Chapter 21
Resonance
2
Series Resonance• Simple series resonant circuit
– Has an ac source, an inductor, a capacitor, and possibly a resistor
• ZT = R + jXL – jXC = R + j(XL – XC)– Resonance occurs when XL = XC
– At resonance, ZT = R
3
Series Resonance• Response curves for a series resonant circuit
4
Series Resonance
5
Series Resonance• Since XL = L = 2fL and XC = 1/C =
1/2fC for resonance set XL = XC
– Solve for the series resonant frequency fs
(Hz)
(rad/sec)
LCf
LC
s
s
21
1
=
=
6
Series Resonance• At resonance
– Impedance of a series resonant circuit is small and the current is large
• I = E/ZT = E/R
7
Series Resonance• At resonance
VR = IRVL = IXL
VC = IXC
8
Series Resonance• At resonance, average power is P = I2R• Reactive powers dissipated by inductor
and capacitor are I2X• Reactive powers are equal and opposite at
resonance
9
The Quality Factor,Q• Q = reactive power/average power
– Q may be expressed in terms of inductor or capacitor
• For an inductor, Qcoil= XL/Rcoil
RL
RX
RIXIQ LL
s
=== 2
2
10
The Quality Factor,Q• Q is often greater than 1
– Voltages across inductors and capacitors can be larger than source voltage
EV
IRIXQs ==
11
The Quality Factor,Q• This is true even though the sum of the
two voltages algebraically is zero
12
Impedance of a Series Resonant Circuit
• Impedance of a series resonant circuit varies with frequency
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
−+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+=
−+=
⎟⎠⎞⎜⎝
⎛
RCLC
RCLC
R
CLCjR
CjLjR
ωωθ
ωω
ωω
ωω
121tan
12 22
T
12
T
T
Z
Z
Z
13
Bandwidth• Bandwidth of a circuit
– Difference between frequencies at which circuit delivers half of the maximum power
• Frequencies, f1 and f2
– Half-power frequencies or the cutoff frequencies
14
Bandwidth• A circuit with a narrow bandwidth
– High selectivity• If the bandwidth is wide
– Low selectivity
15
Bandwidth• Cutoff frequencies
– Found by evaluating frequencies at which the power dissipated by the circuit is half of the maximum power
16
Bandwidth
LCLR
LR
LCLR
LR
II
f
f1
422
142
2
2
2
2
22
2
2
11
maxhpf
++==
++−
==
=
17
Bandwidth• From BW = f2 - f1
• BW = R/L• When expression is multiplied by on top
and bottom– BW = s/Q (rad/sec) or BW = fs/Q (Hz)
18
Series-to-Parallel Conversion• For analysis of parallel resonant circuits
– Necessary to convert a series inductor and its resistance to a parallel equivalent circuit
LP
P
S
LS
LS
LSSLP
S
LSSP
XR
RX
Q
XXR
X
RXR
R
==
+=
+=
22
22
19
Series-to-Parallel Conversion• If Q of a circuit is greater than or equal to 10
– Approximations may be made• Resistance of parallel network is
approximately Q2 larger than resistance of series network– RP Q2RS
– XLP XLS
20
Parallel Resonance• Parallel resonant circuit
– Has XC and equivalents of inductive reactance and its series resistor, XLP and RS
• At resonance– XC = XLP
21
Parallel Resonance• Two reactances cancel each other at
resonance– Cause an open circuit for that portion
• ZT = RP at resonance
22
Parallel Resonance
• Response curves for a parallel resonant circuit
23
Parallel Resonance• From XC = XLP
– Resonant frequency is found to be
LCR
LCf
2
121
−=
24
Parallel Resonance• If (L/C) >> R
– Term under the radical is approximately equal to 1
• If (L/C) 100R– Resonant frequency becomes
LCf
21
=
25
Parallel Resonance• Because reactances cancel
– Voltage is V = IR• Impedance is maximum at resonance
– Q = R/XC
• If resistance of coil is the only resistance present– Circuit Q will be that of the inductor
26
Parallel Resonance• Circuit currents are
IQX
IQX
P
CC
P
LL
R
==
==
=
VI
VI
RV
I
27
Parallel Resonance• Magnitudes of currents through the
inductor and capacitor – May be much larger than the current source
28
Bandwidth• Cutoff frequencies are
LCCRRC
LCCRRC1
41
21
141
21
222
221
++=
+−=
29
Bandwidth• BW = 2 - 1 = 1/RC• If Q 10
– Selectivity curve becomes symmetrical around P
30
Bandwidth• Equation of bandwidth becomes
• Same for both series and parallel circuits
P
P
PC
Q
RX
=
=
BW
BW
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