Impedance
1.
Im
Simple Complex Arithmetic Fact
z
iz
You should be clear that in the complex plane multiplication by i
is the same as rotation by π/2. Likewise division by i is the same
as rotation by −π/2.
The phase difference between two complex numbers a and b is
simply the difference of their arguments, Arg( a) − Arg(b). The
simple arithmetic fact implies
z and iz have a phase difference of π/2.
z and z/i have a phase difference of −π/2.
Re
z/i = −iz
(1)
We will need this when we discuss phasors.
2.
Complex Impedance
We repeat for reference some of the DE’s given in the previous note.
..
.
LQ + RQ +
..
.
L I + RI +
1
Q = Vin
C
(2)
.
1
I = V in
C
(3)
Using complex arithmetic and the Exponential Response formula we
can understand all the statements about impedance and phasors.
First, note that if we remove the inductor and capacitor then (2) is just
.
Ohm’s law, i.e. RQ = RI = Vin .
Now we make the crucial assumption of sinusoidal input (alternating
current):
Vin (t) = V0 sin(ωt).
With this input we will solve equation (3).
I we use prime instead of
First, complexify (3): (Because of the tildes (I)
dot to indicate derivatives.)
LI
I '' + R I
I' +
1 I I'
I = Vin = i ω V0 eiω t ,
C
I = Im( I
I ).
The Exponential Response formula gives the periodic solution:
iωV0 iωt
I
e .
I=
P(iω )
(4)
Impedance
OCW 18.03SC
A little algebra shows that the coefficient of V0 eiωt in (4) is
iω
iω
1
=
=
.
2
P(iω )
− Lω + 1/C + Riω
iLω + 1/(iCw) + R
Accordingly we define the complex impedance as
I = iLω + 1 + R.
Z
iCw
(5)
I depends on the input frequency ω.)
(Notice Z
We can now write the complex version of Ohm’s law (always assuming
I
Vin = V0 eiωt ):
1 I
I
Iin = Z
II
or V
I.
(6)
I = ·V
I in
Z
We can associate a separate impedance to each circuit element:
IL = iLω,
Z
IR = R,
Z
IC =
Z
1
.
iCω
(7)
Comparing (5) and (7) we see that for a set of elements wired in series
the total complex impedance is just the sum of the individual impedances.
That is, impedance behaves just like resistance in series.
What’s more, using the voltage drops across each element we see they
individually satisfy a complex Ohm’s Law.
IL I
IL = L I
I ' = Liω I
I=Z
V
I,
I= 1
IC = 1 Q
V
C
C
IR = R I
I,
V
I
I=
1 I I I
I = ZC I .
iC ω
Note: the formulas involving ω depend crucially on the assumption that
the complex input is V0 eiωt .
3.
Impedance in Parallel
It is also true and easy to show that for circuit elements in parallel the
complex impedances combine like resistors in parallel. That is, if impedances
I1 and Z
I sat­
I2 are in parallel then the total impedance of the pair, call it Z,
Z
1
1
1
isfies =
+ .
I
I
I
Z
Z1
Z2
To see this we use Ohm’s law for a single circuit, KVL and Kirchoff’s cur­
rent law (KCL). They imply
2
Impedance
•
I2 .
I=I
I1 , V
I=I
I2 Z
I = I1 + I2 , V
I1 Z
I
I
V
V
I 1 + 1
I=
⇒ I
+
=V
I1
I2
I1
I2
Z
Z
Z
Z
1
I=
I
⇒ V
I. QED
I
I2
1/Z1 + 1/ Z
4.
OCW 18.03SC
I1_
•
I1
•
V
_
•
I21_
•
•
I1
Z
I2
Z
•
•
•
•
Amplitude-Phase Form and Real Impedance
First we put the expression (5) for complex impedance in the form we
need
I = iLω + 1 + R = i ( Lω − 1 ) + R = iS + R.
Z
iCω
Cω
We call S = Lω − 1/(Cω ) the reactance; note that S = 0 when ω 2 =
1/( LC ).
I |eiφ , where | Z
I | = S2 + R2 and φ = Arg( Z
I ) = tan−1 (S/R).
I = |Z
In amplitude phase form Z
Notice the sign of φ depends on the sign of S = Lω − 1/Cω and also that
φ is between −π/2 and π/2.
Thus,
V0
V0
I
I = √
ei(ωt−φ) =
ei(ωt−φ) .
(8)
2
2
2
2
( Lω − 1/Cω ) + R
S +R
√
I | = ( Lω − 1/Cω )2 + R2 is called the real impedance.
The term S2 + R2 = | Z
Taking imaginary parts in (8) gives
I | = V0 sin(ωt − φ),
I |Z
which is like Ohm’s Law, except with a phase shift.
5.
Phasors
(The term phasor just means eiωt ).
We have seen that each element of an LRC circuit obeys a complex
Ohm’s law:
IL = Z
IL I
V
I,
I = Liω I
IR = RI,
I
V
IC = Z
IC I
V
I=
1 I
I.
iCω
(9)
Each of the complex voltages is some constant factor I
I, which is, in turn,
a multiple of eiωt . If we plot the voltages in the complex plane then as t
increases the entire picture will rotate at frequency ω. We call each of these
voltages a phasor.
3
Impedance
OCW 18.03SC
We want to look at the phase difference between the various voltages.
IL and V
IC imply
By our simple arithmetic fact (1), the factors i and 1/i in V
IL and V
IC are respectively π/2 ahead and π/2 behind V
IR .
1. The phasors V
Equation (8) implies
IR is φ behind V
Iin (if φ is negative then V
IR is ahead of V
Iin .
2. The phasor V
Later we will look at the excellent Series LRC Circuit applet which illus­
trates this.
�V
I Im
Iin
V
� L
I
I
φ
IR
ωt V
Re
�V
IC
6.
Amplitude Response and Practical Resonance
√
The natural frequency of the circuit is ω0 = 1/ LC. This is the fre­
quency of oscillation when the “damping” term R is zero.
The practical resonance of the system (3) is independent
of the value of
√
R and always at the natural frequency ω0 = 1/ LC (This is easy to see in
(8), since | I
I | is clearly maximized when the term ( Lω − 1/C ω )2 = 0.)
That is, practical resonance occurs when
V0 iωt
IC = 0 ⇒ iLω − i/Cω = 0 ⇒ Z
I = R, I
IL + Z
I=
e .
Z
R
IR all line up, i.e.,
Iin , I
I and V
In the phasor picture, at practical resonance V
I
I
lag is 0 and VR = Vin .
This is one case where the corresponding sinusoidal graphs of the real
voltages are neat enough to give a nice picture: the graph of VR is exactly
in phase with Vin ; VL and VC have the same magnitude and are 180◦ out of
phase; increasing R doesn’t change VR , but decreases the amplitude of VL
and VC .
The applet Series LRC Circuit shows all this beautifully.
4
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18.03SC Differential Equations
Fall 2011
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