Resonance in Series & Parallel Circuits | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

Resonance in Series and Parallel Circuits 

Series circuit

The circuit, with resistance R, inductance L, and a capacitor, C in series (Fig. 17.1a) is connected to a single phase variable frequency (f) supply.

The total impedance of the circuit is 

The current in the circuit is maximum, if  The frequency under the above condition is 

This condition under the magnitude of the current is maximum, or the magnitude of the impedance is minimum, is called resonance. The frequency under this condition with the constant values of inductance L, and capacitance C, is called resonant frequency. If the capacitance is variable, and the frequency, f is kept constant, the value of the capacitance needed to produce this condition is

The magnitude of the impedance under the above condition is   with the reactance  as the inductive reactance  is equal to capacitive reactance    The phase angle is  , and the power factor is unity   which means that the current is in phase with the input (supply) voltage.. So, the magnitude of the current  in the circuit is only limited by resistance, R. The phasor diagram is shown in Fig. 17.1b.

The magnitude of the voltage drop in the inductance L/capacitance C (both are equal, as the reactance are equal) is 

The magnification of the voltage drop as a ratio of the input (supply) voltage is 

It is termed as Quality (Q) factor of the coil. The impedance of the circuit with the constant values of inductance L, and capacitance C is minimum at resonant frequency and increases as the frequency is changed, i.e. increased or decreased, from the above frequency. The current is maximum at  and decreases as frequency is changed   The variation of current in the circuit having a known value of capacitance with a variable frequency supply is shown in Fig. 17.2.

Fig. 17.2 Variation of current under variable frequency supply 

The maximum value of the current is (V/R). If the magnitude of the current is reduced to   of its maximum value, the power consumed in R will be half of that with the maximum current, as power is I² R . So, these points are termed as half power points. If the two frequencies are taken as  where  and  , the band width being given by
The magnitude of the impedance with the two frequencies is 

 and the ratio  is small, the magnitude of the reactance of the circuit at these frequencies is   As the current is  of its maximum value, the magnitude of the impedance is    of its minimum value (R) at resonant frequency.

From the above, it can be obtained that 

The band width is given by 

It can be observed that, to improve the quality factor (Q) of a coil, it must be designed to have its resistance, R as low as possible. This also results in reduction of band width and losses (for same value of current). But if the resistance, R cannot be decreased, then Q will decrease, and also both band width and losses will increase.

Example 17.1 

A constant voltage of frequency, 1 MHz is applied to a lossy inductor (r in series with L), in series with a variable capacitor, C (Fig. 17.3). The current drawn is maximum, when C = 400 pF; while current is reduced to   of the above value, when C = 450 pF. Find the values of r and L. Calculate also the quality factor of the coil, and the bandwidth.

Solution 

The quality factor of the coil is 

The band with is

Parallel circuit

The circuit, with resistance R, inductance L, and a capacitor, C in parallel (Fig. 17.4a) is connected to a single phase variable frequency (f) supply. The total admittance of the circuit is

The current is

The current in the circuit is minimum, if 

This condition under which the magnitude of the total (supply) current is minimum, or the magnitude of the admittance is minimum (which means that the impedance is maximum), is called resonance. It may be noted that, for parallel circuit, the current or admittance is minimum (the impedance being maximum), while for series circuit, the current is maximum (the impedance being minimum). The frequency under this condition with the constant values of inductance L, and capacitance C, is called resonant frequency. If the capacitance is variable, and the frequency, f is kept constant, the value of the capacitance needed to produce this condition is 

The magnitude of the impedance under the above condition is  while the magnitude of the admittance is  The reactive part of the admittance is B = 0 as the susceptance (inductive)  is equal to the susceptance (capacitive)  The phase angle is  and the power factor is unity  The total (supply) current is phase with the input voltage. So, the magnitude of the total current  in the circuit is only limited by resistance R. The phasor diagram is shown in Fig. 17.4b.

The magnitude of the current in the inductance, L / capacitance, C (both are equal, as the reactance are equal), is  This may be termed as the circulating current in the circuit with only inductance and capacitance, the magnitude of which is

substituting the value of   This circulating current is smaller in magnitude than the input current or the current in the resistance as

The input current increases as the frequency is changed, i.e. increased or decreased from the resonant frequency

In the two cases of series and parallel circuits described earlier, all components, including the inductance, are assumed to be ideal, which means that the inductance is lossless, having no resistance. But, in actual case, specially with an iron-cored choke coil, normally a resistance r is assumed to be in series with the inductance L, to take care of the winding resistance and also the iron loss in the core. In an air-cored coil, the winding resistance may be small and no loss occurs in the air core. An iron-cored choke coil is connected in parallel to capacitance, and the combination is fed to an ac supply (Fig. 17.5a).

The total admittance of the circuit is

If the magnitude of the admittance is to be minimum, then

The frequency is

This is the resonant frequency. The total admittance is 
The total impedance is

This current is at unity power factor with φ = 0°. The total current can be written as

So, the condition is 

From the above, the condition, as given earlier, can be obtained. The total current is 
The value, as given here, can be easily obtained. The phasor diagram is shown in Fig. 17.5b. It may also be noted that the magnitude of the total current is minimum, while the magnitude of the impedance is maximum.

Example 17.2

A coil, having a resistance of 15 Ω and an inductance of 0.75 H, is connected in series with a capacitor (Fig. 17.6a. The circuit draws maximum current, when a voltage of 200 V at 50 Hz is applied. A second capacitor is then connected in parallel to the circuit (Fig. 17.6b). What should be its value, such that the combination acts like a noninductive resistance, with the same voltage (200 V) at 100 Hz? Calculate also the current drawn by the two circuits.

Solution f₁ = 50 Hz V = 200 V     R = 15 Ω     L = 0.75 H

From the condition of resonance at 50 Hz in the series circuit,

The maximum current drawn from the supply is, 

As the combination is resistive in nature, the total admittance is

From the above expression, 

The total admittance is 
The total impedance is 
The total current drawn from the supply is 

The condition for resonance in both series and parallel circuits fed from single phase ac supply is described. It is shown that the current drawn from the supply is at unity power factor (upf) in both cases. The value of the capacitor needed for resonant condition with a constant frequency supply, and the resonant frequency with constant value of capacitance, have been derived. Also taken up is the case of a lossy inductance coil in parallel with a capacitor under variable frequency supply, where the total current will be at upf. The quality factor of the coil and the bandwidth of the series circuit with known value of capacitance have been determined. This is the final lesson in this module of single phase ac circuits. In the next module, the circuits fed from three phase ac supply will be described.