Define resonating frequency f 0 and obtain the relation f 0 = 1 2 π 1 LC Also explain the impedance corresponding to resonating frequency in L - C - R circuit.

Resonant circuits are of two types : series resonant circuit and parallel resonant circuit. Here, we shall discuss only series resonant circuit : Series Resonant Circuit: A series resonant circuit is the circuit, in which the frequency of the applied voltage is equal to the natural frequency of the circuit. In this condition, the voltage and the current are in the same phase, and the inductive reactance X L is equal to the capacitive reactance X C . The impedance of the circuit is minimum (equal to the resistance) and the current in the circuit is maximum. Let us consider an alternating-current circuit having inductance L, capacitance C and resistance R connected in series. The impedance of the circuit is Z = R 2 + X L − X C 2 = R 2 + ωL − 1 ωC 2 if the phase difference between the applied alternating voltage and the resulting current be ϕ , then tanϕ = X L − X C R = ωL − 1 ωC R If the inductive reactance X L and the capacitive reactance X C of the circuit be equal, then the voltage and the current will be in the same phase ϕ = 0 . In this condition, the impedance Z of the circuit will be minimum (= R) and the current I ( = V / R ) will be maximum. This is the condition of resonance. Thus, for resonance, we have X L = X C ωL = 1 ωC ω = 1 LC or f = ω 2 π = 1 2 π LC f is the frequency of the applied alternating voltage. For resonance, it should be 1 2 π LC which is the natural frequency of an L-C-R circuit (when R is negligible). So, in series resonant circuit, the frequency of the applied voltage is equal to the natural frequency of the circuit. A graph between the frequency / of the applied voltage and the current I in the circuit is shown in (a), in which f 0 = 1 2 π LC is the natural frequency of the circuit. For f = f 0 , the current is maximum ( l max ) . Both for f < f 0 , and f > f 0 , the current is less than its maximum value. Similarly, a graph between the voltage frequency f and the impedance Z will be as shown in Fig. (b). At f = f 0 , the impedance Z is minimum (= R).

Draw circuit diagram and explain the working of a zener diode as a dc voltage regulator with the help of its I-V characteristic.

Circuit diagram for a zener diode as a dc voltage regulator: Working of zener diode as a voltage regulator: The I-V characteristics curve of a Zener diode is shown below. It is seen that when the applied reverse bias voltage V reaches the breakdown voltage V z of the Zener diode, there is a large change in the current. Hence, after the breakdown voltage V z , a large change in the current can be produced by an almost insignificant change in the reverse bias voltage. In other words, zener voltage remains constant, even though current through the zener diode varies over a wide range. This property of the zener diode is used for regulating supply voltages so that they are constant.