Transcript Chapter 21: Resonance

Chapter 21
Resonance
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
2
Series Resonance
• Response curves for a series resonant circuit
3
Series Resonance
4
Series Resonance
• Since XL = L = 2fL and XC = 1/C =
1/2fC for resonance set XL = XC
– Solve for the series resonant frequency fs
s 
fs 
1
LC
1
(rad/sec)
2 LC
(Hz)
5
Series Resonance
• At resonance
– Impedance of a series resonant circuit is
small and the current is large
• I = E/ZT = E/R
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Series Resonance
• At resonance
VR = IR
VL = IXL
VC = IXC
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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
8
The Quality Factor,Q
• Q = reactive power/average power
– Q may be expressed in terms of inductor or
capacitor
I 2 X L X L L
Qs  2 

R
R
I R
• For an inductor, Qcoil= XL/Rcoil
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The Quality Factor,Q
• Q is often greater than 1
– Voltages across inductors and capacitors can
be larger than source voltage
IX V
Qs 

IR E
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The Quality Factor,Q
• This is true even though the sum of the
two voltages algebraically is zero
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Impedance of a Series
Resonant Circuit
• Impedance of a
series resonant
circuit varies with
frequency
Z T  R  j L 
j
C
  2 LC  1 

Z  R  j
T


C


Z
T


2

R


 2 LC  1
RC
  2 LC  1 

1

  tan 
 RC






2
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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
13
Bandwidth
• A circuit with a narrow bandwidth
– High selectivity
• If the bandwidth is wide
– Low selectivity
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Bandwidth
• Cutoff frequencies
– Found by evaluating frequencies at which the
power dissipated by the circuit is half of the
maximum power
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Bandwidth
I hpf
I max

2
R
R2 1
1  2 f 


2
1
2L
4 L LC
R
R2 1
2  2 f  

2
2
2 L 4 L LC
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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)
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Series-to-Parallel Conversion
• For analysis of parallel resonant circuits
– Necessary to convert a series inductor and its
resistance to a parallel equivalent circuit
RS 2  X LS
RP 
RS
RS  X LS

X LS
2
X LP
2
X LS
RP
Q

RS
X LP
2
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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
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Parallel Resonance
• Parallel resonant circuit
– Has XC and equivalents of inductive reactance
and its series resistor, XLP and RS
• At resonance
– XC = XLP
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Parallel Resonance
• Two reactances cancel each other at
resonance
– Cause an open circuit for that portion
• ZT = RP at resonance
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Parallel
Resonance
• Response curves
for a parallel
resonant circuit
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Parallel Resonance
• From XC = XLP
– Resonant frequency is found to be
R 2C
f
1
L
2 LC
1
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Parallel Resonance
• If (L/C) >> R
– Term under the radical is approximately equal
to 1
• If (L/C)  100R
– Resonant frequency becomes
f 
1
2 LC
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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
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Parallel Resonance
• Circuit currents are
V
IR 
R
V
IL 
 QPI
XL
V
IC 
 QPI
XC
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Parallel Resonance
• Magnitudes of currents through the
inductor and capacitor
– May be much larger than the current source
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Bandwidth
• Cutoff frequencies are
1
1
1
1 


2 2
2RC
LC
4R C
1
1
1
2 


2 2
2RC
LC
4R C
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Bandwidth
• BW = 2 - 1 = 1/RC
• If Q  10
– Selectivity curve becomes symmetrical
around P
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Bandwidth
• Equation of bandwidth becomes
X C P
BW 
R
BW 
P
QP
• Same for both series and parallel circuits
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