Introduction:

The ac circuits have comprised only one type of reactance that has been either inductive or capacitive. The impedance of these circuits has as well been analysed and the phase relationships of voltages and currents examined. Whenever both capacitive and inductive types of reactance are combined such relationships become even more fascinating.
     
Ideal Inductor and Capacitor in Series:

Consider an inductor and a capacitor joined in series as shown in figure below. In this case the same current flows via both elements and hence I_L = I_C. If this is taken as reference zero phase, then it can be observe that the voltage across the inductor leads the current and hence it appears 90^o ahead of it in the phasor diagram of figure shown below. On other hand, voltage across the capacitor lags the current and hence it appears 90^o behind on phasor diagram. It can be seen that the voltages across the inductor and capacitor are in anti-phase or 180^o out of phase with one other. The relative magnitudes of the voltages are distinct and depend on the values of inductance and capacitance of such elements at specific frequency of excitation:

Figure: An Inductor and Capacitor joined in Series

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

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

The impedance of series combination can be found in normal way. Since the elements are in series, the currents via both elements are similar and the voltage drop across the series combination is the sum of voltage drops across the individual elements. Then the impedance is as follow:

Z = v(t)/i(t) where v(t) = v_L(t) + v_C(t)

Then, Z = [{v_L(t) + v_C(t)}/{i(t)}] = {v_L(t)}/i(t) + {v_C(t)}/i(t) = Z_L + Z_C

Z = jωL – j(1/ωC) = j[ωL – (1/ ωC)]

This can be seen that the total impedance of the network is purely reactive with no resistance, in case of an ideal inductor and capacitor implied in figure above. Given the nature of above expression, there is evidently a value of frequency for which the expression is zero.

It can be found as:

ωL – (1/ωC) = 0

ωL = 1/ωC

ω² = 1/LC

ω = ω_o = 1/√LC

The value of this frequency, referred to in practice as the resonant frequency, mainly depends entirely of the values of components employed. This outcome implies that at this frequency, the impedance of series combination is zero in case of ideal components. The magnitude of impedance is as follow:

|Z| = ωL – (1/ωC)

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

This is plotted as shown as a function of frequency in general form in figure above. It can be observe that the impedance is very high at low and high frequencies. It becomes zero at resonant frequency ω = ω₀.

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

In essence, at resonant frequency the consequence of inductive reactance cancels the capacitive reactance. Thus the same current develops voltages with equivalent magnitude and opposite polarity at resonant frequency and the total effect is zero voltage or potential drop across the combined series combination, giving the resulting net impedance of zero as shown in figure above. It should be noted, though, that the individual voltages across each and every element are not zero as can be seen in figure above where the phasor diagram and waveforms are shown at resonant frequency. The exact magnitudes of voltages across the capacitor and inductor depend on the values of such components, the inductance in Henrys and the capacitance in Farads.

Resistance, Inductor and Capacitor in Series:

Figure below shows a resistor added in series with the prior inductor and capacitor joined in series. Again, the same current flows via all of the elements and therefore IL = IC = IR. The same relationships hold between current and voltage in the inductor and the capacitor therefore their phase relationships are unmodified. The voltage across the resistor is in phase with current flowing via it therefore their phasors appear superimposed as shown in figure below. Though, in this case the impedance has an added element in the resistor that is present. The total voltage across the circuit is the vector sum of three individual components and hence the impedance is given as:

Z = v(t)/i(t) where v(t) = v_L(t) + v_C(t) + v_R(t)

Figure: A Resistor, Capacitor and Inductor, RLC, Connected in Series

Then,

Z = [v_L(t) + v_C(t) + v_R(t)]/[i(t)] = [v_L(t)/i(t)] + [v_C(t)/i(t)] + [v_R(t)/i(t)]

Z = Z_L + Z_C + Z_R = jωL – j(1/ωC) + R

And hence,

Z = R + j(ωL – 1/ωC)

In this case it can be observe that the impedance is truly complicated, containing a real part and an imaginary part. The real part is the resistance whereas the imaginary part is reactive. The reactive part can be dominated by inductive reactance or the capacitive reactance depending on the values of such components and the frequency of operation. The impedance thus has a related magnitude and phase as:

|Z| = √R² + (ωL – 1/ωC)²

The value of both the phase and magnitude depend on the values of all the components and also the frequency. At frequency ω = ω₀ having similar value as above the impedance has its minimum value. This time, though, while the reactive component becomes zero and the impedance remains finite at the value of resistance, R.

Figure: The Magnitude of the Impedance of the Series RLC Combination

Note that the total resultant voltage developed across the combined series combination is as:

V(t) = v_L(t) + v_C(t) + v_R(t) = i(t)Z = i(t) |Z| ∠ΦZ

When the current phasor is taken as reference zero phase vectors then the phase and magnitude of the resultant voltage across the series RLC combination is as:

v(t) = i(t)Z = I |Z|∠ΦZ

Where I is the magnitude of current. Then the phase and magnitude of the resultant voltage are received as:

A plot of phasor diagram that comprises the resultant voltage across the series combination is as shown in figure below for particular values of components and for particular frequency. Note that the resultant total voltage across the series combination consists of a magnitude and phase that depends on all three components and can lag or lead the current depending on whether the total reactance is capacitive or inductive.

Figure: Phasor Diagram and Waveforms for Series RLC Combination

Application:

One of the applications of a series LC circuit is the IF Trap in a super heterodyne radio receiver as shown in figure below. The standard domestic AM/FM radio is such a receiver. This kind of radio receiver applies a huge amount of gain to the signal picked up at the aerial in an intermediate frequency or IF stage. The intermediate frequency is selected to lay exterior the reception band of the radio. Though, if a signal at this IF frequency is picked up at aerial it can interfere rigorously with reception of the wanted signal. Thus an IF Trap is comprised in the form of a series LC circuit that has a resonant frequency equivalent to the intermediate frequency. Winding of the aerial coil makes the inductance of the series circuit and its resonance with selected capacitor value provides a near zero impedance at IF. Thus any signal at this frequency appearing at aerial is shunted to ground and doesn’t develop any detectable voltage at the input of RF amplifier.

Figure: A Series LC Circuit as an IF Trap in a Radio Receiver

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