Principles and Practice of Engineering Exam (PE) Electrical and Computer Engineering (PE Electrical) Syllabus

Define voltage regulation of a transmission line or transformer.

$\text{VR}\% = \dfrac{|V_{NL}| - |V_{FL}|}{|V_{FL}|}\times 100\%$, the rise in receiving-end voltage from full load to no load.

For an ideal transformer with turns ratio $a = N_1/N_2$, relate voltages, currents, and reflected impedance.

$\dfrac{V_1}{V_2} = a$, $\dfrac{I_1}{I_2} = \dfrac{1}{a}$, and impedance referred to primary $Z_1 = a^{2} Z_2$.

Name the two main loss components in a transformer and how each varies with load.

Core (iron) losses: hysteresis + eddy currents, essentially constant with voltage/frequency, independent of load. Copper ($I^{2}R$) losses: vary with the square of load current.

State the condition for maximum efficiency of a transformer in terms of its losses.

Maximum efficiency occurs when copper (variable) losses equal core (constant) losses: $I^{2}R = P_{core}$.

Give the synchronous speed of an AC machine in terms of frequency and poles, and define slip for an induction motor.

$n_s = \dfrac{120 f}{P}$ rpm. Slip: $s = \dfrac{n_s - n_r}{n_s}$, the fractional difference between synchronous and rotor speed.

What is the relationship between developed torque, power, and speed in a rotating machine?

$P = T\omega$, where $\omega = \dfrac{2\pi n}{60}$ rad/s for speed $n$ in rpm; thus $T = \dfrac{P}{\omega}$.

Compare a synchronous machine and an induction machine in terms of rotor speed relative to the field.

Synchronous machine: rotor rotates exactly at synchronous speed ($s=0$, separately excited rotor). Induction machine: rotor slips below synchronous speed ($s>0$) to induce rotor currents.

List the standard categories of protective relay operating principles by the quantity they sense.

Overcurrent (current magnitude), differential (current in vs out), distance/impedance (V/I ratio), directional (power-flow direction), and over/under-voltage or frequency relays.

Explain the principle of differential (87) protection for a transformer or bus.

It compares current entering and leaving a protected zone; under normal/through-fault conditions they balance, but an internal fault creates a difference current $I_{diff} = |I_{in} - I_{out}|$ that trips the relay.

In overcurrent coordination, what does the coordination time interval (CTI) ensure between upstream and downstream devices?

The CTI (typically 0.2-0.4 s) ensures the downstream (closer-to-fault) device operates first, giving selectivity so only the faulted section is isolated.

What are the three symmetrical components used to analyze unbalanced three-phase faults?

Positive-sequence, negative-sequence, and zero-sequence components. For a single line-to-ground fault the three sequence networks connect in series.

Give the formula for symmetrical three-phase fault current using per-unit Thevenin impedance.

$I_{fault,pu} = \dfrac{V_{pu}}{Z_{th,pu}}$ (often $\approx \dfrac{1.0}{Z_{th}}$), then multiply by base current to get amperes.

Which code governs electrical installations in the U.S., and what is its formal designation?

The National Electrical Code (NEC), published as NFPA 70, which sets minimum requirements for safe electrical wiring and equipment installation.

Per the NEC, define ampacity and the general continuous-load conductor sizing rule.

Ampacity is the maximum current a conductor can carry continuously without exceeding its temperature rating. Conductors/overcurrent devices must be sized for at least 125% of continuous load: $I \geq 1.25\,I_{continuous}$.

What is the purpose of equipment grounding versus system (neutral) grounding?

Equipment grounding bonds non-current-carrying metal to provide a fault-current path that trips protection and limits touch voltage. System grounding references the neutral to earth to stabilize voltage and control overvoltages.

Write the diode Shockley equation and state the ideal-diode approximation.

$I = I_S\left(e^{\frac{V}{nV_T}} - 1\right)$, with $V_T = \dfrac{kT}{q}\approx 26\ \text{mV}$ at room temperature. Ideal approximation: conducts when forward biased, blocks when reverse biased.

State the conditions for a BJT to operate in the active region and the relation between collector and base current.

Active region: base-emitter junction forward biased, base-collector reverse biased. $I_C = \beta I_B$ and $I_E = I_C + I_B = (\beta+1)I_B$.

Give the saturation-region drain current equation for an n-channel MOSFET.

$I_D = \dfrac{1}{2}\mu_n C_{ox}\dfrac{W}{L}\left(V_{GS} - V_{th}\right)^{2}$, valid when $V_{GS} > V_{th}$ and $V_{DS} \geq V_{GS} - V_{th}$.

For an ideal op-amp with negative feedback, state the two 'golden rules' and the inverting-amplifier gain.

Rules: (1) no current flows into the inputs; (2) the inputs are at the same voltage (virtual short). Inverting gain: $\dfrac{V_{out}}{V_{in}} = -\dfrac{R_f}{R_{in}}$.

State the Nyquist-Shannon sampling theorem.

To reconstruct a signal without aliasing, the sampling frequency must exceed twice the highest frequency component: $f_s > 2 f_{max}$ (the Nyquist rate is $2 f_{max}$).

Write the definition of the continuous-time Fourier transform and the Laplace transform.

Fourier: $X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t}\,dt$. Laplace: $X(s) = \int_{0}^{\infty} x(t)e^{-st}\,dt$, with $s = \sigma + j\omega$.

Define the convolution of an LTI system's input with its impulse response, and what it computes.

$y(t) = x(t)*h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)\,d\tau$. It gives the system output for any input from the impulse response $h(t)$.

For a feedback control system with forward gain $G$ and feedback $H$, write the closed-loop transfer function.

$\dfrac{Y(s)}{R(s)} = \dfrac{G(s)}{1 + G(s)H(s)}$. The characteristic equation is $1 + G(s)H(s) = 0$.

State the condition for stability of an LTI system in terms of pole locations (continuous time).

All poles of the transfer function must lie in the open left half of the s-plane (negative real parts). A pole in the right half-plane makes the system unstable.

For a standard second-order system, relate damping ratio $\zeta$ to the response type and write the natural/damped frequency relation.

$\zeta < 1$ underdamped (oscillatory), $\zeta = 1$ critically damped, $\zeta > 1$ overdamped. Damped frequency: $\omega_d = \omega_n\sqrt{1-\zeta^{2}}$.

Define gain margin and phase margin as used in control-loop stability analysis.

Gain margin: additional gain (dB) at the phase-crossover frequency (where phase $=-180^\circ$) before instability. Phase margin: additional phase lag at the gain-crossover frequency (where $|GH|=1$) before reaching $-180^\circ$.

Compare amplitude modulation (AM) and frequency modulation (FM) and give the AM total transmitted power relation.

AM varies carrier amplitude with the message; FM varies carrier frequency. FM resists noise better but uses more bandwidth. AM power: $P_t = P_c\left(1 + \dfrac{m^{2}}{2}\right)$ for modulation index $m$.

State the Shannon-Hartley channel capacity theorem.

$C = B\log_2\left(1 + \dfrac{S}{N}\right)$ bits/s, where $B$ is bandwidth and $S/N$ the signal-to-noise power ratio. It sets the maximum error-free data rate.

Convert a logic gate count: write the Boolean theorems known as De Morgan's laws.

$\overline{A\cdot B} = \overline{A} + \overline{B}$ and $\overline{A + B} = \overline{A}\cdot\overline{B}$. They convert AND to OR (and vice versa) through complementation.

In digital systems, how many distinct states does an $n$-bit register hold, and what is the largest unsigned value it can store?

It holds $2^{n}$ distinct states, with maximum unsigned value $2^{n} - 1$.

State Amdahl's law for the speedup from parallelizing a fraction $p$ of a program across $N$ processors.

$S = \dfrac{1}{(1-p) + \dfrac{p}{N}}$; as $N\to\infty$, speedup is limited to $\dfrac{1}{1-p}$ by the serial fraction.