Two Mark Question and Answers set - 6

126. State maximum power transfer theorem.

Ans: The theorem is defined as below for a D.C. circuit. "Maximum power will be delivered from a voltage source to a load, when the load resistance is equal to the internal resistance of the source."

Refer the figure given. When power delivered to R_L is maximum,

R_L = R_g

127. Write some applications of maximum power transfer theorem.

Ans: 1. In communication circuits.

2. In power networks.

128. The power delivered is maximum if the load impedance is equal to the supply circuit impedance - True or False.

Ans: True.

129. The transients are due to the presence of energy storing elements in the circuit - True or False.

Ans: True.

130. What is a step function?

Ans: A step function is defined mathematically as f (t) = A u(t),

where u (t) = 1 for t > 0

= 0 for  t < 0

Graphically it is shown as below.

131. What is an initial condition?

Ans: Initial conditions of a network are the values of currents and voltage just before switching. The instant just before switching is denoted by t = 0-. The instant exactly at switching is denoted by t = 0 and the instant just after switching is denoted by t = 0+.

132. What is a transient ?

Ans: If a circuit containing one or more energy-storage elements (such as L and C) is excited by a source which abruptly changes its value, the energy state of the circuit is disturbed. This disturbance may also be due to variation of the storage elements and also due to switching conditions.

After certain time, the circuit settles down to a new steady state operation. The time required for the circuit to change from one steady state to another steady state is known as transient period. The values of voltage and current within this period are known as transient response or simply transients.

133. What is the steady state value ?

Ans: A circuit consisting of constant sources is said to be in steady state if the voltages and currents do not change with time.

134. Write the transient current equation when RL series circuit is connected to a step voltage of V volts.

Ans:

I = I [ 1 – e^-1/T]

where I = V / R

T = Time constant = L / R

135. A D.C. voltage of 100 volts is applied to a series RL circuit with R = 25Ω.What will be the current in the circuit at twice the time constant ?

Ans:As the voltage source exists, there is rise of current in the circuit,

i = I [1-e^-t/T ]

Here I = V/R = 100/25 = 4A

Given that t = 2T

i =  4 [1-e^-2T/T ]

= 4 [1-e^-2] = 3.46 A

136. Sketch the current given by i (t)=5 - 4e^-20t.

137. Distinguish between free and forced response.

Ans: When a circuit contains storage elements which are independent of the sources, the response depends upon the nature of circuit. This response is called natural or free response.

The storage elements deliver the energy to the resistances. So, the response changes with time, gets saturated after some time. It is referred to as the transient response. When we consider sources acting on a circuit, the response depends on the nature of such sources. This response is called forced response.

138. Draw the equivalent circuit for inductor and capacitor at t = 0+ when there is no initial energy.

Ans:

139. Define a time constant of a RL circuit.

Ans: The time constant of RL series circuit is defined as the period during which the current rises to 63.22% of its final value.

140. Draw the equivalent circuit for the inductor and capacitor at t = 0+ with presence of initial energy.

Ans :

141. Distinguish between the steady state and the transient response of an electrical circuit.

Ans: A circuit having constant sources is said to be in steady state if the currents and the voltages do not vary with time. In a circuit containing energy storage elements, with change in excitation, the voltage and currents change from one state to other. The behaviour of the voltage or current when it is changed from one state to other is called the transient state.

142. Define a time constant of a RC circuit.

Ans: The time constant for the RC circuit is the time during which the current falls to 36.8% of initial current.

143. Draw the equivalent circuit at t=0+ for a capacitor with initial charge of q₀.

Ans:

144. Sketch the response of RC network for a unit step input.

Ans :

i = 1 e^-1/λ

λ = CR called time constant.

145. What are periodic inputs ?

Ans: Periodic functions are those for which f (t) = f(t + nπ), where n is an integer and T is the period. As example, sine wave form

146. What are critical frequencies? Why are they so called?

Ans: The critical frequencies are poles and zeros. They are so called because zeros of the network function give the values of s which make the function zero and poles of a network function give the values of s which make the function infinite. At every other frequency, the network function has a finite non-zero value.

147. Draw the transient response of R-L circuit for step input.

Ans :

148. Define the time constant of a transient response.

Ans: The time constant of a function (transient) is the time at which the exponent of e is unity. For example, for RL circuit excited by D.C. source,

i = I [1-e^-1/λ ]

λ is called time constant.

At one time constant, t = λ.

So, i = I [1- e^-1] = 0.632 I

149. Find the time constant of RL circuit having R = 10Ω and L = 0.1 mH.

Ans :

R = 10 Ω and L = 0.1 mH

T = L / R = 0.1 × 10^-3 / 10

= 10^-5 sec = 10 μ sec

150. What is meant by critical damping?

Ans: For RLC series circuit excited by a D.C. source, if (R/2L)² = 1/LC, the current is said to  be critically damped.

151. Define logarithmic decrement.

Ans: For RLC series circuit, excited by a D.C. source,if  (R/2L)² < 1/LC, the current i is said to be oscillatory.

It is called under-damping one. We can write that,

log_e v = 2πα / β

v = e 2πα / β

log_e v = ∆

∆ is called logarithmic decrement.

152. Define damping ratio.

Ans: Consider a RLC circuit. The ratio of value of resistance in a circuit to the resistance for critical damping is called damping ratio and is denoted by σ.

153. What is meant by free response?

Ans: Free response is also known as natural response or transient response. It is due to the internal energy stored in the network. It depends upon the type of elements, their size, etc. This response is independent of the source. The natural response dies gradually. i.e., It approaches zero as time becomes infinity.

154. What is meant by a natural frequency and a damped frequency?

Ans: The natural frequency of oscillation is given by

ω_n = 1/ √LC

Damped frequency

= β = ω_n √1- σ²

155. Write the expression for the natural frequency for a RLC series circuit.

Ans: Natural frequency ω_n = 1 / √LC Rad/sec.