1E6 Electrical Engineering
AC Circuit Analysis and Power
Lecture 12: Parallel Resonant Circuits
12.1 Introduction
There are equivalent circuits to the series combinations examined which
exist in parallel configurations. The issues surrounding resistive and reactive
components and their effect on the circuit impedance are similar while the result
can often be the opposite of the case for series configurations. Some comparable
parallel combinations will therefore be examined.
12.2 Ideal Inductor and Capacitor in Parallel
Consider the inductor and capacitor connected in parallel as shown in Fig.
1 below. The reactive elements are taken to be ideal. In this case the end nodes of
both elements are connected together which means that the voltage across the
inductor is identical to the voltage across the capacitor so that VL = VC. If this is
taken as reference zero phase, then it can be seen that the current through the
capacitor leads the voltage so that it appears 900 ahead on the phasor diagram of
Fig. 1, while the current through the inductor lags the voltage so that it appears
900 behind. It can be seen therefore that in this case the currents through the
inductor and the capacitor are in anti-phase or 1800 out of phase with each
other. The relative magnitudes of the currents are different and depend on the
values of inductance and capacitance of these elements at the frequency of
excitation. This situation is similar to the series combination except that in this
case it is the voltage rather than the current which is the common component.
j
IL
+
ω
IC
C
IC
+
L
VL
-
current leads
voltage for C
VL = VC
VC
IL
ω
-
ω
current lags
voltage for L
-j
Fig. 1
An Inductor and Capacitor Connected in Parallel
1
VL , VC
j
ω
IL
IC
VL, VC
t
ω
IL
ω
IC
-j
Fig. 2 Phasor Diagram and Waveforms for Inductor and Capacitor in Parallel
Waveforms are shown for sinusoidal excitation of the circuit in Fig. 2.
From this it is clearly evident that the phasors representing the currents through
the inductor and the capacitor are exactly 1800 out of phase, showing excursions
on opposite sides of the abscissa axis. The difference in the amplitudes depends
on the relative magnitudes of the reactances as functions of frequency and hence
also on the values of inductor and capacitor used.
The impedance of the parallel combination can be found as for the case of
the series combination. As the elements are in parallel, the voltages across both
elements are identical and the current through the parallel combination is the
sum of the voltage drops across the individual elements. Then the impedance is
given as:
Ζ
v(t)
i(t)
i(t)  i L (t)  i C (t)
where
Then:
Ζ
v(t)
v(t)

i(t) i L (t)  i C (t)
or
1 i L (t)  i C (t) i L (t) i C (t)



Z
v(t)
v(t) v(t)
Z  ZL
1 1
1


 C
Z Z L ZC
Z L ZC
2
Then:
ZL ZC
Z
ZL  ZC
 1 

jL
 jC 
Z
1
jL 
j C
Z
jL
jL

j 2 2 LC  1 1   2 LC
 L 
Z  j

2
1


LC


It can be seen that for the parallel combination of ideal inductor and capacitor
the overall impedance of the network is also purely reactive with no resistance.
It can be seen this time that a critical point exists when the demoninator is zero.
This occurs when:
 2 LC  1
or as before when:
  0 
1
LC
The value of this frequency is again referred to as the resonant frequency, and
depends entirely of the values of the components used. This time the result
implies that at this frequency the impedance of the parallel combination is
infinite for ideal components. The magnitude of the impedance is given as:
Z 
L
1   2 LC
3
¦Z¦
∞
ω
ω0
0
ω
Fig. 3 The Magnitude of the Impedance as a Function of Frequency
The magnitude of the impedance is shown as a function of frequency in
Fig. 3. It can be seen that the impedance is very low at high and low frequencies
but tends towards infinity at the resonant frequency ω = ω0. The situation at
resonance can also be shown as in the phasor diagram and waveforms of Fig. 4.
In essence, at the resonant frequency the effect of the inductive reactance
counteracts that of the capacitive reactance. Therefore the same voltage is
present across both elements but the currents flowing through each element
have equal magnitude and opposite polarity at resonance. The net effect of this
is that zero current flows into the parallel combination, giving the resultant
infinite impedance shown in Fig. 3. What happens is that energy is initially
drawn from the source feeding the circuit. This energy then oscillates between
the inductor and capacitor so that when current is flowing into one element it is
flowing out of the other. The precise magnitudes of the individual currents
depend on the values of these components.
VL , VC
j
IC
IL
ω0
IL
ω0
VL ,VC
t
ω0
IC
-j
Fig. 4 Phasors / Waveforms for Series Inductor and Capacitor at Resonance
4
12.3 Resistor, Inductor and Capacitor in Parallel
Fig. 5 shows a resistor added in parallel with the previous inductor and
capacitor already connected in parallel. Again, the same voltage is developed
across all of the elements so that VL = VC = VR. The same relationships hold
between voltage and current in the inductor and the capacitor so their phase
relationships are unaltered. The current flowing through the resistor is in phase
with the voltage across it so that their phasors appear superimposed in Fig. 5.
j
+
IL
IR
+
C
L
VL
ω
IC
R
VR
-
IC
+
VC
-
IL
ω
current leads
voltage for C
VL = VC =VR
IR
IR
ω
current lags
voltage for L
-j
Fig. 5 A Resistor, Capacitor and Inductor, RLC, Connected in Parallel
In this case the impedance has an added element in the resistor which is present.
The net current flowing through the circuit is the vector sum of the three
individual components of current so that the impedance is given as:
v(t)
Ζ
i(t)
where
i(t)  i L (t)  i C (t)  i R (t)
Then:
v(t)
Ζ
i L (t)  i C (t)  i R ( t )
Inverting as before
1 1
1
1



Z ZL ZC R
5
Since the impedance of the parallel combination of the inductor and capacitor
has already been evaluated above then the product over sum rule can be used to
add the resistance in parallel with this which gives:
 L 
j
R
2
1   LC 
Z 
 L 
j
R
2
 1   LC 
Z
jLR
jL  R 1  2 LC 
In order to obtain the impedance in proper complex form this expression must
be multiplied in the numerator and denominator by its complex conjugate. This
gives:






jLR 1   LC    L R
Z
R 1   LC    L
LR
LR 1   LC 
j
1   LC    L R 1   LC    L
jLR
R 1  2 LC  jL
Z

2
R 1   LC  jL R 1  2 LC  jL
2
2
2
Z
2
R2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
This is a much more complicated expression than in the case of the series RLC
combination. It is complex having a real or resistive part and an imaginary or
reactive part. Again, the reactive part can be dominated by the inductive
reactance or the capacitive reactance depending on the values of these
components and the frequency of operation. The impedance therefore has an
associated magnitude and phase which can be found in the traditional manner
but having a more complicated form of algebra:
6

2

2 2

  LR 2 1  2LC 

L
R
 

Z 
 R 2 1  2LC 2  2L2   R 2 1  2LC 2  2L2 

 




 Z  Tan
1

R 1  2 LC
L

2

The values of both the magnitude and phase depend on the values of all of the
components as well as the frequency. Consider the case as before when:
 LC  1
2
  0 
or
1
LC
At this frequency the impedance becomes purely real and reduces to the value of
the resistance alone.
2 L2 R
Z 2 2 R
L
This is shown in the plot of the magnitude of the impedance of the RLC
combination shown in Fig. 6. In effect what has happened here is that the
parallel combination of the inductor and capacitor at resonance has produced
infinite impedance between them, leaving only the resistance present at this
frequency.
¦Z¦
R
ω0
0
ω
Fig.6 The Magnitude of the Impedance of the Series RLC Combination
7
The net resultant current flowing through the parallel combination is given as:
i(t)  i L (t)  i C (t)  i R (t) 
v(t )
v(t )
v(t )


  Z
Z
Z  Z
Z
If the voltage phasor is taken as the reference zero phase vector, then the
magnitude and phase of this and all of the currents involved, including the
resultant current can be shown as in Fig. 7. Note that the resultant net current
has a magnitude and phase which depends on all three components in the
combination and can lead or lag the current depending on whether the net
reactance is inductive or capacitive.
VL, VC, VR
j
IC
I
ω
IR
IL
VL, VC, VR
t
ω
IL
ω
I
IC
-j
Fig. 7 Phasor Diagram and Waveforms for Series RLC Combination
11.4 Application
The parallel RLC combination is often used as a load on the transistor in
the intermediate frequency amplifier of a radio receiver. When this is the case,
the IF amplifier gain takes on a frequency response which mirrors the frequency
dependence of the impedance of the parallel network. This provides a
frequency-selective tuned gain stage which amplifies only the wanted narrow
band of frequencies surrounding the intermediate frequency which is associated
with the radio station that the receiver is tuned to.
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