Circuit Analysis

Three-phase circuits, symmetrical components, the per-unit system, phasor diagrams, single-phase circuits and direct current circuits. At 10–15 of 80 it sits level with Electrical Safety and Protection as the largest area on the PE Power exam, with six subtopics.

What the problems look like

Short calculations and diagram readings, spread evenly across the subtopics. Recurring shapes — three-phase line current, kW and kvar for wye and delta loads, a neutral current, a bank with one element or fuse open; symmetrical components resolved from three phasors or rebuilt into a phase quantity, and the networks a fault connects; per unit bases, a change of base, a fault duty from a one-line; phasor diagrams read for sequence, lead or lag; single-phase R-L-C branches, rms values of rectified or distorted waveforms, 120/240 V three-wire panels; dc switching transients, resistor networks and two-wire feeder drop.

By exam day you should be able to —

• move between line and phase quantities: wye V_L = √3·V_P, I_L = I_P; delta V_L = V_P, I_L = √3·I_P; |S| = √3·V_L·I_L; Z_Δ = 3·Z_Y for equal branches;
• build any line-to-line phasor as V_ab = V_an − V_bn, which for abc sequence and V_an at 0° gives √3·V_P∠30°;
• resolve sequence sets with a = 1∠120°: I_a0 = (I_a + I_b + I_c)/3, I_a1 = (I_a + a·I_b + a²·I_c)/3, I_a2 = (I_a + a²·I_b + a·I_c)/3;
• set and change bases: Z_base = (kV_LL)²/MVA_3φ, I_base = kVA_3φ/(√3·kV_LL), Z_new = Z_old·(kV_old/kV_new)²·(kVA_new/kVA_old);
• solve a single-phase branch: X_L = ωL, |X_C| = 1/(ωC), |Z| = √(R² + X²), P = V·I·cos θ, |S| = √(P² + Q²);
• evaluate a dc transient at a stated time: τ = L/R or RC, i(t) = (V/R)(1 − e^(−t/τ)) for a coil energized from zero.

Where people lose points

• √3, 3 or neither. √3 goes with line-to-line voltage and line current; 3 with phase quantities. The positive-sequence source is V_LL/√3.
• Balanced-only shortcuts. I_L = √3·I_P and V_ab = √3·V_P∠30° hold for a balanced set. With an open element or unequal loads, write Kirchhoff's laws in phasors, and read from the drawing whether a neutral is connected.
• Adding magnitudes. Currents at a node add as phasors. Arithmetic addition fits in-phase quantities only; root-sum-square fits quantities 90° apart. On a 120/240 V three-wire panel, resistive loads on opposite legs subtract in the neutral and a line-to-line load adds nothing.
• Sequence bookkeeping. The 1/3 belongs to the resolving direction only; the positive and negative rows each use a and a² once. Neutral or residual current is 3·I_a0, and a neutral impedance enters the zero-sequence network as 3·Z_n. Single line-to-ground: three networks in series. Line-to-line: positive and negative only. Double line-to-ground: all three in parallel. An ungrounded wye offers no zero-sequence path.
• Per-unit bases. One base MVA for the study; base voltage steps through each transformer by its nameplate ratio. Move each nameplate percent impedance to the study base before combining, and divide ohms by the base impedance of their own zone.
• Switching state. Sources combine in parallel only if their breakers and the tie are drawn closed.
• rms, peak and sign. A phasor magnitude is rms. Sinusoid rms is X_max/√2; half-wave rectified rms is X_max/2. A current drawn clockwise from its voltage lags, and the load's reactance is positive.
• Transients by rule of thumb. Evaluate the exponential at the instant asked, carry inductor current (not voltage) across the switching instant, and use the resistance the element actually sees. On a two-wire dc run, both conductors are in the loop.

Have these at your fingertips

The line-phase relations, the sequence rows, the base and change-of-base forms, and the R-L and R-C step responses. In the Handbook's Circuits chapter they sit under 3-Phase Circuits, Symmetrical Components, Per Unit System, AC Circuits and DC Circuits. The sequence-network connections for each fault type are further on, under Fault Current Analysis. A dc feeder drop can send you to the supplied NEC (NFPA 70-2020), Chapter 9, Table 8, for conductor resistance. The exam opens one chapter of a standard at a time.

Going deeper

• Three-phase circuits, delta-wye and the 30° shift on the PE Power exam — sequence and the shift
• Symmetrical components and unbalanced faults on the PE Power exam — how faults connect the networks
• The per-unit system on the PE Power exam — bases and zones
• Fault current analysis on the PE Power exam — per unit and sequence networks put to use
• The PE Power handbook and code books — what is supplied, and how to be fast in it

Part of the PE Electrical: Power exam. Practice with exam-real problems — Castor, the AI tutor, walks any one you miss. → Start practicing Or take the free PE Power practice exam first.