12.13 SELECTIVITY AND BANDWIDTH OF A
PARALLEL-TUNED CIRCUIT

Figure 12.19(a) shows a voltage source V of width impedance R_g driving the
parallel-tuned circuit. Figure 12.19(b) shows an equivalent circuit in which
the voltage generator has been replaced by the equivalent current generator
by using the source transformation method.

FIGURE 12.19   A parallel-tuned circuit driven from a voltage source and its equivalent
circuit.

We proceed to find the selectivity and bandwidth of the parallel circuit
neglecting the source resistance R_g. Under this condition, a constant current
I flows through the parallel circuit and the output voltage across the parallel
circuit is given by

Current I being constant, the output voltage V₀ varies exactly in the same
fashion as the impedance Z of the parallel circuit. The response of the parallel
circuit, i.e., the output voltage V₀, is maximum at resonance and falls off on
either side in accordance with the curve of Figure 12.18.

At the half-power frequencies f₁ and f₂, the impedance Z falls to a value
. Hence, from Equation (12.82)

or

or

At upper half-power frequency f₂,

At lower half-power frequency f₁,

Hence,

or

This Q_r is a measure of the selectivity of the parallel resonant circuit.

The higher the value of Q_r the more selective the circuit. Q_r may also
be put as

QUESTIONS FOR DISCUSSION

1. What is resonance ?
2. What is the difference between series and parallel resonance ?
3. What do you mean by current resonance ?
4. What is resonant frequency ?
5. What is the value of the power factor at resonance ?
6. What is the phase difference between voltage and current at
   resonance ?
7. What is bandwidth ?
8. What is the relation between time period and frequency ?
9. What is quality factor ?
   

OBJECTIVE QUESTIONS

UNSOLVED PROBLEMS

1. Two impedances Z₁ = a + Jb and Z₂ = c − Jd are connected in (a)
   series and (b) parallel. Determine the condition of resonance in each
   case.
2. A coil has impedance (2 + 2J) connected in series with the capacitor
   10μF. Determine the frequency at which resonance occurs and also
   determine the Q-factor.
3. A series RLC circuit consists of resistance R = 10Ω, L = 0.05 H, and
   C = 0.02μF. Calculate the resonant frequency. Also determine the
   frequencies at which the voltage across the capacitor and inductor are
   maximum (individually).
4. Prove for series RLC circuit the value of the current is maximum.
5. Prove for series resonant circuit Bw = where Bw = Bandwidth.
6. Two impedances Z₁ = 2 + Jb and Z₂ = C + 3J are connected in
   parallel. Determine the condition of resonance.