RESONANT CIRCUITS
•
Resonance occurs in a circuit that has at least one inductor and
one capacitor.
•
Resonance is a condition in an RLC circuit in which the
capacitive and inductive reactances are equal in magnitude,
resulting in a purely resistive total impedance.
•
Can be designed to have a sharp peak in its amplitude
characteristic allowing the circuit to discriminate input
frequencies.
•
Useful for constructing bandpass filters, known as resonators,
whose frequency response is highly frequency selective.
•
There are two types of resonant circuits: series and parallel
resonant circuits.
SERIES RESONANCE
Vs ( jω )
⎛
1 ⎞
= R + j ⎜ωL −
⎟
I ( jω )
ωC⎠
⎝
1
=0
Resonant occurs when Z ( jω ) is purely real → ω L −
ωC
The value of ω that satisfies such condition is called the resonant
The input impedance is
frequency:
1
ω0 =
LC
Z ( jω ) =
rad/s or
f0 =
1
2π LC
Hz
FEATURES OF SERIES RESONANCE
1. Transfer impedance, H ( jω ) = Z ( jω ) = R is purely resistive and
its magnitude is minimum.
2. Voltage and current are in phase (unity power factor).
3. The frequency response is measured in terms of the circuit’s
current.
Vm
I =
2
1 ⎞
2 ⎛
R + ⎜ω L −
⎟
ωC⎠
⎝
1 2
4. The average power dissipated by the circuit is P(ω ) = I R
2
5. The average power reaches its maximum at resonance (ω = ω0 )
1 Vm2
Vm
where I =
and P(ω0 ) =
R
2 R
6. The half-power frequencies are obtained by
2
⎛
⎞
⎜
⎟
⎟
Vm
1
1 ⎛ 1 Vm2 ⎞
1 2
1⎜
⎟
⎜
P(ω ) = I R = ⎜
R
P
ω
(
)
=
=
⎟
0
⎟
⎜
2
R
2
2
2
2
2⎜
⎠
⎝
1 ⎞ ⎟
2 ⎛
⎜ R + ⎜ω L − ω C ⎟ ⎟
⎝
⎠ ⎠
⎝
2
⎛
1 ⎞
R + ⎜ω L −
⎟ = 2R2
ωC ⎠
⎝
2
ω1 = −
2
2
R
R
1
R
R
1
and ω2 =
+ ⎛⎜ ⎞⎟ +
+ ⎜⎛ ⎟⎞ +
2L
2L
⎝ 2 L ⎠ LC
⎝ 2 L ⎠ LC
(Vm / 2 ) 2 Vm2
P(ω1 ) = P(ω2 ) =
=
and ω0 = ω1ω2
2R
4R
7. The width of the response curve is
called the half-power bandwidth
and defined as the difference
between the two half-power
frequencies:
B = ω2 − ω1 =
R
= ω02 RC
L
8. The height of the curve depends on
the value of R
9. If R is made smaller with a fixed
L and C, the bandwidth decreases
and the selectivity increases.
10. If the ratio L/C increases with
fixed R, the bandwidth decreases
with an increase in selectivity.
10. The circuit’s selectivity is the ability to respond to a certain
frequency and discriminate against other frequencies
11. It refers to the “sharpness” of the resonance which depends on
the height and the width of the response curve.
12. Quantitaively, selectivity is measured by the quality factor Q.
Peak energy stored in the circuit
Q = 2π
Energy dissipated in one period at resonance
1 2
LI
2π f 0 L
Q = 2π 2
=
R
1 2 ⎛1⎞
I R⎜⎜ ⎟⎟
2
⎝ f0 ⎠
ω0 L
1
=
Q=
R ω0 RC
→
B=
R ω0
=
L Q
13. QUALITY FACTOR is the ratio of the resonance frequency to
the bandwidth. The higher the Q the smaller the bandwidth.
14. For high-Q circuits (Q ≥ 10) ,
ω1 and ω2 are symmetrical
around ω0 .
B
ω1 ≈ ω0 −
2
ω 2 ≈ ω0 +
B
2
Ex. (Practice problem 14.7)
A series-connected circuit has R = 4Ω and L = 25 mH.
(a) Calculate the value of C that will produce a quality factor
of 50.
(b) Find the half-power frequencies and the bandwidth
(c) Determine the average power dissipated at ω0 , ω1 , ω2 .
Take Vm=100 V.
PARALLEL RESONANT CIRCUITS
⎛
I ( jω ) 1
1 ⎞
= + j ⎜ω C −
The input admittance is Y ( jω ) =
⎟
Vs ( jω ) R
ωL ⎠
⎝
1
=0
Resonant occurs when Y ( jω ) is purely real → ω L −
ωC
The value of ω that satisfies such condition is called the resonant
frequency:
ω0 =
1
LC
rad/s or
f0 =
1
2π LC
Hz
The frequency response is measured in terms of the circuit’s
common voltage.
Im
V =
2
2
1 ⎞
⎛ 1 ⎞ +⎛
−
ω
C
⎟
⎜ ⎟ ⎜
ωL ⎠
⎝R⎠ ⎝
ω1 = −
2
1
1 ⎞
1
+ ⎛⎜
+
⎟
2 RC
⎝ 2 RC ⎠ LC
2
1
1 ⎞
1
ω2 =
+ ⎛⎜
+
⎟
2 RC
⎝ 2 RC ⎠ LC
B = ω2 − ω1 =
1
L
= ω02
RC
R
Q=
ω0
B
= ω0 RC =
B
B
For high-Q circuits, ω1 ≈ ω0 − and ω2 ≈ ω0 +
2
2
R
ω0 L
For both the series and parallel resonant circuits, the halfpower frequencies can be expressed in terms of Q as
2
ω1 = ω0
⎛ 1 ⎞ ω0
1+ ⎜
⎟ −
⎝ 2Q ⎠ 2Q
2
⎛ 1 ⎞ ω0
⎟ +
⎝ 2Q ⎠ 2Q
ω1 = ω0 1 + ⎜
Ex. Practice Problem 14.8
A parallel resonant circuit has R = 100 kΩ, L = 20 mH, and
C = 5 nF. Calculate ω0 , ω1 , ω2 , Q, B.
Ex. Practice Problem 14.9
Calculate the resonant frequency of the circuit shown.