Network Theory — Complete Revision Notes

Thevenin, Norton & Maximum Power

• Dependent Sources Thevenin Resistance: Deactivate independent sources (voltage short, current open). Connect a test source ($1\text{ V}$ or $1\text{ A}$) at load terminals. Then $R_{th} = \frac{V_{test}}{I_{test}}$. If there are no independent sources, $V_{th} = 0$.
• Maximum Power Transfer:
  • DC: $R_L = R_{th} \implies P_{max} = \frac{V_{th}^2}{4 R_{th}}$
  • AC (Variable $Z_L = R_L + jX_L$): $Z_L = Z_{th}^* = R_{th} - jX_{th} \implies P_{max} = \frac{|V_{th}|^2}{4 R_{th}}$
  • AC (Variable $R_L$ only): $R_L = |Z_{th}| = \sqrt{R_{th}^2 + X_{th}^2}$

Universal Step Response Formula

$$x(t) = x(\infty) + [x(0^+) - x(\infty)] e^{-t/\tau}, \quad t \ge 0$$

• $RC$ Circuit Time Constant: $\tau = R_{eq} C$
• $RL$ Circuit Time Constant: $\tau = \frac{L}{R_{eq}}$
• Initial Conditions ($t = 0^+$):
  • Uncharged capacitor behaves as a Short Circuit ($v_C(0^+) = 0$).
  • Unenergized inductor behaves as an Open Circuit ($i_L(0^+) = 0$).
• DC Steady State ($t = \infty$): Inductor is a Short Circuit ($V_L=0$); Capacitor is an Open Circuit ($I_C=0$).

Series vs Parallel Resonance Comparison