Theory of Parallel Resonant Circuits and its applicatication

Introduction:
   
There are equivalent circuits to series combinations examined that exist in parallel configurations. The issues surrounding reactive and resistive components and their consequence on the circuit impedance are identical whereas the result can frequently be the opposite of the case for series configurations. Various comparable parallel combinations will thus be examined. 

Ideal Inductor and Capacitor in Parallel:


Consider the capacitor and inductor connected in parallel as shown in figure below. The reactive elements are taken to be ideal. In this case, the end nodes of both elements are joined altogether that means that the voltage across the inductor is similar to the voltage across the capacitor and hence VL = VC. If this is taken as reference zero phase, then it can be observe that the current via the capacitor leads the voltage and hence it appears 900 ahead on the phasor diagram of figure shown below, while the current via the inductor lags the voltage and hence it appears 900 behind. This can be seen thus that in this case the currents via the inductor and the capacitor are in anti-phase or 1800 out of phase with one other. The relative magnitudes of currents are distinct and depend on the values of capacitance and inductance of such elements at the frequency of excitation. This condition is similar to the series combination apart from that in this case it is the voltage instead of the current that is the common component.

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  Figure: An Inductor and Capacitor Connected in Parallel

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Figure: Phasor Diagram and Waveforms for Inductor and Capacitor in Parallel

Waveforms are shown for the sinusoidal excitation of circuit as shown in figure above. From this it is obviously evident that the phasors representing the currents via the inductor and capacitor are exactly 1800 out of phase, displaying excursions on the opposite sides of abscissa axis. The difference in the amplitudes mainly depends on the relative magnitudes of the reactances as functions of frequency and therefore as well on the values of capacitor and inductor used.

The impedance of parallel combination can be determined as for the case of series combination. Since the elements are in parallel, the voltages across both elements are similar and the current via the parallel combination is the sum of voltage drops across individual elements. Then the impedance is as shown:

Z = v(t)/i(t) where i(t) = iL(t) + iC(t)

Then,

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This can be seen that for parallel combination of ideal capacitor and inductor the overall impedance of the network is as well purely reactive with no resistance. This can be seen this time that a critical point exists whenever the denominator is zero. This takes place if:

ω2LC = 1

Or as before when,

ω = ωo = 1/√LC

The value of this frequency is again termed to as the resonant frequency, and depends totally of the values of components employed. This time the outcome implies that at this frequency, the impedance of parallel combination is infinite for ideal components. Magnitude of the impedance is as shown below:

|Z| = (ωL)/ (1- ω2LC)

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Figure: The Magnitude of the Impedance as a Function of Frequency

The magnitude of impedance is as shown as a function of frequency in figure above. This can be seen that the impedance is much low at low and high frequencies however tends towards infinity at resonant frequency ω = ω0. The condition at resonance can as well be as shown as in the phasor diagram and waveforms of figure below. In essence, at resonant frequency the consequence of inductive reactance counteracts that of the capacitive reactance. Thus the similar voltage is present across both elements however the currents flowing via each element have equivalent magnitude and opposite polarity at resonance. The total effect of this is that, zero current flows into the parallel combination, providing the resultant infinite impedance shown in figure above. What happens is that energy is originally drawn from the source feeding circuit. This energy then oscillates among the capacitor and inductor and hence when current is flowing into one element it is flowing out of the other. The accurate magnitudes of the individual currents depend on the values of such components.

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Figure: Phasors or Waveforms for Series Inductor and Capacitor at Resonance

Resistor, Inductor and Capacitor in Parallel:

Figure below shows a resistor added in parallel with the preceding inductor and capacitor already joined in parallel. Again, the similar voltage is developed across all elements and hence VL = VC = VR. The same relationships hold between current and voltage in the inductor and the capacitor therefore their phase relationships are unmodified. The current flowing via the resistor is in phase with the voltage across it and hence their phasors appear superimposed in figure shown below.

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Figure: A Resistor, Capacitor and Inductor, RLC, joined in Parallel

In case, the impedance has an added element in the resistor that is present. The total current flowing via the circuit is the vector sum of three individual components of current and hence the impedance is as follow:

Z = v(t)/i(t) where i(t) = iL(t) + iC(t) + iR(t)

Then,

Z = v(t)/[ iL(t) + iC(t) + iR(t)]

Inverting as before:

1/Z = (1/ZL) + (1/ZC) + (1/R)

As the impedance of parallel combination of the capacitor and inductor has already been computed above then the product over sum rule can be employed to add the resistance in parallel that gives:

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In order to receive the impedance in appropriate complex form this expression must be multiplied in the numerator and denominator by its respective complex conjugate. It gives:

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It is a much more complicated expression than in the case of series RLC combination. This is complex containing a real or resistive part and an imaginary or reactive part. Again, the reactive portion can be dominated by the inductive reactance or capacitive reactance based on the values of such components and the frequency of operation. The impedance thus has a related magnitude and phase that can be found in the traditional way however having a more complicated form of algebra:

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The values of both magnitude and phase based on the values of all components and also the frequency. Consider the case as prior to when:

ω2LC = 1 or ω = ωo = 1/√LC
      
At this frequency, the impedance becomes purely real and deceases to the value of resistance alone.

Z = ω2L2R/ω2L2 = R

This is as shown in the plot of magnitude of the impedance of RLC combination as shown in figure below. In consequence what has happened here is that the parallel combination of inductor and capacitor at resonance has generated infinite impedance between them, leaving just the resistance existed at this frequency.

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Figure: The Magnitude of Impedance of Series RLC Combination

The total resultant current flowing via the parallel combination is as follows:

i(t) = iL(t) + iC(t) + iR(t) = v(t)/Z = v(t)/[|Z| ∠ΦZ] = [v(t)/|Z|] ∠-ΦZ

When the voltage phasor is taken as reference zero phase vector, then the phase and magnitude of this and all the currents involved, comprising the resultant current can be shown in figure below. Note that the resultant total current has a phase and magnitude that depends on all the three components in combination and can lag or lead the current depending on whether the total reactance is capacitive or inductive.

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Figure: Phasor Diagram and Waveforms for Series RLC Combination

Application:

The parallel RLC combination is frequently used as a load on the transistor in an intermediate frequency amplifier of a radio receiver. When this is a case, then the IF amplifier gain takes on a frequency response that mirrors the frequency dependence of the impedance of parallel network. This gives a frequency-selective tuned gain stage that amplifies just the wanted narrow band of frequencies surrounding the intermediate frequency that is related with the radio station which the receiver is tuned to.

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