L
R
(
ν
-
1
ν
)
.
k(ν)=
ω
0
L
R
(ν-
1
ν
).
(33)Equation 32 brings very important geometrical insights. First, even though
the phasor voltage
V
R
V
R
in the RLC circuit is complex, the terms
V
R
V
R
and
jk(ν)VRjk(ν)VR are out of phase by π2π2 radians. This means that, for every allowable value of
V
R
V
R
, the corresponding jk(ν)VRjk(ν)VR must add in a right triangle to produce the source voltage
V
V. This is illustrated in Figure 4(a). As the frequency
ν
ν changes, then k(ν)k(ν) changes, producing other values of
V
R
V
R
and jk(ν)VRjk(ν)VR that sum to
V
V. Several such solutions for
V
R
V
R
and jk(ν)VRjk(ν)VR are illustrated in Figure 3.15(b). From the figure we gain the clear impression that the phasor voltage
V
>R
V
>R
lies on a circle of radius V2V2 centered at V2V2 Let's try this solution,
V
R
=
V
2
+
V
2
e
j
ψ
=
V
2
(
1
+
e
j
ψ
)
,
V
R
=
V
2
+
V
2
e
j
ψ
=
V
2
(
1
+
e
j
ψ
)
,
(34)and explore its consequences. When this solution is substituted into Equation 32, the result is
V
=
V
2
(
1
+
e
j
ψ
)
+
j
k
(
ν
)
V
2
(
1
+
e
j
ψ
)
V
=
V
2
(
1
+
e
j
ψ
)
+
j
k
(
ν
)
V
2
(
1
+
e
j
ψ
)
(35)or
2
=
(
1
+
e
j
ψ
)
[
1
+
j
k
(
ν
)
]
.
2
=
(
1
+
e
j
ψ
)
[
1
+
j
k
(
ν
)
]
.
(36)If we multiply the left-hand side by its complex conjugate and the right-hand side by its complex conjugate, we obtain the identity
4=2(1+cosψ)[1+k2(ν)].4=2(1+cosψ)[1+k2(ν)].
(37)This equation tells us how the angle
ψ
ψ depends on k(ν)k(ν) and, conversely, how
k(ν)k(ν) depends on ψψ:
cos
ψ
=
1
-
k
2
(
ν
)
1
+
k
2
(
ν
)
cos
ψ
=
1
-
k
2
(
ν
)
1
+
k
2
(
ν
)
(38)
k
2
(
ν
)
=
1
-
cos
ψ
1
+
cos
ψ
k
2
(
ν
)
=
1
-
cos
ψ
1
+
cos
ψ
(39)The number cosψcosψ lies between -1-1
and +1+1, so a circular solution does indeed
work.
Check -1≤cosψ≤1-1≤cosψ≤1 for -∞<k<∞-∞<k<∞ and -∞<k<∞-∞<k<∞
for -π≤ψ≤π-π≤ψ≤π. Sketch k
k versus ψψ and ψψ versus kk.
The equation VR=V2(1+ejψ)VR=V2(1+ejψ) is illustrated in Figure 5. The angle
that
V
R
V
R
makes with
V
V is determined from the equation
2
φ
+
π
-
ψ
=
π
⇒
φ
=
ψ
2
2
φ
+
π
-
ψ
=
π
⇒
φ
=
ψ
2
(40)In the study of power systems,
cos
φ
cosφ is a "power factor" that determines how much power is delivered to the resistor. We may denote the power factor as
η=cosφ=cosψ2.η=cosφ=cosψ2.
(41)But cosψcosψ may be written as
η=cosφ=cosψ2η=cosφ=cosψ2
(42)But cosψcosψ may be written as
cos
ψ
=
cos
(
φ
+
φ
)
=
cos
2
φ
-
-
sin
2
φ
=
cos
2
φ
-
(
1
-
cos
2
φ
)
=
2
cos
2
φ
-
1
=
2
η
2
-
1
.
cos
ψ
=
cos
(
φ
+
φ
)
=
cos
2
φ
-
-
sin
2
φ
=
cos
2
φ
-
(
1
-
cos
2
φ
)
=
2
cos
2
φ
-
1
=
2
η
2
-
1
.
(43)Therefore the square of the power factor η is
η2=cosψ+12=11+k2(ν)η2=cosψ+12=11+k2(ν)
(44)The power factor is a maximum of 1 for k(ν)=0k(ν)=0, corresponding to ν=1ν=1(ω=ω0)(ω=ω0). It is a minimum of 0 for k(ν)=±∞k(ν)=±∞, corresponding to ν=0,∞ν=0,∞(ω=0,∞)(ω=0,∞).
With
k
k defined as k(ν)=ω0LR(ν-1ν)k(ν)=ω0LR(ν-1ν), plot k2(ν)k2(ν), cosψcosψ, and
η
2
η
2
versus
ν
ν.
Find the value of
ν
ν that makes the power factor η=0.707η=0.707.
(What does "Endorsed by" mean?)
"Reviewer's Comments: 'I recommend this book as a "required primary textbook." This text attempts to lower the barriers for students that take courses such as Principles of Electrical Engineering, […]"