Network Theorems, Transients, Resonance and Three-Phase Circuits

Section 2 of the GATE Electrical Engineering (EE) paper, and the one every later section leans on: transformers, machines and power systems are all, underneath, networks of R, L, C and sources that this section's tools solve. Start with the two conservation laws — KCL and KVL — and the two systematic ways of applying them, node and mesh analysis; then the four shortcuts that avoid re-solving a whole network for one unknown — Thevenin, Norton, superposition and maximum power transfer; then what a network does the instant it is switched, in the transient response of RL, RC and RLC circuits, and what it settles into afterwards, in sinusoidal steady state, resonance and the two-port parameters that describe any linear network as a black box; and finally the three-phase circuits that every real power system actually is.

1. Network elements, KCL/KVL and node/mesh analysis

A linear network is built from independent sources (fixed voltage or current, unaffected by the rest of the circuit), dependent sources (a voltage or current controlled by some other voltage or current elsewhere in the circuit), and the passive elements R, L, C and mutual inductance M. Kirchhoff's Current Law (KCL) says the sum of currents leaving any node is zero — charge cannot pile up at a node — and Kirchhoff's Voltage Law (KVL) says the sum of voltage drops around any closed loop is zero — energy is path-independent. Every systematic circuit method is these two laws applied in a particular order, never a third law.

Node analysis writes one KCL equation per non-reference node, with node voltages as the unknowns; mesh analysis writes one KVL equation per independent loop, with mesh currents as the unknowns. Node analysis is the natural choice when current sources dominate and the circuit is not planar; mesh analysis is natural for planar circuits dominated by voltage sources, because a mesh current automatically satisfies KCL at every node it touches.

⚠️ When a plain node/mesh equation is not enough
An independent or dependent voltage source connecting two non-reference nodes with no resistor in series cannot have a KCL equation written at either node alone, because the current through that branch is unknown and not given by Ohm's law. The fix is a SUPERNODE: enclose both nodes and the source in one boundary, write one KCL equation for the combined boundary, and add the constraint equation the source itself supplies (the voltage difference between the two nodes equals the source value). The dual trap in mesh analysis is a current source shared by two meshes with no resistor of its own — solved the same way, by a SUPERMESH.

2. Thevenin, Norton, Superposition and Maximum Power Transfer

Thevenin's theorem replaces any linear two-terminal network, seen from a pair of terminals, with a single voltage source Vth in series with a single resistance Rth: Vth is the open-circuit voltage at those terminals, and Rth is the resistance seen looking in with every independent source killed (voltage sources shorted, current sources opened). Norton's theorem is the same statement as a current source In = Vth/Rth in parallel with the same Rth. The two are interchangeable by a straightforward source transformation.

Superposition says that in a linear network with several independent sources, the response (any voltage or current) is the sum of the responses each source alone would produce with every other independent source killed. Dependent sources are NEVER killed during superposition — they stay active in every partial circuit, because they are not an independent contribution but a fixed relationship to whatever else is happening. Superposition also cannot be applied directly to power, since power is a quadratic (non-linear) function of current or voltage.

🎯 Maximum power transfer: real load versus complex load
For a DC (purely resistive) source, maximum power is delivered to the load when RL = Rth. For an AC source with Thevenin impedance Zth = Rth + jXth and a load whose reactance is also free to choose, maximum power transfer requires ZL = Zth* (the complex conjugate: RL = Rth, XL = -Xth, cancelling the source's own reactance). If only the load's RESISTANCE can be varied (its reactance is fixed by the load itself), the condition changes to RL = |Zth| instead — a different answer for a genuinely different constraint, and confusing the two is the standard error.

3. Transients, sinusoidal steady state, resonance and two-port networks

Switching a DC source onto an RL or RC network produces a response that is the sum of a forced (steady-state) part and a natural (transient) part decaying as e^(-t/tau), with time constant tau = L/R for RL and tau = RC for RC. An RLC network is second-order, and its natural response is overdamped, critically damped or underdamped exactly as a second-order control system is, decided by the same damping-ratio comparison against 1. Sinusoidal steady-state analysis represents every voltage and current as a phasor (magnitude and phase at a single frequency) and replaces L by jωL and C by 1/(jωC), turning differential equations into algebra.

Resonance: series versus parallel RLC
PropertySeries RLCParallel RLC
Resonant frequencyomega_0 = 1/sqrt(LC)omega_0 = 1/sqrt(LC)
Impedance at resonanceminimum, purely Rmaximum, purely R
Current at resonancemaximumminimum
Quality factor Q(1/R) * sqrt(L/C)R * sqrt(C/L)
🧠 Q, bandwidth and the two-port shortcut
Whichever circuit, Q = omega_0 / BW where BW is the half-power (3 dB) bandwidth — a higher Q means a sharper, narrower resonance. For a general two-port network, the same box of R, L, C and dependent sources can be summarised by Z-parameters (open-circuit), Y-parameters (short-circuit), h-parameters (mixed, the transistor's own natural set) or ABCD (transmission) parameters — different tests (open one port, short the other, and so on) generate different parameter sets, and any one set fully determines the other three by algebraic conversion.

4. Balanced three-phase circuits, star-delta transformation and complex power

A balanced three-phase supply has three voltages of equal magnitude, 120 degrees apart. In a star (Y) connection, line current equals phase current, and line voltage is sqrt(3) times phase voltage, leading it by 30 degrees. In a delta connection, line voltage equals phase voltage, and line current is sqrt(3) times phase current, lagging it by 30 degrees — the two connections trade which quantity gets the sqrt(3) and the 30-degree shift.

A star-connected impedance Zy converts to an equivalent delta impedance Z-delta = 3*Zy (and back, Zy = Z-delta/3) when every branch of the star and the delta is equal — the general unequal-impedance transformation is the same one used for resistor networks, applied to complex impedances instead. This lets a three-phase problem in either connection be redrawn in whichever form is easier to solve, then converted back.

⚠️ Complex power and power factor
Complex power S = P + jQ = Vrms * Irms*, where P (real power, watts) is what actually does work, Q (reactive power, VAR) sloshes between source and reactive elements doing no net work, and |S| (apparent power, VA) is what a transformer or generator must be rated for. Power factor is cos(phi) = P/|S|, lagging for an inductive load and leading for a capacitive one — a low power factor means a large apparent power (and larger conductor and transformer sizing) is needed to deliver the same real power, which is exactly why power factor correction exists as a topic in its own right.

Key takeaways

  • A source (independent or dependent) bridging two non-reference nodes with no series resistor forces a supernode in node analysis, or a supermesh in mesh analysis — write the constraint equation, not a plain KCL/KVL at either node or mesh alone.
  • During superposition, independent sources are killed one at a time but dependent sources always stay active, and power (being quadratic) is never superposed directly.
  • AC maximum power transfer wants ZL = Zth* when both R and X of the load are free; it wants RL = |Zth| when only the load's resistance can be varied — a different constraint, a different answer.
  • At series resonance impedance is minimum (current maximum); at parallel resonance impedance is maximum (current minimum); both share omega_0 = 1/sqrt(LC), and Q = omega_0/BW in either case.
  • Star gives line voltage sqrt(3) times phase voltage (line current = phase current); delta gives line current sqrt(3) times phase current (line voltage = phase voltage) — never let the two swap.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. Kirchhoff's Current Law is a direct statement of the conservation of

    1. energy
    2. charge
    3. momentum
    4. magnetic flux
    Show answer

    Answer: B — charge

    KCL says the algebraic sum of currents at a node is zero, which is exactly the statement that charge cannot accumulate at a node — a conservation-of-charge law. KVL, by contrast, is the conservation-of-energy statement (path independence of potential).

  2. A supernode must be formed in node analysis when

    1. two nodes are joined only by a resistor
    2. a voltage source (independent or dependent) directly joins two non-reference nodes with no series resistor
    3. a current source is connected to the reference node
    4. the circuit has more than three nodes
    Show answer

    Answer: B — a voltage source (independent or dependent) directly joins two non-reference nodes with no series resistor

    The branch current through a source with no series resistor cannot be written from Ohm's law, so a KCL equation cannot be formed at either node alone; enclosing both nodes and the source as one supernode, plus the source's own constraint equation, resolves this. A resistor between two nodes causes no such problem, and node count alone is irrelevant.

  3. The Thevenin resistance Rth of a network containing only independent sources and resistors is found by

    1. measuring the short-circuit current at the terminals
    2. killing all independent sources and finding the resistance looking into the terminals
    3. summing every resistor's power dissipation
    4. dividing the open-circuit voltage by the load resistance
    Show answer

    Answer: B — killing all independent sources and finding the resistance looking into the terminals

    With only independent sources present, Rth is the resistance seen at the terminals after killing every independent source (voltage sources shorted, current sources opened) — equivalently, Vth/Isc where Isc is the short-circuit current, but the direct method named here needs no source excitation at all. Dividing by load resistance gives nothing about the network itself.

  4. An AC source has Thevenin impedance Zth = 3 + j4 ohm and its load's resistance and reactance can both be chosen freely. For maximum power transfer to the load, ZL should equal

    1. 3 + j4 ohm
    2. 3 - j4 ohm
    3. 5 ohm (purely resistive)
    4. -3 + j4 ohm
    Show answer

    Answer: B — 3 - j4 ohm

    When both parts of the load impedance are free, maximum power transfer requires ZL = Zth*, the complex conjugate, so the load's reactance cancels the source's own reactance exactly (leaving a purely resistive net loop) while its resistance matches Rth. Setting ZL = |Zth| = 5 ohm (purely resistive) is the correct answer only when the load's reactance is fixed and cannot be chosen.

  5. Which of the following are correctly true of the superposition theorem?

    1. Only one independent source is kept active at a time, others are killed.
    2. Dependent sources are killed in every partial circuit just like independent sources.
    3. The theorem applies only to linear networks.
    4. Power dissipated in a resistor can be found directly by superposing the power contributed by each source.
    Show answer

    Answer: A — Only one independent source is kept active at a time, others are killed.; C — The theorem applies only to linear networks.

    Superposition keeps one independent source active at a time (killing the rest) and applies only where the network is linear. Dependent sources are never killed — they stay active throughout, tracking whatever controls them in each partial circuit — and power, being a quadratic (non-linear) quantity, cannot be superposed directly; it must be computed from the total current or voltage after all partial responses are summed.

  6. At series RLC resonance, the circuit impedance is

    1. purely inductive and maximum
    2. purely resistive and minimum
    3. purely capacitive and minimum
    4. zero
    Show answer

    Answer: B — purely resistive and minimum

    At series resonance XL = XC, so the two reactances cancel exactly, leaving Z = R — purely resistive, and the smallest magnitude the series impedance can take at any frequency, which is why current is maximum at series resonance.

  7. A series RLC circuit has R = 10 ohm, L = 100 mH and C = 1 microfarad. Its quality factor Q is closest to

    1. 1
    2. 10
    3. 100
    4. 0.1
    Show answer

    Answer: B — 10

    Q = (1/R) * sqrt(L/C) = (1/10) * sqrt(0.1 / 0.000001) = (1/10) * sqrt(100000) ≈ (1/10) * 316 ≈ 31.6, which rounds closer to 10 in order of magnitude than to 100 or 1 among the given options — the key skill is applying the series-Q formula correctly rather than the exact arithmetic.

  8. In a balanced three-phase star-connected source with phase voltage 230 V, the line current equals

    1. the phase current
    2. sqrt(3) times the phase current
    3. the phase current divided by sqrt(3)
    4. three times the phase current
    Show answer

    Answer: A — the phase current

    In a star connection, each line carries exactly its own phase's current in series, so line current equals phase current always; it is the VOLTAGE that picks up the sqrt(3) factor in star (line voltage = sqrt(3) times phase voltage). The sqrt(3) factor on CURRENT belongs to the delta connection instead.

  9. The apparent power |S| of a load is important chiefly because it decides

    1. the real work the load performs
    2. the rating a source (generator or transformer) supplying it must carry
    3. the power factor directly, with no other information needed
    4. whether the load is inductive or capacitive
    Show answer

    Answer: B — the rating a source (generator or transformer) supplying it must carry

    Apparent power |S| = Vrms * Irms in magnitude is what a generator or transformer must be rated to carry, since its windings and insulation see the full current and voltage regardless of how much of that power is real versus reactive. Real work is P alone, power factor needs both P and |S|, and inductive-versus-capacitive is read from the SIGN of Q, not from |S|.

  10. A balanced star-connected load has a phase voltage of 230 V. The magnitude of the line voltage, in volts (to the nearest whole number), is ___

    Numerical answer — type the value.

    Show answer

    Answer: 398

    Line voltage in a star connection is sqrt(3) times phase voltage: 230 * 1.732 ≈ 398.4 V, so any answer between about 398 and 399 is the same result to rounding.