LECTURE 24: Resonance and Coupling for
Maximum Power Transfer
Definition: Input and output voltages/signals are in phase.
1. RLC Admittance/Impedance Transfer Functions
CONSEQUENCE: Yin ( jω ) AND Z in ( jω ) ARE REAL AT ω = ω r , THE
RESONANT FREQUENCY.
EXAMPLE 1: Parallel RLC Yin ( jω ) =
ω rC −
1
= 0 ⇒ ωr =
ωr L
⎛
1
1
1
1 ⎞
+
+ jω C = + j ⎜ ω C −
.
R jω L
R
ω L ⎟⎠
⎝
1
LC
At resonance, the parallel LC part is an open circuit and Yin ( jω r ) =
1
.
R
Lecture 24 Sp 15
2
© R. A. DeCarlo
Example 2: Series RLC
1. Z in ( jω ) = R + jω L +
2. ω r L −
1
= R+
jω C
1
= 0 ⇒ ωr =
ω rC
⎛
1 ⎞
j⎜ω L −
ω C ⎟⎠
⎝
1
LC
3. At ω r ,
Example 3. Find Yin ( jω ) , ω r , and Yin ( jω r ) .
Yin ( jω ) = jω C +
R − jω L
1
= jω C + 2L
RL + jω L
RL + ω 2 L2
Lecture 24 Sp 15
=
3
© R. A. DeCarlo
⎡
⎤
L
+
j
ω
C
−
⎢
⎥
2
2 2
RL2 + ω 2 L2
R
+
ω
L
⎢⎣
⎥⎦
L
RL
At resonance
C−
L
RL2 + ω r2 L2
= 0 ⇒ RL2 + ω r2 L2 =
L
C
It follows that
ω r2
2
2
1 RL
1 RL
=
−
⇒ ωr =
−
=
CL L2
CL L2
CRL2
1−
L
LC
1
CRL2
Observations: (a) ω r exists (is real) only when
< 1.
L
(b) The effect of RL is to make the resonant frequency smaller than
(c) At ω = ω r , Yin ( jω r ) =
RL
RL2
+ ω r2 L2
1
LC
which depends on both L and ω r .
This observation can be used to establish maximum power transfer between a
source and a load at resonant frequencies, e.g., an antenna (source) and its
load, the receiver.
EXAMPLE 4. Design a coupling network to achieve maximum power transfer
from the source to the load RL = 5 Ω. The source is vin = 12 Vrms at 10
rad/s.
Lecture 24 Sp 15
4
© R. A. DeCarlo
From the circuit, the design specs are: (i) ω r = 10 rad/s, and
(ii) Yin ( jω r ) = 0.1 for maximum power transfer.
Objective: choose values for L and C to meet the design specs.
Using design spec (ii):
0.1 =
5
25 + 100L2
⇒ L = 0.5 H
Using design spec (i),
RL2 2
1
25
2
= 100 =
− 2 = −
= − 100
LC L
C 0.25 C
2
in which case C =
= 0.01 F.
200
ω r2
EXAMPLE 5. FIND THE INPUT IMPEDANCE AND ω r .
Lecture 24 Sp 15
5
Zin ( jω ) = jω L +
© R. A. DeCarlo
G − jω C
1
= jω L + 2L
GL + jω C
G L + ω 2C 2
⎛
⎞
C
= 2
+ jω ⎜ L − 2
⎟
2 2
GL + ω C
G L + ω 2C 2 ⎠
⎝
GL
L−
C
GL2
+ ω r2C 2
=
0 ⇒ ω r2
At resonance: Z in ( jω r ) =
2
1 GL
=
−
⇒ ωr =
LC C 2
GL
GL2
2
2
2
1 GL
−
LC C 2
. This can be used for maximum power
+ω C
transfer when the load is larger than the source resistance at a specific
resonant frequency.