Network analysis: transients, resonance and two-port networks

In short

Network analysis here deals with how a circuit behaves in time (transients, when a switch operates), how it behaves with frequency (resonance, when inductive and capacitive reactances cancel), and how a whole network can be reduced to a black box with four parameters (two-port networks). These three ideas are the mathematical backbone of filters, tuners, matching circuits, oscillator tanks, amplifier models and transmission-line calculations. They carry heavy weight in GATE EC/EE (Networks is a guaranteed section), ESE, ISRO/BARC/DRDO/BEL entrance papers, PSU and BSNL JTO exams, and in university "Circuit Theory / Network Analysis" papers. In interviews, RC time constants, Q factor and h/S-parameters are the standard first-round screening questions at Qualcomm, Texas Instruments, Analog Devices, Intel, Tejas Networks, Bharat Electronics and most power/embedded firms.

Explained

1. Initial conditions and the idea of a transient

• A transient is the response during the interval after a sudden change (switching, step input, fault) until the circuit settles into its steady state.
• Complete response = natural (free/transient) response + forced (steady-state) response. Natural response is decided by circuit elements only; forced response follows the source.
• Inductor: current cannot change instantaneously (flux linkage is continuous), so i_L(0⁺) = i_L(0⁻). At t = 0⁺ an initially relaxed inductor behaves as an open circuit; at t → ∞ (DC) it behaves as a short circuit.
• Capacitor: voltage cannot change instantaneously (charge is continuous), so v_C(0⁺) = v_C(0⁻). At t = 0⁺ an initially relaxed capacitor behaves as a short circuit; at t → ∞ (DC) as an open circuit.
• An inductor with initial current I₀ is modelled at t = 0⁺ as a current source I₀; a capacitor with initial voltage V₀ as a voltage source V₀.
• Exception: an impulsive source or an illegal switching (two unequal charged capacitors suddenly paralleled, or breaking a current-carrying inductor) can make v_C or i_L jump; then conservation of charge / flux linkage is applied instead of continuity.
• Standard exam sequence: find x(0⁻) in old steady state → apply continuity → draw the t = 0⁺ circuit → find x(∞) → write the exponential.

2. First-order circuits (one L or one C)

• Any single-energy-storage circuit obeys x(t) = x(∞) + [x(0⁺) − x(∞)]e^−t/τ — the "initial–final–time-constant" method. This one line answers most numerical questions.
• Time constant: τ = L/R for RL, τ = RC for RC, where R is the Thevenin resistance seen *by* the element with sources deactivated.
• τ is the time to complete 63.2 % of the change; after 2τ 86.5 %, 3τ 95 %, 4τ 98.2 %, 5τ ≈ 99.3 % — the practical "settled" rule used in electronics design.
• Initial slope of the exponential = (change)/τ; the tangent at t = 0 reaches the final value at t = τ (a standard graphical question).
• RC charging: v_C rises exponentially, current decays; RC discharging: both decay. RL growth: current rises, v_L = V e^−t/τ decays.
• Source-free (natural) response: i_L(t) = I₀e^−Rt/L, v_C(t) = V₀e^−t/RC.
• Energy: an inductor stores ½Li², a capacitor ½Cv². In an RC charging circuit only 50 % of the energy drawn from a DC source reaches the capacitor; the rest is dissipated in R, independent of R's value.
• Applications of the time constant: 555-timer and relay delays, RC snubbers, debounce networks, coupling/bypass capacitor sizing, oscilloscope probe compensation.

3. Second-order circuits (both L and C)

• Series RLC with a step: L(d²i/dt²) + R(di/dt) + i/C = 0 for the natural part. Characteristic equation s² + 2αs + ω₀² = 0.
• Neper frequency / damping coefficient α = R/2L (series) or 1/2RC (parallel); undamped natural frequency ω₀ = 1/√(LC); damping ratio ζ = α/ω₀.
• Roots s = −α ± √(α² − ω₀²) give three cases:
  1. Overdamped (ζ > 1, α > ω₀): two real distinct roots; response = A₁e^s₁t + A₂e^s₂t. No oscillation, slow.
  2. Critically damped (ζ = 1): repeated roots s = −α; response = (A₁ + A₂t)e^−αt. Fastest approach without overshoot — the design target for many control and driver circuits.
  3. Underdamped (ζ < 1): complex roots −α ± jω_d, with damped frequency ω_d = √(ω₀² − α²); response = e^−αt(A₁cos ω_d t + A₂sin ω_d t) — a decaying ring.
• Critical resistance for a series RLC: R_cr = 2√(L/C); for a parallel RLC: R_cr = ½√(L/C).
• Undamped (ζ = 0, R = 0) gives a sustained oscillation at ω₀ — the ideal LC tank.
• Peak overshoot M_p = exp(−πζ/√(1−ζ²)) and logarithmic decrement relate successive ring peaks; ringing on a PCB trace or in a MOSFET gate drive is exactly this underdamped response, killed by an RC snubber or a gate resistor.
• Dual relationship: series RLC (current response) and parallel RLC (voltage response) have identical mathematics with R ↔ G, L ↔ C.

4. Laplace transform method and network functions

• Transform the circuit once: R → R, L → sL (series source LI₀), C → 1/sC (series source V₀/s); initial conditions enter automatically, so no separate constants need evaluating.
• Solve algebraically, then invert by partial fractions. Key pairs: 1 ↔ δ(t), 1/s ↔ u(t), 1/s² ↔ t, 1/(s+a) ↔ e^−at, ω/(s²+ω²) ↔ sin ωt, s/(s²+ω²) ↔ cos ωt, and shifting for e^−atsin ωt.
• Initial value theorem x(0⁺) = lim s→∞ sX(s); final value theorem x(∞) = lim s→0 sX(s) (valid only if all poles lie in the left half plane, except a single pole at origin).
• Network functions: driving-point impedance Z(s) = V(s)/I(s) at the same port; transfer functions V₂/V₁, I₂/I₁, V₂/I₁ between ports.
• Poles (roots of denominator) decide the natural response and stability — LHP poles ⇒ decaying, imaginary axis ⇒ sustained oscillation, RHP ⇒ unstable. Zeros decide where transmission is blocked.
• For passive networks, poles and zeros of a driving-point function lie in the closed left half plane, with any on the jω axis simple, and the function is positive real — the test for realisability (Hurwitz/positive-real conditions; used in filter synthesis).
• Impulse response h(t) = L⁻¹{H(s)}; step response is its integral. Convolution in time = multiplication in s.

5. Resonance

• Resonance is the condition where the driving-point impedance (or admittance) is purely resistive — the reactive parts cancel, so voltage and current are in phase and the power factor is unity.
• Series RLC resonance (current resonance): at ω₀ = 1/√(LC), X_L = X_C, impedance is minimum = R, current is maximum = V/R. Voltage magnification across L and C = Q·V, which can be hundreds of volts from a 10 V source — the basis of tuned pre-selectors and the reason ratings matter.
• Below ω₀ the series circuit is capacitive (leading current); above ω₀ it is inductive.
• Quality factor Q = (energy stored per cycle × 2π)/(energy dissipated per cycle) = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C) for a series circuit.
• Half-power (−3 dB, cut-off) frequencies f₁, f₂: current falls to 1/√2 of peak, power to half, impedance = √2 R, phase = ∓45°. Bandwidth BW = f₂ − f₁ = f₀/Q = R/2πL. Also f₀ = √(f₁f₂) (geometric mean).
• Selectivity = f₀/BW = Q. High Q ⇒ sharp, selective response; it is raised by lowering R or raising L/C ratio.
• Parallel RLC (anti-resonance / voltage resonance): impedance is maximum at resonance, line current minimum. For an ideal parallel GLC, ω₀ = 1/√(LC) and Q = R/ω₀L = ω₀RC = R√(C/L) — note it is the inverse form of the series expression.
• Practical tank (coil resistance R in series with L, C across it): resonant frequency f_r = (1/2π)√(1/LC − R²/L²), slightly below 1/2π√(LC); dynamic impedance Z_d = L/CR (pure resistance). Resonance exists only if R < √(L/C).
• A practical coil's own quality factor Q_coil = ωL/R; circuit Q is degraded by source and load resistance (loaded Q), which widens the bandwidth.
• Uses: radio/IF tuning, tank circuits of LC oscillators, antenna matching, induction heating, wireless power transfer, notch filters at 50 Hz, and (undesirably) series resonance in power systems with capacitor banks.

6. Two-port networks

• A two-port has an input port (V₁, I₁) and an output port (V₂, I₂), with both port currents taken as flowing into the network. Two of the four variables are independent, giving six parameter sets.
• Z (open-circuit impedance) parameters: V₁ = Z₁₁I₁ + Z₁₂I₂; V₂ = Z₂₁I₁ + Z₂₂I₂. Z₁₁ = V₁/I₁ with output open, Z₁₂ = V₁/I₂ with input open (reverse transfer impedance). Units: ohms. Natural for T-networks and series connection.
• Y (short-circuit admittance) parameters: I₁ = Y₁₁V₁ + Y₁₂V₂; I₂ = Y₂₁V₁ + Y₂₂V₂, measured with ports shorted. Units: siemens. Natural for π-networks, parallel connection, and FET small-signal models.
• h (hybrid) parameters: V₁ = h₁₁I₁ + h₁₂V₂; I₂ = h₂₁I₁ + h₂₂V₂ — h₁₁ in Ω, h₂₂ in S, h₁₂ and h₂₁ dimensionless. The standard low-frequency transistor model (h_fe is h₂₁).
• g parameters: the inverse-hybrid set, I₁ = g₁₁V₁ + g₁₂I₂; V₂ = g₂₁V₁ + g₂₂I₂.
• ABCD (transmission) parameters: V₁ = AV₂ − BI₂; I₁ = CV₂ − DI₂ (the minus signs account for current leaving the output). A, D dimensionless; B in ohms; C in siemens. Used for cascades, transmission lines and power-system line models.
• Reciprocity (no dependent sources, no non-reciprocal elements): Z₁₂ = Z₂₁, Y₁₂ = Y₂₁, h₁₂ = −h₂₁, AD − BC = 1.
• Symmetry (the two ports can be interchanged): Z₁₁ = Z₂₂, Y₁₁ = Y₂₂, A = D, Δh = h₁₁h₂₂ − h₁₂h₂₁ = 1.
• Interconnections: series → add Z matrices; parallel → add Y matrices; cascade → multiply ABCD matrices in order; series–parallel → add h; parallel–series → add g. (Series/parallel addition is valid only if Brune's test is satisfied, i.e. no port-current violation; an ideal transformer at the junction guarantees it.)
• Image impedances Z_i1, Z_i2: terminating each port in its image impedance makes both ports look matched. For a symmetrical network they are equal and called the characteristic (iterative) impedance Z₀ = √(B/C) = √(Z_oc·Z_sc).
• Propagation constant for a symmetrical two-port: cosh γ = A = √(AD); attenuation in nepers (1 Np = 8.686 dB) — the classical image-parameter design route for attenuators, T/π pads and LC filters.
• At RF, Z/Y/h parameters cannot be measured (open and short circuits are unrealisable and may make the device oscillate), so S-parameters on a 50 Ω reference (S₁₁ return loss, S₂₁ insertion gain) are used instead, measured on a vector network analyser.
• Indian practice and exams use SI units and IS/IEC symbol conventions (IS 1885 series, electrotechnical vocabulary); impedance in Ω, admittance in S, frequency in Hz, phase in degrees or radians. Reactive power/power-factor rules for industrial capacitor banks are governed by CEA regulations and state DISCOM tariffs, where unwanted resonance with harmonics is a real design constraint.

Key points

• Inductor continuity - i_L cannot jump; open at t = 0⁺ if initially relaxed, short at DC steady state.
• Capacitor continuity - v_C cannot jump; short at t = 0⁺ if initially relaxed, open at DC steady state.
• First-order master formula - x(t) = x(∞) + [x(0⁺) − x(∞)]e^−t/τ; τ = L/R or RC with R = Thevenin resistance at the element.
• 5τ rule - a first-order circuit is treated as settled (99.3 %) after five time constants.
• Damping ratio ζ = α/ω₀ decides overdamped (>1), critically damped (=1) or underdamped (<1) behaviour.
• Critical resistance - 2√(L/C) for series RLC, ½√(L/C) for parallel RLC.
• Damped frequency ω_d = ω₀√(1 − ζ²) is always less than the undamped ω₀ = 1/√(LC).
• Laplace method - replace L by sL and C by 1/sC with initial-condition sources; initial conditions need no extra constants.
• Pole location decides stability and the shape of the natural response; zeros decide transmission nulls.
• Series resonance - minimum impedance R, maximum current, unity power factor, voltage magnification Q across L and C.
• Parallel resonance - maximum impedance (L/CR for a practical tank), minimum line current, used as a tuned load.
• Quality factor - Q = ω₀L/R = (1/R)√(L/C) series; Q = R√(C/L) parallel; equals selectivity f₀/BW.
• Bandwidth BW = f₀/Q = R/2πL; half-power points give |Z| = √2R and ±45° phase.
• f₀ = √(f₁f₂) - resonant frequency is the geometric, not arithmetic, mean of the half-power frequencies.
• Reciprocity test - Z₁₂ = Z₂₁, or AD − BC = 1; symmetry test - Z₁₁ = Z₂₂, or A = D.
• Cascade = ABCD product; series = Z sum; parallel = Y sum; series–parallel = h sum.
• h-parameters model BJTs at low frequency; Y suits FETs; S-parameters replace all of them at RF.
• Characteristic impedance Z₀ = √(B/C) = √(Z_open·Z_short) for a symmetrical two-port.

Formulas and examples

Example 1 - RL transient. A 20 V DC source is switched onto R = 10 Ω in series with L = 2 H (initially relaxed).

• τ = L/R = 2/10 = 0.2 s; final current i(∞) = 20/10 = 2 A; i(0⁺) = 0.
• i(t) = 2(1 − e^−t/0.2) A. At t = 0.1 s: e^−0.5 = 0.6065 → i = 2(0.3935) = 0.787 A.
• v_L(t) = 20e^−t/0.2; at t = 0.1 s, v_L = 20 × 0.6065 = 12.13 V.
• Time to reach 1.9 A (95 %): t = τ ln 20 = 0.2 × 2.996 = 0.599 s (≈ 3τ).
• Final stored energy = ½ × 2 × 2² = 4 J.

Example 2 - RC discharge. A 10 µF capacitor charged to 100 V is discharged through 100 kΩ at t = 0.

• τ = RC = 10⁵ × 10⁻⁵ = 1 s; v_C(t) = 100e^−t V; i(0⁺) = 100/10⁵ = 1 mA.
• At t = 2 s: v_C = 100e⁻² = 13.53 V; at t = 5 s, 0.67 V (effectively discharged).
• Initial stored energy = ½ × 10⁻⁵ × 100² = 0.05 J, all dissipated in R.

Example 3 - Second-order classification and response. Series R = 200 Ω, L = 1 H, C = 1 µF, 10 V step, relaxed.

• ω₀ = 1/√(1 × 10⁻⁶) = 1000 rad/s (f₀ = 159.2 Hz); α = R/2L = 200/2 = 100; ζ = 100/1000 = 0.1 → underdamped.
• R_cr = 2√(L/C) = 2√(10⁶) = 2000 Ω; since 200 Ω ≪ 2000 Ω, it rings heavily.
• ω_d = √(10⁶ − 10⁴) = 994.99 rad/s (158.4 Hz).
• i(t) = (V/ω_dL)e^−αt sin ω_d t = 0.01005 e^−100t sin(994.99t) A — first peak ≈ 8.6 mA, decaying with a 10 ms envelope time constant (1/α).
• Raising R to 2 kΩ gives ζ = 1: i(t) = (V/L)t e^−1000t, no overshoot.

Example 4 - Series resonance design. R = 10 Ω, L = 100 mH, C = 1 µF, V = 10 V rms.

• ω₀ = 1/√(0.1 × 10⁻⁶) = 3162.3 rad/s → f₀ = 503.3 Hz.
• Q = ω₀L/R = 3162.3 × 0.1/10 = 31.6; check: (1/10)√(0.1/10⁻⁶) = (1/10)(316.2) = 31.6 ✓.
• I₀ = 10/10 = 1 A; V_L = V_C = Q × V = 316 V (capacitor must be rated far above 10 V).
• BW = f₀/Q = 503.3/31.6 = 15.9 Hz; f₁ = 495.4 Hz, f₂ = 511.3 Hz; √(495.4 × 511.3) = 503.3 Hz ✓.
• Same components as a practical parallel tank: f_r = (1/2π)√(10⁷ − 10⁴) = 503.0 Hz; Z_dynamic = L/CR = 0.1/(10⁻⁶ × 10) = 10 kΩ — the same parts give 10 Ω in series and 10 kΩ in parallel at resonance.

Example 5 - Two-port T network. Series 10 Ω, shunt 20 Ω, series 30 Ω.

• Z₁₁ = 10 + 20 = 30 Ω; Z₂₂ = 30 + 20 = 50 Ω; Z₁₂ = Z₂₁ = 20 Ω → reciprocal (Z₁₂ = Z₂₁) but not symmetrical (Z₁₁ ≠ Z₂₂).
• ΔZ = 30 × 50 − 20 × 20 = 1100.
• Y₁₁ = 50/1100 = 45.45 mS, Y₂₂ = 30/1100 = 27.27 mS, Y₁₂ = Y₂₁ = −20/1100 = −18.18 mS.
• A = 30/20 = 1.5, B = 1100/20 = 55 Ω, C = 1/20 = 0.05 S, D = 50/20 = 2.5; check AD − BC = 3.75 − 2.75 = 1 ✓ (reciprocal).
• Cascading this with an identical network: multiply matrices → A′ = 1.5(1.5) + 55(0.05) = 5.0, B′ = 1.5(55) + 55(2.5) = 220 Ω, C′ = 0.05(1.5) + 2.5(0.05) = 0.2 S, D′ = 0.05(55) + 2.5(2.5) = 9.0; A′D′ − B′C′ = 45 − 44 = 1 ✓.

Where it is used

• RF and wireless design (Qualcomm, MediaTek, Tejas Networks, Signalchip, ISRO/SAC Ahmedabad): LC tanks, matching networks, antenna tuning and S-parameter work on VNAs are resonance and two-port theory applied directly.
• Analog/mixed-signal and embedded hardware (Texas Instruments, Analog Devices, Cypress/Infineon Bengaluru): RC time constants for debounce, filtering and reset circuits; snubbers and gate resistors to damp underdamped ringing.
• Power electronics and power systems (ABB, Siemens, BHEL, Hitachi Energy, state DISCOMs): switching transients, resonant converters (LLC), inrush and capacitor-bank resonance with harmonics; ABCD parameters for transmission-line models in load-flow studies.
• Communication and instrumentation: IF tuned amplifiers, 50 Hz notch filters in ECG and biomedical front-ends, metal detectors, RFID and wireless-power coils, induction heaters and cooktops.
• Test, verification and application-engineering roles at BEL, DRDO labs, HAL and C-DAC: step-response and bandwidth measurement, Q measurement with an LCR meter or Q-meter, rise-time/settling-time checks.
• Exam and placement screening: GATE EC/EE and IES Networks sections, PSU written tests, and first-round interviews at nearly every semiconductor and hardware firm in India.

Interview questions

• Q: Why can an inductor current not change instantaneously? A: It would require an infinite voltage, since v = L di/dt; flux linkage must be continuous unless an impulsive source is applied.
• Q: What does an uncharged capacitor look like at t = 0⁺ and at steady state? A: A short circuit at t = 0⁺ (voltage still zero) and an open circuit at DC steady state (no current).
• Q: How do you find the time constant of a circuit with several resistors? A: Deactivate the independent sources and find the Thevenin resistance seen by the L or C; τ = L/R_th or R_th·C.
• Q: What is critical damping and why is it often the design target? A: ζ = 1, repeated real roots; it gives the fastest response with no overshoot or ringing, which matters for drivers, step inputs and control loops.
• Q: Is the damped oscillation frequency higher or lower than 1/√(LC)? A: Lower — ω_d = √(ω₀² − α²), so resistance always reduces the ringing frequency.
• Q: At series resonance, how can a 10 V source produce 300 V across a capacitor? A: The L and C voltages are equal and opposite and each equals Q times the source voltage; they cancel in the loop but are individually large, so component voltage ratings must allow for it.
• Q: Define Q factor in two different ways. A: Energy-based: 2π × (energy stored)/(energy dissipated per cycle); circuit-based: ω₀L/R for a series circuit, equal to the selectivity f₀/BW.
• Q: How does loading affect the bandwidth of a tuned circuit? A: Source and load resistance lower the effective Q, so the bandwidth f₀/Q widens and selectivity falls.
• Q: What are the tests for reciprocity and symmetry of a two-port? A: Reciprocal if Z₁₂ = Z₂₁ (equivalently AD − BC = 1); symmetrical if Z₁₁ = Z₂₂ (equivalently A = D).
• Q: Why are ABCD parameters preferred for cascaded networks? A: The overall matrix is simply the product of the individual ABCD matrices in cascade order, which no other set gives directly.
• Q: Why are S-parameters used instead of h or Z parameters at microwave frequencies? A: Ideal opens and shorts cannot be realised at those frequencies and may make an active device oscillate; S-parameters are measured with matched 50 Ω terminations.
• Q: Can a passive two-port have h₁₂ = h₂₁? A: Not for a reciprocal passive network — reciprocity there requires h₁₂ = −h₂₁; equality would mean a non-reciprocal element.

Common mistakes

• Treating an inductor as a short or a capacitor as an open at t = 0⁺ — those are the DC steady-state models, used only at t → ∞.
• Using the raw source resistance instead of the Thevenin resistance at the element when computing τ, or forgetting that τ for RL is L/R while for RC it is R·C (unit check: H/Ω = s, Ω·F = s).
• Taking f₀ as the arithmetic mean of the half-power frequencies; it is the geometric mean √(f₁f₂) (the two coincide only for high Q).
• Using the series Q expression (1/R)√(L/C) for a parallel circuit, where it inverts to R√(C/L), and ignoring that a practical tank resonates slightly below 1/2π√(LC).
• Dropping the minus signs in the ABCD definition (I₂ taken as leaving the port) and then finding AD − BC ≠ 1 for an obviously reciprocal network.
• Adding Y-matrices for a cascade or multiplying Z-matrices for a parallel connection — each interconnection has one correct parameter set.
• Mixing rad/s with Hz (forgetting the 2π) in ω₀, Q and bandwidth calculations, and quoting mH/µF values without converting to H and F.

Related topics

Network theorems and circuit analysis · Active filters and op-amp applications · Oscillators and power supplies · Signals and systems: Laplace and Fourier analysis · Transmission lines and RF matching · Control systems: time response and stability