DC Circuit Analysis

Node Voltage Method

Applies KCL at each non-reference node to find all node voltages.

Procedure:

1. Select 1 node as reference (ground, $V = 0$).
2. Assign voltages $V_1, V_2, \ldots$ to the remaining nodes.
3. Write KCL at each non-reference node: sum of currents leaving = 0.
4. Express each current as $(V_i - V_j)/R$.
5. Solve the linear system.

Mesh Current Method

Applies KVL around each independent loop.

Procedure:

1. Identify all meshes (innermost loops of a planar circuit).
2. Assign a clockwise mesh current $I_1, I_2, \ldots$ to each.
3. Write KVL for each mesh: $\sum V_{\text{drops}} = 0$.
4. Solve for mesh currents; branch currents are algebraic sums of contributing mesh currents.

Superposition Theorem

The response (voltage or current) at any element equals the sum of responses due to each independent source acting alone.

To isolate a source:

• Replace all other voltage sources with short circuits.
• Replace all other current sources with open circuits.

Thevenin’s Theorem

Any linear 2-terminal network can be replaced by a voltage source $V_{th}$ in series with a resistance $R_{th}$.

$V_{oc}$: open-circuit voltage at the terminals.

$I_{sc}$: short-circuit current at the terminals.

Alternatively: deactivate all independent sources and compute the equivalent resistance seen at the terminals.

Norton’s Theorem

Dual of Thevenin’s theorem. Any linear 2-terminal network can be replaced by a current source $I_N$ in parallel with $R_N$.

The 2 forms are interconvertible:

DC Time Constant

Time for a capacitor to charge through a resistor from $0\,\text{V}$ to $63.2\%$ of the applied DC voltage.

Alternatively: time to discharge to $36.8\%$ of the initial charge voltage.

Here:

• $R$: resistance (ohms)
• $C$: capacitance (farads)
• $L$: inductance (henries)