DC circuits & Thevenin / Norton

Reference notes

This is the foundational toolkit for any linear DC (and by extension, any sinusoidal steady-state AC) circuit. Use Next → to walk through Ohm + Kirchhoff, the two systematic methods (mesh / nodal), Thevenin and Norton equivalents, the maximum power transfer theorem, and superposition.

Three fundamental laws

• Ohm's law: V = I · R for a resistor.
• Kirchhoff's Current Law (KCL): ΣI = 0 at any node. Conservation of charge.
• Kirchhoff's Voltage Law (KVL): ΣV = 0 around any closed loop. Conservation of energy.

Resistor combinations

• Series: same current, voltages add → R_total = R_1 + R_2 + ...
• Parallel: same voltage, currents add → 1/R_total = 1/R_1 + 1/R_2 + ...; for two resistors R_total = R_1 · R_2 / (R_1 + R_2).
• Voltage divider: V_R = V_total · R / R_total (series chain).
• Current divider: I_R = I_total · R_other / (R_other + R) (two-resistor parallel).

Power in circuit elements

P > 0 when absorbed; P < 0 when supplied. Sum of supplied = sum of absorbed (conservation of energy).

Systematic methods

• Mesh analysis: define a clockwise mesh current in each window; apply KVL around each mesh; sum voltage drops = 0. M-1 equations for an M-mesh circuit. Best for voltage-source-heavy networks.
• Nodal analysis: choose a reference node (ground); define node voltages; apply KCL at each non-reference node. N-1 equations for an N-node circuit. Best for current-source / parallel-branch-heavy networks.
• Modified Nodal Analysis (MNA) — what SPICE uses; generalizes nodal to handle voltage sources and dependent sources at scale.

Thevenin's theorem

Any linear 2-terminal network → equivalent ONE voltage source V_th in series with ONE resistance R_th:

• V_th = open-circuit voltage at the terminals.
• R_th = equivalent resistance with all independent sources zeroed (V-source → short, I-source → open). Dependent sources stay active.
• Or compute as R_th = V_oc / I_sc (short-circuit current method).

Once V_th and R_th are known, terminal behavior for ANY load R_L is:

Norton's theorem

Dual of Thevenin: same network → I_N in PARALLEL with R_N.

• I_N = short-circuit current at terminals = V_th / R_th.
• R_N = R_th.
• Source transformations: V_th ↔ I_N · R_N freely interconverts Thevenin and Norton forms.

Maximum power transfer

For fixed V_th, R_th, the R_L that absorbs maximum power equals R_th.

• Important context: max-power-transfer is NOT the same as high efficiency. It maximizes power delivery, accepting that half the source power dissipates in R_th.
• Appropriate for fixed-source applications: antenna match, audio amplifier output, signal coupling.
• NOT appropriate for utility power transmission, where η near 100 % is desired (R_load >> R_th).

Superposition

In a LINEAR circuit with multiple independent sources, the V (or I) response at any element = sum of responses caused by each source ALONE, with all other independent sources zeroed (V → short, I → open).

• Applies to LINEAR responses (V, I).
• Does NOT apply to power (P is quadratic / nonlinear in sources). Compute V or I via superposition first, then square to get P.
• Foundation of small-signal analysis and frequency-domain phasor analysis.