EE111 Electrical Circuit Analysis
Lecture 21
Dr. Oliver Faust1
November 13, 2014
1
School of Science & Engineering
Habib University
Motivation
Coordinate transformation simplifies the AC analysis.
Contents
1 The Complex Forcing Function
1.1 Avoiding differential equations . . . . . . . . . . . . . . . . . . .
1
2
2 The Phasor
3
1
The Complex Forcing Function
The basic idea is that sinusoids and exponentials are related through complex
numbers. Euler’s identity, for example, tells us that
ejθ = cos θ + j sin θ
(1)
Whereas taking the derivative of a cosine function yields a (negative) sine
function, the derivative of an exponential is simply a scaled version of the same
exponential. If at this point the reader is thinking, “All this is great, but there
are no imaginary numbers in any circuit I ever plan to build!” that may be
true. What we’re about to see, however, is that adding imaginary sources to
our circuits leads to complex sources which (surprisingly) simplify the analysis
process. It might seem like a strange idea at first, but a moment’s reflection
should remind us that superposition requires any imaginary sources we might
add to cause only imaginary responses, and real sources can only lead to real
responses. Thus, at any point, we should be able to separate the two by simply
taking the real part of any complex voltage or current.
1
1.1
Avoiding differential equations
RL circuit with an AC forcing function
R
i(t)
vs (t)=Vm cos ωt
L
Let’s try out this idea on the simple RL series circuit shown in the figure above.
The real source Vm cos ωt is applied; the real response i(t) is desired. Since
n
o
Vm cos ωt = < {Vm cos ωt + jVm sin ωt} = < Vm ejωt
(2)
the necessary complex source is
Vm ejωt
(3)
We express the complex response that results in terms of an unknown amplitude
Im and an unknown phase angle φ:
Im ej(ωt+φ)
(4)
Writing the differential equation for this particular circuit,
Ri+L
di
= vs
dt
(5)
we insert our complex expressions for vs and i:
R Im ej(ωt+φ) + L
d
Im ej(ωt+φ) = Vm ejωt
dt
(6)
take the indicated derivative:
R Im ej(ωt+φ) + jω L Im ej(ωt+φ) = Vm ejωt
(7)
and obtain an algebraic equation. In order to determine the values of Im and
φ, we divide throughout by the common factor ejωt :
R Im ejφ + jω L Im ejφ = Vm
(8)
Im ejφ (R + jω L) = Vm
(9)
factor the left side:
rearrange:
Im ejφ =
Vm
R + jω L
2
(10)
and identify Im and φ by expressing the right side of the equation in exponential
or polar form:
−1
Vm
ej[− tan (ωL/R)]
Im ejφ = √
(11)
2
2
2
R +ω L
Thus,
Im = √
Vm
R2 + ω 2 L2
and
φ = − tan−1
ωL
R
(12)
(13)
In polar notation, this may be written as
Im φ
or
√
R2
Vm
− tan−1 (ωL/R)
+ ω 2 L2
(14)
(15)
The complex response is given by Eq. 11. Since Im and φ are readily identified,
we can write the expression for i(t) immediately. However, if we feel like using
a more rigorous approach, we may obtain the real response i(t) by reinserting
the ejωt factor on both sides of Eq. 11 and taking the real part. Either way, we
find that
Vm
−1 ωL
i(t) = Im cos(ωt + φ) = √
cos ωt − tan
(16)
R
R2 + ω 2 L2
which agrees with the response obtained in the previous lecture for the same
circuit.
2
The Phasor
In the last section, we saw that the addition of an imaginary sinusoidal source
led to algebraic equations which describe the sinusoidal steady-state response
of a circuit. An intermediate step of our analysis was the “canceling” of the
complex exponential term–once its derivative was taken, we apparently had no
further use for it until the real form of the response was desired. Even then, it
was possible to read the magnitude and phase angle directly from our analysis,
and hence skip the step where we overtly take the real part. Another way of
looking at this is that every voltage and current in our circuit contain the same
factor ejωt , and the frequency, although relevant to our analysis, does not change
as we move through the circuit. Dragging it around, then, is a bit of a waste of
time.
These complex quantities are usually written in polar form rather than exponential form in order to achieve a slight additional saving of time and effort.
For example, a source voltage
v(t) = Vm cos ωt = Vm cos(ωt + 0◦ )
3
(17)
we now represent in complex form as
Vm 0◦
(18)
i(t) = Im cos(ωt + φ)
(19)
Im φ
(20)
and its current response
becomes
This abbreviated complex representation is called a phasor. Let us review the
steps by which a real sinusoidal voltage or current is transformed into a phasor,
and then we will be able to define a phasor more meaningfully and to assign a
symbol to represent it. A real sinusoidal current
i(t) = Im cos(ωt + φ)
(21)
is expressed as the real part of a complex quantity by invoking Euler’s identity
n
o
i(t) = < Im ej(ωt+φ)
(22)
We then represent the current as a complex quantity by dropping the instruction
<{. . .}, thus adding an imaginary component to the current without affecting
the real component; further simplification is achieved by suppressing the factor
ejωt :
n o
I = Im ejφ
(23)
and writing the result in polar form:
I = Im φ
Outlook
Lecture 22 will cover:
• Impedance and Admittance,
• Nodal and Mesh Analysis.
4
(24)