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If the value of current i(t) for the circuit shown below is i(t) 20 e^{2t}, then the voltage source v(t) will be given by
v_{R}(t) = i(t) x 1 = 20 e^{2t} = v_{c}(t)
= 80 e ^{2t}
Now, i'(t) = i(t) + i_{c}(t)
= 20 e^{2t} + 80 e^{2t} = 60 e^{2t}
So,
or,
v(t) = (60  30  20) e^{2t} = 10 e^{2t}
The circuit shown in figure is critically damped.
The value of R is
For an RLC series circuit to be critically damped
or,
or, 120 R = 40 ( R + 120)
or, 3 R = R + 120 or R = 60 Ω
The voltage across a passive element in an electric circuit is given by It is given that V(0^{+})
The value of v(t) is
Given,
Taking Laplace transform on both sides,
or,
or,
Using partial fraction expansion,
A = 1, B = 1,
C =  3
So, V(t) = (t + cost  3 sint)
A voltage is given by V(s) = 1/S(S + a). If V(∞) = 2 volts, then V(1) is
Given,
= 1/a = 2
or, a = 0.5
Thus,
or v(t) = 2(1  e^{  0.5t})
Hence, v(1) = 2(1  e^{0.5})
Transient current in an RLC circuit oscillatory when
Characteristic equation of series RLC circuit is
Here,
and
or,
For oscillatory response,
or,
Which of the following conditions are necessary for the validity of initial value theorem
The poles and zeros of the transfer function for the circuit shown below are located a
Applying KCL at the given node in sdomain, we have
or,
or,
Hence, there is nozero and poles are at 4 s^{2} + s + 2 = 0
or,
Assertion (A): Laplace transformation finds its application in solving the transient behaviour of the electric circuits.
Reason (R): The transient response of an electrical circuit can best be described by a linear differential equation.
Because the transient response of an electrical circuit can best be described by a differential equation hence, Laplace transformation finds its application in solving the transient behaviour of the electric circuits.
The final value of f(t) = e^{t} (sin 2t + cos 5t) is
f(t) = e^{t} sin 2t + e^{2t} cos 5t
∴
∴ Find value of f(t) is
At t = 0^{+} with zero initial condition, the voltage across 20 Ω resistor is
At t = 0^{+}, capacitor will act as short circuit while inductor an open circuit
∴ V_{20Ω} = 10 volt
Assuming zeroinitial condition v_{c}(t) in the given circuit will be given by
Applying node analysis,
or,
or,
Here
∴
or,
The initial and final value of current through the resistor Rina series RLC circuit with unit step input and zero initial condition are respectively
At t = 0, inductor act as open circuit i(0^{+}) = 0 A
At t = capacitor act as open circuit i(∞) = 0 A
At t = ∞, inductor will act as short circuit anc capacitor open circuit.
Consider the following statements for a series RLC circuit excited with a voltage v(t)
1. For ξ, = 0, there is noloss in the circuit.
2. The damping ratio of the circuit is independent of resistance R of the circuit.
3. The response of the circuit is oscillator if ξ value is more than unity.
Q. Which of the above statements is/are true?
(characteristic equation)
Therefore
If R = 0, ξ = 0 (viceversa) and for ξ < 1, response will be oscillatory.
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