Download Resonant Circuit

Resonant Circuit
Series Behavior
 The behavior of the series
i
R
RLC circuit is governed by the
impedance.
• Magnitude and phase
v
L
1 

Z  R 2   L 

C 

C
1

 L 
C
  arctan 
R









2
Perfect Match
 There is special behavior when
XC = XL.
• Vectors cancel
• Impedance only from resistor
VL=IXL
VR=IR
 This is called resonance.
VC=IXC
Resonant Frequency
 The requirements for
XC  X L
1
 0 L
0C
1
 
LC
2
0
0 
1
LC
0
1
f0 

2 2 LC
resonance come from the
reactances.
 There is a resonant frequency
0 associated with the circuit.
• Angular frequency 
• Can be converted into
frequency f in Hz
Vector Sum
 The total impedance is the
magnitude of Z.
XC
XL
Z

 The phase between the current
and voltage is the angle 
between Z and the x-axis.
R
Z  R2  X L  X C 
2
1 

2
Z  R   L 

C 

2
X L  XC
tan  
R
1


L


C
  arctan 
R









Peak Performance
 At resonance the current is at maximum for the voltage.
Circuit Example
 Find the resonant frequency in
the following circuit in Hz.
 The problem requires the
formula for the frequency f.
f0 
1
2 LC
100 W
 Only the inductance and
capacitance matter.
10 V
250 mH
0.1 mF
• 1/2 (0.25 H 10-7 F)1/2 = 1 kHz
Circuit Example
 The behavior of the series
100 W
RLC circuit is governed by the
impedance.
• Magnitude and phase
10 V
250 mH
0.1 mF
1 

Z  R 2   L 

C 

1

 L 
C
  arctan 
R









2
Resonant Reactance
 In the preceding circuit the
voltage across each
component can be found.
• Current due to resistor only
 The voltage across the
inductor has an amplitude of
158 V.
• So does the capacitor
I  V / R  0.1A
VL  2f 0 IL  158 V
I
VC 
 158 V
2f 0C
 They are each 90° out of
phase and cancel out.
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