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AExE0102 Network theorems

The four core theorems

Theorem What it does Key procedure or condition
Superposition Adds individual source contributions Linear circuit; one independent source at a time
Thevenin Replaces a two-terminal network by \(V_{th}\) in series with \(R_{th}\) \(V_{th}=V_{oc}\)
Norton Replaces it by \(I_N\) in parallel with \(R_N\) \(I_N=I_{sc}\)
Maximum power transfer Chooses the load for greatest delivered power Resistive case: \(R_L=R_{th}\)

Superposition theorem

For a linear circuit, the voltage or current response caused by several independent sources equals the algebraic sum of the responses caused by each source acting alone.

When considering one independent source:

Deactivated ideal source Replacement
Voltage source Short circuit
Current source Open circuit
  • Keep dependent sources active because their values depend on circuit variables.
  • Add signed voltages or currents, including direction and polarity.
  • Do not superpose power directly because \(P=I^2R\) or \(V^2/R\) is nonlinear.
  • The theorem applies to linear DC and linear AC phasor circuits.

Thevenin and Norton equivalents

At the chosen load terminals:

\[ V_{th}=V_{oc}, \qquad I_N=I_{sc}, \qquad R_N=R_{th}. \]

Conversion:

\[ V_{th}=I_NR_{th}, \qquad I_N=\frac{V_{th}}{R_{th}}. \]

To find \(R_{th}\):

  1. Remove the load.
  2. Deactivate independent voltage and current sources.
  3. Find the resistance looking into the terminals.

If dependent sources are present, keep them active and apply a test source at the terminals; then \(R_{th}=V_{test}/I_{test}\). When valid and finite, \(R_{th}=V_{oc}/I_{sc}\) is another route.

Recognition: Thevenin is voltage-source plus series resistance. Norton is current-source plus parallel resistance.

Maximum power transfer

For a resistive Thevenin source,

\[ P_L=\frac{V_{th}^2R_L}{(R_{th}+R_L)^2}. \]

Maximum load power occurs at

\[ R_L=R_{th}, \qquad P_{L,\max}=\frac{V_{th}^2}{4R_{th}}. \]

At this matched point in the elementary Thevenin model, \(R_L\) and \(R_{th}\) dissipate equal power, so load efficiency is exactly \(50\%\). Maximum power does not mean maximum efficiency. Power systems normally use \(R_L\gg R_{th}\) to avoid large source loss.

For AC networks, the general condition is conjugate matching:

\[ Z_L=Z_{th}^{*}. \]

Power-reference condition: at exact resistive matching in the ideal Thevenin model, the load receives one half of the power supplied by the ideal Thevenin source; half is dissipated in \(R_{th}\).

RC and RL transients

Circuit Time constant Rising response from zero Natural decay
RC \(\tau=RC\) \(v_C(t)=V_f(1-e^{-t/\tau})\) \(v_C(t)=V_0e^{-t/\tau}\)
RL \(\tau=L/R\) \(i_L(t)=I_f(1-e^{-t/\tau})\) \(i_L(t)=I_0e^{-t/\tau}\)

At one time constant:

  • a rising quantity reaches \(1-e^{-1}=63.2\%\) of its final change;
  • a decaying quantity remains at \(e^{-1}=36.8\%\) of its initial value;
  • after roughly \(5\tau\), a first-order transient is effectively settled.

Continuity rules:

  • capacitor voltage cannot jump instantaneously in an ideal circuit;
  • inductor current cannot jump instantaneously in an ideal circuit;
  • at DC steady state, an ideal capacitor is open and an ideal inductor is short.

Sinusoidal RLC relations

\[ X_L=\omega L, \qquad X_C=\frac{1}{\omega C}. \]

For a series RLC circuit,

\[ Z=R+j(X_L-X_C), \]
\[ |Z|=\sqrt{R^2+(X_L-X_C)^2}, \qquad \phi=\tan^{-1}\left(\frac{X_L-X_C}{R}\right). \]

Phase recognition:

  • resistor: voltage and current in phase;
  • inductor: current lags voltage by \(90^\circ\);
  • capacitor: current leads voltage by \(90^\circ\);
  • net \(X_L>X_C\): inductive, current lags;
  • net \(X_C>X_L\): capacitive, current leads.

Resonance

For the ideal/simple LC resonance condition,

\[ X_L=X_C, \qquad \omega_0=\frac{1}{\sqrt{LC}}, \qquad f_0=\frac{1}{2\pi\sqrt{LC}}. \]
Feature Series resonance Parallel resonance
Input impedance Minimum; \(Z=R\) in series RLC Maximum in the ideal parallel case
Source current Maximum Minimum
Power factor Unity Unity at the input
Main use Pass/select a frequency Reject/select or tank behavior

At series resonance, \(V_L\) and \(V_C\) can each greatly exceed the source voltage but are opposite in phase and cancel in the sum.

For a series RLC circuit,

\[ Q=\frac{\omega_0L}{R}=\frac{f_0}{\Delta f}. \]

High \(Q\) means narrow bandwidth and sharp resonance. Parallel-resonant formulas depend on where practical loss resistance is placed; do not blindly interchange \(R/X\) and \(X/R\) formulas.

Active, reactive, and apparent power

Use RMS voltage and current:

\[ P=VI\cos\phi, \qquad Q=VI\sin\phi, \qquad S=VI. \]
Quantity Symbol Unit Meaning
Active/true power \(P\) watt (W) Average energy converted to work or heat
Reactive power \(Q\) volt-ampere reactive (VAR) Energy exchanged with \(L\) and \(C\)
Apparent power \(S\) volt-ampere (VA) RMS voltage-current product
\[ S^2=P^2+Q^2, \qquad \text{power factor}=\cos\phi=\frac{P}{S}. \]

Under the usual load convention, inductive \(Q\) is positive and capacitive \(Q\) is negative. An ideal inductor or capacitor has zero average active power even though it can carry current and exchange reactive power.

Network-theorem examples

  1. If \(V_{th}=12\ \text{V}\) and \(R_{th}=3\ \Omega\), maximum power occurs at \(R_L=3\ \Omega\) and is \(12^2/(4\times3)=12\ \text{W}\).
  2. If \(R=10\ \text{k}\Omega\) and \(C=10\ \mu\text{F}\), then \(\tau=RC=0.1\ \text{s}\).
  3. If \(L=10\ \text{mH}\) and \(C=10\ \mu\text{F}\), then \(f_0\approx503\ \text{Hz}\).

Network-theorem revision box

  • Superposition: linear; one independent source at a time; never add powers.
  • Thevenin: \(V_{oc}\) + series \(R\). Norton: \(I_{sc}\) + parallel \(R\).
  • Maximum resistive load power: \(R_L=R_{th}\); matched efficiency \(50\%\).
  • Series resonance: low \(Z\), high source current.
  • Parallel resonance: high \(Z\), low source current.
  • \(P\) in W, \(Q\) in VAR, \(S\) in VA.