3.1 DC Circuit Analysis

Why DC analysis anchors the exam

The NCEES FE Electrical and Computer specification gives Circuit Analysis (DC and AC) 10-15 of 110 questions, and DC fundamentals also feed Electronics, Power, and Control Systems items. Most DC questions are short once the model is correct, so the goal is a reliable setup routine rather than clever tricks.

Ohm's law and the two Kirchhoff laws

Ohm's law relates voltage, current, and resistance: V = IR. Power dissipated in a resistor is P = VI = I^2 R = V^2 / R.

Kirchhoff's Voltage Law (KVL): the algebraic sum of voltages around any closed loop is zero. Walk the loop in one direction and assign a sign to each rise or drop consistently.

Kirchhoff's Current Law (KCL): the algebraic sum of currents entering a node equals the sum leaving. KCL is the basis of the node-voltage method.

Declare a reference current direction and a ground node before writing equations; a negative result simply means the true direction is opposite your assumption.

Series and parallel reduction

Resistors in series carry the same current and add directly: R_eq = R_1 + R_2 + ... For resistors in parallel the voltage is shared and conductances add:

1/R_eq = 1/R_1 + 1/R_2 + ...

For exactly two resistors in parallel, the product-over-sum shortcut is fastest: R_eq = R_1 R_2 / (R_1 + R_2). Two equal resistors in parallel give half the value.

Voltage divider: for series resistors across a source, V_x = V_s * R_x / R_total.

Current divider: for two parallel resistors, the current through R_1 is I_1 = I_total * R_2 / (R_1 + R_2) (the opposite resistor appears in the numerator).

Note that capacitors combine the opposite way to resistors.

Thevenin, Norton, and source transformation

Any linear two-terminal network reduces to a Thevenin equivalent: an ideal voltage source V_th in series with R_th. The dual is the Norton equivalent: a current source I_N in parallel with the same R_th, where I_N = V_th / R_th.

Procedure to find the equivalent seen from two terminals:

1. V_th = open-circuit voltage across the terminals.
2. I_N = short-circuit current through the terminals.
3. R_th = V_th / I_N, or deactivate all independent sources (short voltage sources, open current sources) and compute the resistance looking back into the terminals.

When dependent sources are present, you cannot simply zero them; instead apply a 1 A or 1 V test source at the terminals and compute R_th = V_test / I_test.

Superposition

For a linear circuit with multiple independent sources, the response equals the sum of responses to each source acting alone. Zero the others: replace voltage sources with shorts and current sources with opens. Superposition is convenient when sources have different types or frequencies, but it does not apply to power (power is nonlinear in voltage/current).

Maximum power transfer

For a fixed Thevenin source, maximum power is delivered to the load when the load matches the source resistance: R_L = R_th. At that match,

P_max = V_th^2 / (4 R_th).

At the match the source and load each dissipate half the total, so the transfer efficiency is only 50 percent. In AC the matching condition becomes a complex conjugate: Z_L = Z_th*.

Node-voltage and mesh-current methods

The node-voltage method applies KCL at each non-reference node, writing each branch current in terms of node voltages and conductances. Pick the node touching the most branches (often a source terminal) as ground.

The mesh-current method applies KVL around each independent loop, solving for loop currents. Mesh analysis is efficient for planar circuits dominated by voltage sources; node analysis is efficient when current sources and many parallel branches dominate. Choose whichever yields fewer simultaneous equations.