UPSC ESE E&T Network Theory Syllabus

Define the resonant frequency of a series RLC circuit.

At resonance XL = XC, giving f₀ = 1/(2π√(LC)). At resonance the series RLC impedance is minimum and purely resistive (= R), so current is maximum.

Define the quality factor Q of a series RLC circuit at resonance.

Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C). It measures selectivity; Q = f₀/BW (resonant frequency divided by bandwidth).

What is the bandwidth of a series RLC resonant circuit?

BW = R/L = f₀/Q (in rad/s, BW = R/L; in Hz, BW = R/(2πL)). It is the difference between the two half-power (−3 dB) frequencies.

Compare the impedance behavior of series vs parallel RLC circuits at resonance.

A series RLC circuit has minimum impedance (= R) and maximum current at resonance; a parallel RLC (tank) circuit has maximum impedance and minimum line current (acts as a rejector/anti-resonant circuit).

Classify the damping of a second-order RLC circuit by damping ratio ζ.

ζ > 1: overdamped (two real roots, no oscillation); ζ = 1: critically damped (fastest non-oscillatory response); ζ < 1: underdamped (oscillatory decay); ζ = 0: undamped (sustained oscillation).

Define the Laplace transform of a function f(t).

F(s) = ∫₀^∞ f(t)·e^(−st) dt, where s = σ + jω is the complex frequency variable. It converts time-domain differential equations into algebraic equations in s.

Give the Laplace transforms of a unit step u(t), unit impulse δ(t), and ramp t·u(t).

L{u(t)} = 1/s; L{δ(t)} = 1; L{t·u(t)} = 1/s².

Give the Laplace transforms of e^(−at) and sin(ωt).

L{e^(−at)} = 1/(s+a); L{sin(ωt)} = ω/(s²+ω²); and L{cos(ωt)} = s/(s²+ω²).

State the Laplace differentiation property for f'(t).

L{f'(t)} = sF(s) − f(0⁻), and L{f''(t)} = s²F(s) − s·f(0⁻) − f'(0⁻). This incorporates initial conditions directly.

State the Laplace integration property.

L{∫₀ᵗ f(τ)dτ} = F(s)/s.

State the Initial Value and Final Value Theorems of the Laplace transform.

Initial Value: f(0⁺) = lim(s→∞) sF(s). Final Value: f(∞) = lim(s→0) sF(s), valid only if all poles of sF(s) lie in the left half plane (system is stable).

What are the s-domain impedances of R, L, and C (with zero initial conditions)?

Resistor: R; Inductor: sL; Capacitor: 1/(sC).

What conditions (Dirichlet conditions) must a periodic signal satisfy to have a valid Fourier series?

Over one period the function must be single-valued, have a finite number of maxima/minima, a finite number of finite discontinuities, and be absolutely integrable (∫|f(t)|dt finite).

Write the trigonometric Fourier series of a periodic signal f(t).

f(t) = a₀ + Σ[aₙ cos(nω₀t) + bₙ sin(nω₀t)], where a₀ is the average (DC) value, ω₀ = 2π/T, and aₙ, bₙ are the harmonic coefficients.

Which Fourier coefficients vanish for even and for odd functions?

Even function: only cosine terms (bₙ = 0). Odd function: only sine terms (a₀ = 0 and aₙ = 0). Half-wave symmetry: only odd harmonics are present.

Write the exponential (complex) form of the Fourier series and its coefficient.

f(t) = Σ Cₙ e^(jnω₀t), with Cₙ = (1/T)∫_T f(t)e^(−jnω₀t) dt. The Cₙ are generally complex with conjugate symmetry for real signals.

Define the Fourier Transform and its inverse.

F(ω) = ∫_{−∞}^{∞} f(t)e^(−jωt) dt; inverse f(t) = (1/2π)∫_{−∞}^{∞} F(ω)e^(jωt) dω. It extends Fourier analysis to non-periodic (aperiodic) signals.

What is the key difference between the Fourier Series and the Fourier Transform?

The Fourier series represents periodic signals as a discrete sum of harmonics (line/discrete spectrum); the Fourier transform represents aperiodic signals with a continuous spectrum F(ω).

How are the Fourier and Laplace transforms related?

The Fourier transform is the Laplace transform evaluated on the imaginary axis: F(ω) = F(s)|_{s=jω}, valid when the region of convergence includes the jω-axis (i.e., σ = 0).

State Parseval's theorem for a periodic signal.

The average power equals the sum of the powers of its Fourier components: (1/T)∫_T |f(t)|² dt = Σ |Cₙ|² (exponential form), i.e., total power is distributed among the harmonics.

Define the Z-parameters (open-circuit impedance parameters) of a two-port network.

They relate port voltages to port currents: V₁ = Z₁₁I₁ + Z₁₂I₂ and V₂ = Z₂₁I₁ + Z₂₂I₂. Each Zᵢⱼ is found with one port open-circuited (current = 0).

How is Z₁₁ of a two-port measured?

Z₁₁ = V₁/I₁ with I₂ = 0 (output port open-circuited). It is the input driving-point impedance with the output open.

Define the Y-parameters (short-circuit admittance parameters) of a two-port.

They relate port currents to port voltages: I₁ = Y₁₁V₁ + Y₁₂V₂ and I₂ = Y₂₁V₁ + Y₂₂V₂. Each Yᵢⱼ is found with one port short-circuited (voltage = 0).

How is Y₂₁ of a two-port measured?

Y₂₁ = I₂/V₁ with V₂ = 0 (output short-circuited). It is the forward transfer admittance.

Define the hybrid (H) parameters of a two-port network.

V₁ = h₁₁I₁ + h₁₂V₂ and I₂ = h₂₁I₁ + h₂₂V₂. Here h₁₁ is input impedance (V₂=0), h₁₂ reverse voltage gain (I₁=0), h₂₁ forward current gain (V₂=0), h₂₂ output admittance (I₁=0).

Why are h-parameters widely used for transistors (BJTs)?

Because h₁₁ (input impedance) and h₂₁ (forward current gain β) are easily measured with the output short-circuited, and h₁₂, h₂₂ with the input open — matching how transistor characteristics are naturally specified.

Define the transmission (ABCD) parameters of a two-port network.

They relate input port to output port: V₁ = A·V₂ − B·I₂ and I₁ = C·V₂ − D·I₂ (with I₂ taken as flowing out of port 2). A and D are dimensionless, B is impedance (Ω), C is admittance (S).

How are the ABCD parameters individually defined/measured?

A = V₁/V₂ (I₂=0, output open); B = −V₁/I₂ (V₂=0, output short); C = I₁/V₂ (I₂=0, output open); D = −I₁/I₂ (V₂=0, output short).

What is the main advantage of transmission (ABCD) parameters?

For two-ports connected in cascade, the overall ABCD matrix is the matrix product of the individual ABCD matrices, making cascade analysis straightforward.

What is the reciprocity condition expressed in Z, Y, and ABCD parameters?

A reciprocal two-port satisfies Z₁₂ = Z₂₁, Y₁₂ = Y₂₁, h₁₂ = −h₂₁, and for ABCD parameters AD − BC = 1.

What is the symmetry condition for a two-port network in Z, Y, and ABCD parameters?

A symmetrical two-port satisfies Z₁₁ = Z₂₂, Y₁₁ = Y₂₂, and A = D for the transmission parameters.