18.03 Complex Impedance and Phasors
Jeremy Orloff
Impedance: Generalizes Ohm’s law V = IR to capacitors and inductors.
Recall: Two resistances R1 and R2 combine to give an equivalent resistance R.
1
1
1
=
+
.
For R1 , R2 in series R = R1 + R2 and in parallel
R
R1 R2
We are going to use the exponential response formula and complex arithmetic to
understand the notions of complex impedance and phasor diagrams.
Simple circuit physics
The picture at right shows an inductor, capacitor and resistor in series with a driving
voltage source.
VL
I(t) is the current in the circuit in amps.
L is the inductance in henries.
R is the resistance in ohms.
C is the capacitance in farads.
Vin
Vin is the input voltage to the circuit.
Q(t) is the charge on the capacitor, so I(t) = Q0 (t).
•
•
L
•
∼
VC
C
•
jt
I
R
•
•
VR
From physics we get that the voltage drops across each of the circuit elements.
Q
.
C
The amazing thing is that this and Kirchhoff’s voltage law (KVL) is all the physics we
need to understand this circuit. The rest is linear CC DE’s and complex arithmetic.
VL = LI 0 = LQ00 ,
Summary:
VR = RI = RQ0 ,
VC =
We start with a summary of our results. Individual items will be explained below.
Compatible units: current = amps, voltage = volts, resistance = ohms,
inductance = henries, capacitance = farads.
1
Voltage drops: LI 0 , RI,
Q.
C
1
1
DEs: LQ00 + RQ0 + Q = Vin ; LI 00 + RI 0 + I = Vin0 .
C
C
Complex impedance (valid when Ṽin = eiωt ):
Z̃L = iLω, Z̃R = R, Z̃C =
total impedance Z̃ = Z̃R + Z̃L + Z̃C = R + i(ωL − 1/(ωC)).
(continued)
1
1
,
iCω
18.03 Complex Impedance and Phasors
˜ ṼL = Z̃L I,
˜ ṼR = Z̃R I,
˜ ṼC = Z̃C I.
˜
Complex Ohm’s Law: Ṽin = Z̃ I,
Phasors: All the output voltages are plotted in the complex plane as a rigid set of
vectors that rotate at frequency ω. VR and I˜ point in the same direction, ṼL leads I˜
by π/2, ṼC lags I˜ by π/2. I˜ either leads or lags Ṽin by φ = atan2(Lω − 1/(Cω), R).
Reactance and real impedance:
Reactance = S = ωL + 1/(ωC). ⇒ Z̃ = R + iS.
p
√
Real impedance = |Z̃| = R2 + S 2 = R2 + (ωL − 1/(ωC))2 .
E0
If Vin = E0 sin ωt then from Ṽin = Z̃ I˜ we get I =
sin(ωt − φ),
|Z̃|
with the phase angle φ = tan−1 (S/R).
I.e., the amplitude of the current is given by E0 /real impedance.
√
Im
Practical resonance: Always at the natural frequency ω0 = 1/ LC
izWg
Simple complex arithmetic fact
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.
7G
z
Re
' z/i = −iz
The basic differential equations
KVL implies the total voltage drop around the circuit has to be 0. Which leads to
the following second order CC linear DE.
LQ(t)00 + RQ(t)0 +
1
Q(t) = Vin (t).
C
Differentiating this equation we can write
LI(t)00 + RI(t)0 +
1
I(t) = Vin0 (t).
C
Complex Impedance
Now complex arithmetic and the exponential response formula will allow us to understand all about phasors and impedance.
First, note that if we remove the inductor and capacitor the first DE is just Ohm’s
law, i.e. RQ0 = RI = Vin .
Now make the crucial assumption of sinusoidal input (alternating current):
Vin (t) = V0 sin(ωt).
1˜
˜
I = Ṽin0 = iωV0 eiωt , I = Im(I).
C
iωV0 iωt
e .
The exponential response formula gives the periodic solution: I˜ =
P (iω)
Complexify the second equation:
LI˜00 + RI˜0 +
(continued)
2
18.03 Complex Impedance and Phasors
A little algebra shows
Define Z̃ = iLω +
quency ω).
iωV0
iωV0
V0
=
=
.
P (iω)
−Lω 2 + 1/C + Riω
iLω + 1/(iCw) + R
1
+ R = complex impedance (which depends on the input freiCw
We can now write the complex version of Ohm’s law (always assuming Ṽin = V0 eiωt ):
1
˜
I˜ = · Ṽin or Ṽin = Z̃ I.
Z̃
Notice that we can associate a separate impedance to each element.
1
Z̃L = iLω, Z̃R = R, Z̃C =
iCω
We have seen that for a set of elements wired in series the total complex impedance
is just the sum of the individual impedances. (Just like resistances in series.)
What’s more, using the voltage drops across each element we see they individually
satisfy a Complex Ohm’s Law.
Z
1 ˜
1
1
0
˜
˜
˜
˜
˜
I˜ =
ṼL = LI = Liω I = Z̃L I, ṼR = RI, ṼC = Q̃ =
I = Z̃C I.
C
C
iCω
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 Z̃1 and Z̃2 are in parallel then the total impedance of the pair, call it Z̃,
1
1
1
=
+ .
satisfies
Z̃
Z̃1 Z̃2
To see this we use the complex Ohm’s law, KVL and
Kirchhoff’s current law (KCL). They imply
I˜*4
•
•
I˜ = I˜1 + I˜2 , Ṽ = Z̃1 I˜1 , Ṽ = Z̃2 I˜2 .
I˜
1
Ṽ
1
Ṽ
1
•
⇒ I˜ =
+
= Ṽ
+
Z̃1 Z̃2
Z̃1 Z̃2
Z̃1
Ṽ
1
˜
⇒ Ṽ =
I. QED
•
1/Z̃1 + 1/Z̃2
jt
•
•
Amplitude-phase form
1
1
First rewrite Z̃ = iLω +
+ R = i(Lω −
) + R = iS + R.
iCω
Cω
Here, S = Lω − 1/(Cω) is the reactance. Note S = 0 when ω 2 = 1/(LC).
√
In amplitude phase form Z̃ = reiφ , where r = S 2 + R2 and φ = atan2(S, R).
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.
V0
V0
Thus I˜ = √
ei(ωt−φ) = p
ei(ωt−φ) .
2
2
2
2
S +R
(Lω − 1/Cω) + R
p
√
The term S 2 + R2 = |Z̃| = (Lω − 1/Cω)2 + R2 is called the real impedance.
(continued)
3
I˜*42
•
•
Z̃2
•
•
18.03 Complex Impedance and Phasors
Taking imaginary parts: I|Z̃| = V0 sin(ωt − φ). Which is like Ohm’s law,except with
a phase shift.
Phasors (phasor just means eiωt )
Im
@L Ṽ
ṼR^ L
in
2>
φ
2>
ṼR
ωt
I˜
Note: the entire picture is rotating at frequency ω and the real values of each of the
voltages are given by the y coordinate (the
imaginary part) of their respective phasors.
Re
Note: the phasor ṼR is φ behind Ṽin (if φ is
negative then ṼR is ahead of Ṽin ).
Note: the phasors ṼL and ṼC are respectively
π/2 ahead and π/2 behind ṼR .
ṼC
In class we will look at the lovely SeriesLRC
applet.
Amplitude response and practical resonance
√
Natural frequency = w0 = 1/ LC (Clearly when S = 0.)
√
Practical Resonance: Always at the natural frequency ω0 = 1/ LC (This is easy
˜ since I˜ is clearly maximized when the
to see in the amplitude-phase form of I,
term (Lω − 1/Cω)2 = 0.)
V0
I.e, practical resonance is when Z̃L + Z̃C = 0 ⇒ iLω − i/Cω = 0 ⇒ Z̃ = R, I˜ = eiωt .
R
In the phasor picture, at practical resonance Ṽin , I˜ and ṼR all line up, i.e., lag is 0
and ṼR = Ṽin .
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’ (link is given below) shows all this beautifully.
Applet A nice d’AIMP applet showing all of this is at
http://www-math.mit.edu/daimp (there is a link on the class website), click on Series
RLC Circuit.
Suggested applet exercise Set it to show you all four voltages and the current I.
Set L = 500 mH, C = 100 µF , R = 250 ohms.
Compute the resonant frequency of the system.
Move ω to the resonant frequency, watch the phasors and the sinusoidal plots as you
do this.
With ω set at ω0 watch the amplitudes of the 3 output voltages and the output current as R increases. Explain everything you see in terms of the complex Ohm’s laws.
˜
(And the exponential response formula solution for I.)
4