JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II PRINCIPLES OF ELECTRICAL ENGINEERING Previous Question Paper (July/August - 2021 - R15)

The time constant of a circuit is defined as the time taken by the response (current or voltage) to reach approximately 63.2% of its final steady-state value during a transient process. It represents the rate at which the energy storage elements (inductor or capacitor) charge or discharge.

For a series RL circuit with a DC source applied at t=0, the governing differential equation is:

The solution for the current is:

The exponential term is of the form e^-t/τ, where τ is the time constant. By comparison, for the RL circuit:

For a series RC circuit with a DC source, the governing equation is:

Differentiating with respect to time leads to the voltage across the capacitor:

Again, comparing the exponential term to e^-t/τ, the time constant for the RC circuit is:

In both cases, a smaller time constant indicates a faster response, while a larger time constant indicates a slower transition to the steady state.

For a series RLC circuit excited by a step input voltage V, the circuit is governed by the second-order differential equation:

The characteristic equation is s² + (R/L)s + (1/LC) = 0, which has roots s_1,2 = -α ± √(α² - ω₀²), where α = R/(2L) is the damping factor and ω₀ = 1/√(LC) is the resonant frequency. The transient response depends on the damping ratio ζ = α/ω₀:

• Overdamped (ζ > 1): The roots are real and distinct. The current response is i(t) = A₁e^s₁t + A₂e^s₂t.
• Critically Damped (ζ = 1): The roots are real and equal (s₁ = s₂ = -α). The current response is i(t) = (A₁ + A₂t)e^-αt.
• Underdamped (ζ < 1): The roots are complex conjugates s_1,2 = -α ± jω_d, where ω_d = √(ω₀² - α²). The current response is i(t) = e^-αt(A₁cos(ω_dt) + A₂sin(ω_dt)).

The constants A₁ and A₂ are determined by the initial conditions of the circuit, specifically the initial inductor current i(0) and the initial capacitor voltage v_C(0). For a step input, the transient response represents the decaying component of the current as the circuit approaches its steady-state condition.