Derive the expression of power in LCR circuit
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Answer:
Phasor diagram is shown for a series LCR circuit.
Using definition of reactance,
Voltage across capacitor is V
C
=IX
C
=
ωC
I
Voltage across inductor is V
L
=IX
L
=IωL
Voltage across resistor is V
R
=IR
From definition of impedance, V
S
=IZ
From the phasor diagram,
V
S
2
=V
R
2
+(V
L
−V
C
)
2
(
2
V
m
)
2
=I
2
(R
2
+(ωL−
ωC
1
)
2
)
I=
2(R
2
+(ωL−
ωC
1
)
2
)
V
m
Power dissipated in the circuit is:
P=I
2
R
P=
2(R
2
+(ωL−
ωC
1
)
2
)
V
m
2
R
At resonance,
ωL=
ωC
1
Denominator of power is minimum and hence, power is maximum.
Maximum power is given by:
P
max
=
2R
V
m
2
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