FE ECE Formula Sheet: Key Equations by Topic

Mathematics
11% of the exam

Algebra & trig, complex numbers, calculus, ODEs, linear algebra, vectors, discrete math.

Complex Numbers and Phasors

The three equivalent forms of a complex number, Euler's identity, complex arithmetic and roots, and the phasor representation that turns AC circuits into algebra.

Three equivalent forms: $z=a+jb=c\,\angle\theta=c\,e^{j\theta}$; Rectangular ($a$ real, $b$ imag), polar ($c$ magnitude, $\theta$ angle), exponential. Same point in the complex plane.
Magnitude and angle: $c=\sqrt{a^2+b^2},\qquad \theta=\arctan\!\big(\tfrac{b}{a}\big)\;(\text{quadrant-corrected})$; Rectangular to polar. Quadrant-correct $\theta$ from the signs of $a$ and $b$ (or use $\operatorname{atan2}(b,a)$).
Rectangular components: $a=c\cos\theta,\qquad b=c\sin\theta$; Polar to rectangular: real and imaginary parts of $c\angle\theta$.
Euler's identity: $e^{\pm j\theta}=\cos\theta\pm j\sin\theta$; Links exponential and trigonometric forms; the basis of phasors and Fourier analysis.
Addition / subtraction: $z_1\pm z_2=(a_1\pm a_2)+j(b_1\pm b_2)$; Use rectangular form. Combine real with real, imaginary with imaginary.
Multiplication / division: $z_1 z_2=c_1 c_2\,\angle(\theta_1+\theta_2),\qquad \tfrac{z_1}{z_2}=\tfrac{c_1}{c_2}\,\angle(\theta_1-\theta_2)$; Use polar form. Multiply/divide magnitudes, add/subtract angles.
Conjugate and magnitude product: $z^{*}=a-jb,\qquad z\,z^{*}=a^2+b^2=c^2$; Conjugate reflects across the real axis; product gives squared magnitude. Used to rationalize division.
De Moivre (powers): $z^{n}=c^{n}\,\angle(n\theta)=c^{n}(\cos n\theta+j\sin n\theta)$; Raise to integer power $n$ in polar form: magnitude to the power, angle times the power.
k-th roots: $z^{1/k}=c^{1/k}\,\angle\dfrac{\theta+360^\circ n}{k},\;\; n=0,\dots,k-1$; Every nonzero $z$ has exactly $k$ distinct roots, evenly spaced by $360^\circ/k$.
Sinusoid to phasor (cosine reference): $V_m\cos(\omega t+\phi)\;\leftrightarrow\;\mathbf{V}=V_m\angle\phi$; Phasor = complex amplitude. Convert sines via $\sin\theta=\cos(\theta-90^\circ)$ first.
Element impedances: $Z_R=R,\;\; Z_L=j\omega L,\;\; Z_C=\dfrac{1}{j\omega C}$; Phasor-domain impedances; $\omega=2\pi f$ in rad/s, $R$ in $\Omega$, $L$ in H, $C$ in F.
Phasor Ohm's law: $\mathbf{V}=\mathbf{I}\,Z$; Complex Ohm's law. $\angle Z$ is the phase by which voltage leads current.

Where it lives: NCEES FE Reference Handbook — Mathematics: Algebra of Complex Numbers and Polar Coordinate System · NCEES FE Reference Handbook — Mathematics: Euler's Identity and Roots · NCEES FE Reference Handbook — Electrical and Computer Engineering: AC Circuits

Read the Mathematics chapter

Probability and Statistics
4% of the exam

Central tendency & dispersion, probability distributions, expected value.

Central Tendency and Dispersion

Mean, median, and mode plus the variance, standard deviation, and coefficient of variation that quantify spread — and the population-versus-sample distinction the FE punishes.

Arithmetic mean: $\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i$; Balance point of the data. $X_i$ = ith observation, $n$ = number of observations; units match the data.
Weighted mean: $\bar{X}_w = \frac{\sum_i w_i X_i}{\sum_i w_i}$; Use when observations have unequal importance. $w_i$ = weight on $X_i$. Divide by the sum of weights, not the count.
Population variance: $\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(X_i-\mu)^2$; Spread of a complete population of size $N$ about its true mean $\mu$. Units are (data units)$^2$.
Sample variance: $s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2$; Unbiased estimate of $\sigma^2$ from a sample. The $n-1$ divisor (Bessel's correction) accounts for estimating $\mu$ by $\bar{X}$.
Standard deviation: $\sigma=\sqrt{\sigma^2},\qquad s=\sqrt{s^2}$; Square root of variance; same units as the data, which is why it is the reportable spread measure.
Coefficient of variation: $CV = \frac{s}{\bar{X}}$; Dimensionless relative dispersion; multiply by 100 for percent. Use $\sigma/\mu$ for a population. Requires a meaningful nonzero mean.
Computational sum-of-squares identity: $\sum_{i=1}^{n}(X_i-\bar{X})^2=\sum_{i=1}^{n}X_i^2-n\bar{X}^2$; One-pass shortcut for variance using $\sum X_i$ and $\sum X_i^2$; divide by $N$ or $n-1$ as appropriate.
Median (sorted data): $\tilde{X}=\begin{cases}X_{(n+1)/2}, & n\text{ odd}\\[2pt]\tfrac{1}{2}\!\left(X_{n/2}+X_{n/2+1}\right), & n\text{ even}\end{cases}$; Middle value of the ascending-sorted data; for even $n$ average the two central items. Resistant to outliers.
Sample range: $R = X_{\max}-X_{\min}$; Crudest dispersion measure: largest minus smallest value. Sensitive to a single outlier.

Where it lives: NCEES FE Reference Handbook — Engineering Probability and Statistics: Dispersion, Mean, Median, and Mode Values · Montgomery & Runger, Applied Statistics and Probability for Engineers

Read the Probability and Statistics chapter

Ethics and Professional Practice
4% of the exam

Codes of ethics, NCEES Model Law & Rules, intellectual property, safety.

Codes of Ethics and Professional Obligations

The NCEES Model Law/Rules hierarchy of duties — public safety paramount, competence, disclosure, and the seal — that every FE ethics question turns on.

Paramount duty: $\text{Obligation to the public} \succ \text{client/employer} \succ \text{self}$; The ordering that resolves nearly every dilemma. Safeguarding public health, safety, and welfare is first and foremost (Model Rules 240.15.A.1).
Competence rule: $\text{Accept work} \iff \text{qualified by education or experience}$; A licensee takes only assignments within their specific technical field (Model Rules 240.15.B.1).
Seal / responsible charge rule: $\text{Seal} \iff \text{prepared by or under responsible charge}$; Stamp only work you prepared or directly supervised with authority to change and full responsibility. Plan stamping is a disciplinary violation (Model Law 150.10.A.10).
Conflict-of-interest rule: $\text{Conflict} \Rightarrow \text{disclose to all affected parties}$; Disclose any known or potential conflict that could influence or appear to influence judgment (Model Rules 240.15.B.6).
Single-project compensation rule: $\text{Pay from} > 1 \text{ party} \Rightarrow \text{written consent of all}$; No compensation from more than one party for the same project unless fully disclosed and agreed in writing (Model Rules 240.15.B.7).
Confidentiality rule: $\text{Disclose client data} \iff \text{consent or required by law}$; Do not reveal professionally obtained facts/data without prior consent, except as authorized or required by law (Model Rules 240.15.B.4).
Public-opinion rule: $\text{Public opinion} \iff \text{adequate facts} \wedge \text{competent evaluation}$; Speak publicly on engineering matters only on an adequate factual basis (Model Rules 240.15.A.5); separately, identify interested parties who sponsor a statement (240.15.A.6).
Licensure path: $\text{Degree} \to \text{FE} \to \text{EI} \to \text{experience} \to \text{PE} \to \text{seal}$; Model Law 130.10 sequence: board-accepted (ABET) degree, pass FE to become Engineer Intern, board-set progressive experience (four years after an EAC/ABET bachelor's), five references, pass PE; only a PE may take responsible charge and seal.

Where it lives: NCEES FE Reference Handbook — Ethics and Professional Practice: Code of Ethics (Model Rules §240.15, Rules of Professional Conduct) · NCEES FE Reference Handbook — Ethics and Professional Practice: Model Law §110.20 (Definitions: Responsible Charge), §130.10 (General Requirements for Licensure), §150.10 (Grounds for Disciplinary Action) · NSPE Code of Ethics for Engineers

Read the Ethics and Professional Practice chapter

Engineering Economics
5% of the exam

Time value of money, cost estimation, risk, break-even and benefit-cost analysis.

Time Value of Money

Move money through time with the six compound-interest factors, convert nominal to effective rates, and read a cash-flow diagram the way the FE wants.

Single payment compound amount (F/P): $F = P(1+i)^n$; Grows a present sum $P$ to future $F$. $i$ = rate per period, $n$ = number of periods.
Single payment present worth (P/F): $P = F(1+i)^{-n}$; Discounts a future sum $F$ to present $P$. Reciprocal of $F/P$.
Capital recovery (A/P): $\left(\tfrac{A}{P},i,n\right)=\dfrac{i(1+i)^n}{(1+i)^n-1}$; Level payment $A$ that repays present $P$ over $n$ periods; the loan-payment factor.
Uniform series present worth (P/A): $\left(\tfrac{P}{A},i,n\right)=\dfrac{(1+i)^n-1}{i(1+i)^n}$; Present worth of $n$ equal end-of-period amounts $A$.
Sinking fund (A/F): $\left(\tfrac{A}{F},i,n\right)=\dfrac{i}{(1+i)^n-1}$; Deposit per period needed to accumulate future target $F$.
Series compound amount (F/A): $\left(\tfrac{F}{A},i,n\right)=\dfrac{(1+i)^n-1}{i}$; Future worth of $n$ equal deposits $A$.
Capital recovery identity: $\left(\tfrac{A}{P},i,n\right)=\left(\tfrac{A}{F},i,n\right)+i$; Capital recovery equals sinking-fund deposit plus interest on principal; a fast EUAC builder and an algebra check.
Uniform gradient to series (A/G): $\left(\tfrac{A}{G},i,n\right)=\dfrac{1}{i}-\dfrac{n}{(1+i)^n-1}$; Converts an arithmetic gradient $G$ (0, G, 2G, ...) into an equivalent uniform series.
Effective annual rate: $i_e=\left(1+\dfrac{r}{m}\right)^{m}-1$; $r$ = nominal annual rate, $m$ = compounding periods per year. Rate per period is $r/m$.
Continuous compounding: $i_e=e^{r}-1$; Limit of effective rate as $m\to\infty$; use when interest compounds continuously.
Capitalized cost (perpetuity): $P=\dfrac{A}{i}$; Present worth of a perpetual uniform series $A$ at rate $i$; used for endless maintenance/replacement.

Where it lives: NCEES FE Reference Handbook — Engineering Economics: Factor Table and Nomenclature · NCEES FE Reference Handbook — Engineering Economics: Non-Annual Compounding · NCEES FE Reference Handbook — Engineering Economics: Interest Rate Tables

Read the Engineering Economics chapter

Properties of Electrical Materials
4% of the exam

Semiconductors, conductivity & resistivity, permittivity, permeability, thermal properties.

Electrical Properties of Materials

Resistivity and conductivity, temperature coefficients, permittivity and dielectrics, permeability and magnetic cores, and the conductor-vs-insulator scale the FE leans on.

Resistance from resistivity: $R = \frac{\rho L}{A}$; $\rho$ = resistivity $(\Omega\!\cdot\!\text{m})$, $L$ = length (m), $A$ = cross-section (m²). Gives $R$ in $\Omega$.
Conductivity: $\sigma = \frac{1}{\rho}$; Reciprocal of resistivity, in S/m. Copper $\approx5.8\times10^7\,\text{S/m}$.
Temperature dependence of resistivity: $\rho = \rho_0\bigl[1 + \alpha(T - T_0)\bigr]$; $\rho_0$ at reference $T_0$; $\alpha$ = temperature coefficient $(/^\circ\text{C})$. Same form for $R$.
Temperature dependence of resistance: $R = R_0\bigl[1 + \alpha(T - T_0)\bigr]$; Direct device form; copper $\alpha\approx0.00393\,/^\circ\text{C}$ referenced to $20\,^\circ\text{C}$.
Parallel-plate capacitance: $C = \frac{\varepsilon A}{d} = \frac{\varepsilon_r\varepsilon_0 A}{d}$; $\varepsilon$ = permittivity (F/m), $A$ = plate area (m²), $d$ = separation (m).
Permittivity: $\varepsilon = \varepsilon_r\varepsilon_0, \qquad \varepsilon_0 = 8.85\times10^{-12}\ \text{F/m}$; $\varepsilon_r$ (= dielectric constant $\kappa$) is dimensionless; air $\approx1$.
Coil inductance: $L = \frac{N^2\mu A}{\ell} = \frac{N^2}{\mathcal{R}}$; $N$ = turns, $\mu$ = core permeability (H/m), $A$ = core area, $\ell$ = mean path length.
Permeability: $\mu = \mu_r\mu_0, \qquad \mu_0 = 4\pi\times10^{-7}\ \text{H/m}$; $\mu_r$ = relative permeability; $\approx1$ for non-magnetic, $10^3$–$10^5$ for iron/ferrite.
Reluctance: $\mathcal{R} = \frac{\ell}{\mu A}$; Magnetic-circuit resistance $(\text{H}^{-1})$; flux $\phi = \mathcal{F}/\mathcal{R}$ with mmf $\mathcal{F}=NI$.
Sheet resistance: $R = R_s\frac{L}{W}, \qquad R_s = \frac{\rho}{t}$; Thin-film/diffused/trace layer: $R_s$ in $\Omega/\square$, $t$ = thickness, $L/W$ = number of squares.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Resistivity, Capacitors and Inductors · NCEES FE Reference Handbook — Materials Science/Structure of Matter: Electrical Properties · IEEE/ASTM standard conductor tables

Read the Properties of Electrical Materials chapter

Circuit Analysis
11% of the exam

KCL/KVL, series-parallel, Thevenin & Norton, node/loop analysis, waveforms, phasors, impedance.

Phasors, Impedance, and AC Power

Phasor analysis, R/L/C impedance, series-parallel combination, and the P/Q/S power triangle — the core AC toolkit for the FE-ECE exam.

Phasor transform: $X_{\max}\cos(\omega t+\phi)\;\leftrightarrow\;X\angle\phi$; Maps a sinusoid at frequency $\omega$ to a complex phasor. Magnitude is peak or RMS by context; use RMS for power.
Impedance: $Z=\frac{\mathbf{V}}{\mathbf{I}}=R+jX$; Ratio of voltage to current phasors. $R$ = resistance, $X$ = reactance, both in ohms.
Inductor / capacitor reactance: $X_L=\omega L,\qquad X_C=\frac{1}{\omega C}$; $Z_L=+jX_L$ (current lags $90^\circ$); $Z_C=-jX_C$ (current leads $90^\circ$). $\omega=2\pi f$ in rad/s.
Admittance: $Y=\frac{1}{Z}=G+jB$; Reciprocal of impedance. $G$ = conductance, $B$ = susceptance, in siemens. Adds in parallel.
Series / parallel impedance: $Z_s=\sum Z_k,\qquad Z_p=\frac{Z_1 Z_2}{Z_1+Z_2}$; Impedances add in series; product-over-sum for two in parallel (or sum admittances).
Real power: $P=V_{\text{rms}}I_{\text{rms}}\cos\theta=I_{\text{rms}}^2 R$; Average power dissipated (W). $\theta$ = angle from voltage to current = impedance angle.
Reactive power: $Q=V_{\text{rms}}I_{\text{rms}}\sin\theta=I_{\text{rms}}^2 X$; Reactive power (var). Positive for inductive (lagging), negative for capacitive (leading) loads.
Apparent power: $S=V_{\text{rms}}I_{\text{rms}}=\sqrt{P^2+Q^2}$; Magnitude of complex power (VA). Always $S\ge P$.
Complex power: $\mathbf{S}=\mathbf{V}\mathbf{I}^*=P+jQ$; $\mathbf{I}^*$ = conjugate of current phasor. Real part is $P$, imaginary part is $Q$.
Power factor: $\text{pf}=\cos\theta=\frac{P}{S}$; Lagging (inductive) when current lags voltage; leading (capacitive) when it leads. Dimensionless, 0 to 1.
Power from impedance: $\mathbf{S}=I_{\text{rms}}^2\,Z=\frac{V_{\text{rms}}^2}{Z^*}$; Direct complex power from load impedance; convenient when only $Z$ and one of $V$ or $I$ is known.
Power-factor correction: $Q_C=P\left(\tan\theta_1-\tan\theta_2\right)$; Capacitor reactive power (kvar) to move from $\theta_1$ to a better $\theta_2$ at fixed real power $P$.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: AC Circuits (Phasor Transforms, Impedance, Complex Power) · NCEES FE Reference Handbook — Mathematics: Complex Numbers / Euler's Identity · Hayt, Kemmerly & Durbin, Engineering Circuit Analysis

Read the Circuit Analysis chapter

Linear Systems
5% of the exam

Frequency and transient response, resonance, Laplace transforms, transfer functions.

Laplace Transforms and Transfer Functions

Move circuits and systems into the s-domain, build the transfer function H(s), read its poles and zeros, and pull endpoints out with the initial- and final-value theorems.

Laplace transform definition: $F(s) = \int_0^\infty f(t)\,e^{-st}\,dt$; Unilateral transform for $t\ge0$. $s=\sigma+j\omega$ is complex frequency.
Core transform pairs: $\delta(t)\!\to\!1,\;\; u(t)\!\to\!\tfrac{1}{s},\;\; t\,u(t)\!\to\!\tfrac{1}{s^2},\;\; e^{-at}\!\to\!\tfrac{1}{s+a}$; Impulse, unit step, ramp, decaying exponential. The building blocks for partial-fraction inversion.
Sinusoid pairs: $\sin(bt)\!\to\!\dfrac{b}{s^2+b^2}, \quad \cos(bt)\!\to\!\dfrac{s}{s^2+b^2}$; Damped versions shift $s\to s+a$: $e^{-at}\sin bt\to b/((s+a)^2+b^2)$.
Differentiation property: $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$; Turns an ODE into algebra and carries the initial condition. Second derivative: $s^2F(s)-sf(0)-f'(0)$.
Integration and time shift: $\mathcal{L}\Big\{\!\int_0^t f\,d\tau\Big\}=\dfrac{F(s)}{s}, \;\; \mathcal{L}\{f(t-a)u(t-a)\}=e^{-as}F(s)$; Integration divides by $s$; a delay $a$ multiplies by $e^{-as}$ (the dead-time factor).
s-domain impedances: $Z_R=R, \quad Z_L=sL, \quad Z_C=\dfrac{1}{sC}$; Generalized impedances. Apply KVL/KCL/dividers as in DC; $s\to j\omega$ recovers phasor impedance.
Transfer function: $H(s)=\dfrac{Y(s)}{X(s)}\Big|_{\text{zero IC}}$; Output-to-input ratio with zero initial conditions. Output for any input: $Y(s)=H(s)X(s)$.
Pole-zero (factored) form: $H(s)=K\dfrac{\prod(s-z_i)}{\prod(s-p_j)}$; Zeros $z_i$ from numerator roots, poles $p_j$ from denominator roots; poles set stability and natural response.
Initial-value theorem: $f(0^+)=\lim_{s\to\infty} sF(s)$; Gives the starting value directly from $F(s)$. Valid when $F(s)$ is strictly proper (no impulse at $t=0$, so $sF(s)$ has a finite $s\to\infty$ limit).
Final-value theorem: $f(\infty)=\lim_{s\to 0} sF(s)$; Gives the steady-state value — ONLY if all poles of $sF(s)$ are in the open left half-plane.
DC gain: $H(0)=\lim_{s\to0}H(s)$; Steady-state gain to a constant input; equals the final value of the step response of a stable system.
Frequency response: $H(j\omega)=H(s)\big|_{s=j\omega}$; Steady-state sinusoidal transfer: magnitude scales amplitude, angle shifts phase. Basis of Bode plots.

Where it lives: NCEES FE Reference Handbook — Mathematics: Laplace Transforms · NCEES FE Reference Handbook — Electrical and Computer Engineering: AC Circuits · NCEES FE Reference Handbook — Instrumentation, Measurement, and Control: Transfer Functions · Nilsson & Riedel, Electric Circuits

Read the Linear Systems chapter

Signal Processing
5% of the exam

Sampling and aliasing, Nyquist, analog filters, digital filters and Z-transforms.

Sampling and the Nyquist Theorem

Why sampling above twice the highest frequency is mandatory, how aliasing folds energy down, and how anti-alias filtering and reconstruction recover the original signal.

Sampling interval and rate: $f_s = \dfrac{1}{T_s}$; $T_s$ = time between samples (s); $f_s$ = sample rate (samples/s = Hz). The fundamental link for every sampling problem.
Sampling theorem (Nyquist criterion): $f_s > 2 f_{max}$; Sample strictly faster than twice the highest frequency $f_{max}$ present in the band-limited signal to avoid aliasing. Handbook 'Sampling' note: $f_s > 2 f_N$.
Nyquist rate: $f_{NR} = 2 f_{max}$; Minimum acceptable sample rate (Hz). A property of the signal. Used when a problem asks for the smallest $f_s$.
Folding (Nyquist) frequency: $f_{fold} = \dfrac{f_s}{2}$; Highest frequency a chosen $f_s$ can represent without ambiguity (Hz). A property of the sampler. Frequencies above it alias.
Alias frequency: $f_{alias} = \left| f - k f_s \right|, \;\; k = \operatorname{round}\!\left(\tfrac{f}{f_s}\right)$; Apparent (baseband) frequency that a component $f > f_s/2$ takes after sampling. $k$ is the nearest integer multiple of $f_s$ to $f$.
First-zone alias (folding): $f_{alias} = f_s - f \quad (f_s/2 < f < f_s)$; Quick form when $f$ lies in the first zone above the fold: the apparent frequency is the mirror image about $f_s/2$.
Anti-alias filter cutoff: $f_{c} \le \dfrac{f_s}{2}$; Analog low-pass cutoff placed before the ADC; attenuates content above the folding frequency so it cannot alias into the band of interest.
Ideal reconstruction (sinc interpolation): $x(t) = \sum_{n} x[n]\,\operatorname{sinc}\!\left(\dfrac{t-nT_s}{T_s}\right)$; Whittaker–Shannon formula; exactly recovers a band-limited $x(t)$ from its samples when the sampling theorem was satisfied.
ADC voltage resolution: $\Delta V = \dfrac{V_H - V_L}{2^{n}}$; Quantization step of an $n$-bit converter over range $[V_L, V_H]$ (V). Sampling sets the time axis of the digitized signal; this companion handbook formula sets the amplitude axis.

Where it lives: NCEES FE Reference Handbook — Instrumentation, Measurement, and Control: Sampling · NCEES FE Reference Handbook — Electrical and Computer Engineering: Sampled Messages · Oppenheim & Schafer, Discrete-Time Signal Processing

Read the Signal Processing chapter

Electronics
7% of the exam

Diodes and transistors, amplifiers, operational amplifiers, instrumentation, power electronics.

Operational Amplifiers

Use the two golden rules of the ideal op-amp to solve inverting, noninverting, summing, difference, integrator, and differentiator circuits, then bound them with gain-bandwidth and slew rate.

Ideal op-amp golden rules: $i_+=i_-=0,\qquad v_+=v_-$; Linear region with negative feedback: no input current, zero differential input (virtual short). Solve via KCL at the inverting node.
Open-loop relation: $v_o=A(v_1-v_2),\quad A\to\infty$; Defines the device; finite differential input × huge gain ⇒ $v_1\approx v_2$ when not saturated. $v_1$ = noninverting, $v_2$ = inverting.
Inverting amplifier: $v_o=-\frac{R_f}{R_1}\,v_{in}$; Noninverting input grounded; gain set by resistor ratio, negative sign. Input resistance to source = $R_1$.
Noninverting amplifier: $v_o=\left(1+\frac{R_f}{R_1}\right)v_{in}$; Signal at noninverting input; gain ≥ 1, ideally infinite input impedance. $R_f=0$ gives a unity buffer.
Inverting summer: $v_o=-R_f\left(\frac{v_1}{R_1}+\frac{v_2}{R_2}+\cdots\right)$; Weighted inverted sum; each input scaled by $R_f/R_k$. Inputs are isolated by the virtual ground.
Difference amplifier: $v_o=\frac{R_f}{R_1}\,(v_b-v_a)$; Valid with matched ratios ($R_f/R_1$ on both inverting and noninverting sides). Rejects common-mode signal.
Integrator: $v_o(t)=-\frac{1}{RC}\int v_{in}\,dt + v_o(0)$; Capacitor in feedback. Output = inverted time-integral; $RC$ sets the time constant.
Differentiator: $v_o(t)=-RC\,\frac{dv_{in}}{dt}$; Capacitor at input, resistor feedback. Output ∝ rate of change; noise-prone, usually damped.
Gain-bandwidth product: $\text{GBW}=A_{CL}\,f_{-3\text{dB}}=f_t$; Constant equal to unity-gain frequency $f_t$. Bandwidth $=f_t/A_{CL}$; higher gain ⇒ less bandwidth (small-signal).
Slew rate limit: $\left.\frac{dv_o}{dt}\right|_{\max}=SR$; Maximum output rate of change (V/μs). A large-signal limit independent of GBW; exceeding it distorts the waveform.
Full-power bandwidth: $f_{\max}=\frac{SR}{2\pi V_p}$; Highest frequency at which a sinusoid of peak $V_p$ can be output without slew distortion.
CMRR: $\text{CMRR}=\left|\frac{A_d}{A_{cm}}\right|,\quad \text{CMRR}_{dB}=20\log_{10}\left|\frac{A_d}{A_{cm}}\right|$; Ratio of differential to common-mode gain; infinite for an ideal op-amp ($A_{cm}=0$). Measures rejection of common-mode input.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Operational Amplifiers · NCEES FE Reference Handbook — Electrical and Computer Engineering: Electronics · Sedra & Smith, Microelectronic Circuits

Read the Electronics chapter

Power Systems
8% of the exam

Power theory, transmission & distribution, transformers, motors and generators.

AC Power and Three-Phase Systems

Real, reactive, apparent, and complex power; power factor and correction; wye/delta and the √3 relations that drive nearly every power-systems question on the FE.

Real (average) power: $P=V_{rms}I_{rms}\cos\theta=\tfrac{1}{2}V_{max}I_{max}\cos\theta$; Average power in watts; $\theta$ is the angle from $\mathbf{V}$ to $\mathbf{I}$ (the impedance angle). Use RMS values unless peak values are explicitly given.
Reactive power: $Q=V_{rms}I_{rms}\sin\theta$; In vars. Positive (absorbed) for inductive/lagging loads, negative for capacitive/leading loads.
Complex and apparent power: $\mathbf{S}=\mathbf{V}\mathbf{I}^{*}=P+jQ,\qquad S=|\mathbf{S}|=V_{rms}I_{rms}$; $\mathbf{I}^{*}$ is the conjugate of the phasor current; $S$ in VA is what conductors and transformers are sized to.
Power triangle / power factor: $S=\sqrt{P^{2}+Q^{2}},\qquad pf=\cos\theta=\frac{P}{S}$; Lagging pf = inductive ($I$ lags $V$); leading pf = capacitive ($I$ leads $V$).
Power factor correction: $Q_C=P\left(\tan\theta_1-\tan\theta_2\right)$; Capacitor reactive power (var) to move from initial angle $\theta_1$ to target $\theta_2$ at constant real power $P$.
Wye line-phase relations: $V_{LL}=\sqrt{3}\,V_{LN}\angle{+30^\circ},\qquad I_L=I_\phi$; Star connection; line-to-line voltage leads line-to-neutral by $30^\circ$ (positive/ABC sequence).
Delta line-phase relations: $V_{LL}=V_\phi,\qquad I_L=\sqrt{3}\,I_\phi\angle{-30^\circ}$; Delta connection; line current is $\sqrt{3}$ times phase current and lags it by $30^\circ$ (positive sequence).
Balanced three-phase power: $S=\sqrt{3}\,V_{LL}I_L=3V_\phi I_\phi$; Same total from line or phase quantities; $P=S\cos\theta$, $Q=S\sin\theta$ with $\theta$ the load impedance angle.
Delta-to-wye impedance: $Z_Y=\frac{Z_\Delta}{3}$; Balanced loads only; lets you reduce a delta load to per-phase wye analysis.
Per-phase voltage: $V_{LN}=\frac{V_{LL}}{\sqrt{3}}$; $480\,\text{V}\to277\,\text{V}$; $208\,\text{V}\to120\,\text{V}$; $13.8\,\text{kV}\to7.97\,\text{kV}$. Used for single-phase (per-phase) equivalents.
Resistive power: $P=\frac{V_{rms}^{2}}{R}=I_{rms}^{2}R$; For a pure resistor $\theta=0$, so $Q=0$ and $S=P$; useful as a sanity check.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: AC Power (Complex Power) · NCEES FE Reference Handbook — Electrical and Computer Engineering: Balanced Three-Phase (3-φ) Systems · Irwin & Nilsson, Electric Circuits

Read the Power Systems chapter

Electromagnetics
4% of the exam

Electrostatics, magnetostatics, Maxwell's equations, wave propagation, transmission lines.

Electrostatics and Magnetostatics

Coulomb and Gauss for E-fields, potential and capacitance, then Biot-Savart and Ampere for B-fields, inductance, and the energy stored in each field.

Coulomb force: $\mathbf{F}_2=\dfrac{Q_1 Q_2}{4\pi\varepsilon r^{2}}\,\mathbf{a}_{r12}$; Force on charge 2 from charge 1; $r$ in m, $Q$ in C, $\varepsilon=\varepsilon_r\varepsilon_0$ (F/m). Positive product = repulsive (along $\mathbf{a}_{r12}$).
Point-charge E field: $\mathbf{E}=\dfrac{Q}{4\pi\varepsilon r^{2}}\,\mathbf{a}_r,\quad \mathbf{F}=Q\mathbf{E}$; $E$ in V/m, radial, $\propto 1/r^2$. Multiply by a test charge to get force.
Line and sheet fields: $E_{\text{line}}=\dfrac{\rho_L}{2\pi\varepsilon r},\quad E_{\text{sheet}}=\dfrac{\rho_s}{2\varepsilon}$; Infinite line charge $\rho_L$ (C/m) falls as $1/r$; infinite sheet $\rho_s$ (C/m$^2$) is uniform.
Gauss's law: $\oint_S \mathbf{D}\cdot d\mathbf{S}=Q_{\text{enc}},\quad \mathbf{D}=\varepsilon\mathbf{E}$; Flux of $\mathbf{D}$ (C/m$^2$) through a closed surface equals enclosed charge (C). Use with symmetric Gaussian surfaces.
Parallel-plate E field: $E=\dfrac{V}{d}$; Uniform field between plates, directed from $+$ to $-$; $V$ in volts, $d$ in m, $E$ in V/m.
Parallel-plate capacitance: $C=\dfrac{\varepsilon A}{d}=\dfrac{\varepsilon_r\varepsilon_0 A}{d}$; $A$ = plate area (m$^2$), $d$ = spacing (m), $\varepsilon_r$ = dielectric constant; $C$ in farads.
Capacitor stored energy: $W=\tfrac{1}{2}Cv^{2}=\dfrac{q^{2}}{2C}=\tfrac{1}{2}qv$; Three equivalent forms in joules with $q=Cv$.
Electric / magnetic field energy: $W_E=\int_V \tfrac{1}{2}\varepsilon E^{2}\,dV,\quad W_H=\int_V \tfrac{1}{2}\mu H^{2}\,dV$; Energy as field integrals over a volume; energy densities $\tfrac12\varepsilon E^2$ (J/m$^3$) and $\tfrac12\mu H^2$.
Long-wire magnetic field: $H=\dfrac{I}{2\pi r},\quad \mathbf{B}=\mu\mathbf{H}$; $H$ in A/m circles the wire (right-hand rule); $B=\mu H$ in tesla; $r$ from the wire axis (m).
Ampere's law: $\oint \mathbf{H}\cdot d\mathbf{l}=I_{\text{enc}}$; Circulation of $\mathbf{H}$ around a closed loop equals enclosed current; the fast path for symmetric geometries.
Force on a conductor: $\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}$; Force on length $\mathbf{L}$ of conductor carrying current $I$ in flux density $\mathbf{B}$; magnitude $BIL\sin\theta$ in N.
Inductance: $L=\dfrac{N^{2}\mu A}{\ell}=\dfrac{N^{2}}{\mathcal{R}},\quad \mathcal{R}=\dfrac{\ell}{\mu A}$; $N$ turns, core area $A$ (m$^2$), mean path $\ell$ (m), permeability $\mu=\mu_r\mu_0$; reluctance $\mathcal{R}$ in H$^{-1}$.
Inductor stored energy: $W=\tfrac{1}{2}Li^{2}$; Magnetic energy in joules; the dual of $\tfrac{1}{2}Cv^2$.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Electrostatics / Electrostatic Fields · NCEES FE Reference Handbook — Electrical and Computer Engineering: Magnetic Fields and Capacitors/Inductors · Hayt & Buck, Engineering Electromagnetics

Read the Electromagnetics chapter

Control Systems
6% of the exam

Block diagrams, Bode plots, closed-loop stability, steady-state error, transient performance.

Closed-Loop Response and Stability

Find pole locations from the characteristic equation, test stability with Routh-Hurwitz and root locus, and tie BIBO stability to margins.

Closed-loop characteristic equation: $1 + G_1(s)G_2(s)H(s) = 0$; Roots are the closed-loop poles. For unity feedback, $1+G(s)=0$. Set the denominator of $T(s)$ to zero.
Closed-loop transfer function: $T(s) = \dfrac{G(s)}{1+G(s)H(s)}$; Poles = roots of $1+GH$; these (not the open-loop poles) determine the closed-loop response.
BIBO stability condition: $\text{Re}(p_i) < 0 \quad \forall\, i$; All closed-loop poles strictly in the left half-plane. A $j\omega$-axis pole is marginal; any RHP pole is unstable.
Routh array entry: $b_i = \dfrac{a_{n-1}a_{n-2i} - a_n a_{n-2i-1}}{a_{n-1}}$; Each entry is the negative of the $2\times 2$ determinant of the two rows above, divided by the pivot $a_{n-1}$ — i.e. cross-multiply in the order shown in the latex (pivot-row column term minus top-row column term). Continue until the $s^0$ row.
Routh stability test: $\text{stable} \iff \text{first column all same sign}$; Number of first-column sign changes = number of right-half-plane roots.
Necessary coefficient condition: $a_n, a_{n-1}, \dots, a_0 > 0 \ \text{(all present)}$; Necessary (not sufficient): any missing or non-positive coefficient means at least one non-LHP root.
Auxiliary equation (marginal case): $A(s) = c\,s^2 + d = 0 \;\Rightarrow\; \omega_{\text{osc}} = \sqrt{d/c}$; Formed from the row above a zeroed first-column entry; gives the sustained-oscillation frequency at $K_{\text{marg}}$.
Root-locus condition: $1 + K\dfrac{N(s)}{D(s)} = 0$; Loci start at open-loop poles ($K=0$), end at open-loop zeros or infinity ($K\to\infty$).
Gain margin (dB): $\text{GM}_{\text{dB}} = -20\log_{10}\big|G(j\omega_{180})\big| = 20\log_{10}\dfrac{K_{\text{marg}}}{K_{\text{design}}}$; Handbook dB form, evaluated at the phase-crossover $\omega_{180}$. The linear gain margin is the bare factor $K_{\text{marg}}/K_{\text{design}}$; the dB value is $20\log_{10}$ of that ratio (e.g. a factor of $5\approx 14$ dB). Positive dB = stable, minimum-phase.

Where it lives: NCEES FE Reference Handbook — Instrumentation, Measurement, and Control: Control Systems · NCEES FE Reference Handbook — Electrical and Computer Engineering: Decibels and Bode Plots · Ogata, Modern Control Engineering

Read the Control Systems chapter

Communications
5% of the exam

AM/FM/PCM modulation, Fourier methods, multiplexing, digital communications.

Analog Modulation

AM and FM/PM, modulation index, Carson-rule bandwidth, and power distribution — the formula family behind nearly every analog-comms question on the FE.

Conventional AM signal: $x_{AM}(t)=A_c\bigl[1+a\,m_n(t)\bigr]\cos(2\pi f_c t)$; $a$ = modulation index ($\le 1$ to avoid overmodulation); $m_n(t)=m(t)/\max|m(t)|$ is the normalized message; $A_c$, $f_c$ = carrier amplitude and frequency.
Modulation index from envelope: $a=\frac{E_{max}-E_{min}}{E_{max}+E_{min}}$; Reads $a$ directly off the maximum and minimum envelope amplitudes of a single-tone AM waveform (dimensionless).
AM total power: $P_t=P_c\!\left(1+\frac{a^2}{2}\right)$; Single-tone AM. $P_c$ = unmodulated carrier power (W); the two sidebands together carry $P_c a^2/2$.
AM efficiency: $\eta=\frac{a^2}{2+a^2}$; Fraction of transmitted power in the sidebands (information). Maximum $33.3\%$ at $a=1$.
DSB-SC signal: $x_{DSB}(t)=A_c\,m(t)\cos(2\pi f_c t)$; Suppressed carrier; all power in sidebands; needs synchronous (coherent) demodulation. Bandwidth $2W$.
Bandwidths of AM variants: $B_{AM}=B_{DSB}=2W,\qquad B_{SSB}=W$; $W$ = message bandwidth (Hz). SSB occupies half the spectrum of DSB/AM.
Angle-modulated carrier: $x_{Ang}(t)=A_c\cos\!\bigl(2\pi f_c t+\phi(t)\bigr)$; $\phi(t)$ = phase deviation set by the message. PM: $\phi=k_P m(t)$; FM: $\phi=k_F\!\int m\,d\lambda$.
Instantaneous frequency (FM): $f_i(t)=f_c+\frac{1}{2\pi}\frac{d\phi}{dt}$; Peak excursion of $f_i$ from $f_c$ is the peak frequency deviation $\Delta f$ (Hz).
FM deviation ratio: $D=\frac{\Delta f}{W}$; $\Delta f$ = peak frequency deviation (Hz); $W$ = message bandwidth (Hz). For single tone, $D$ equals the modulation index $\beta$.
Carson's rule (wideband FM): $B\approx 2(D+1)\,W=2(\Delta f+W)$; 98%-power bandwidth for $D>1$. Reduces to $B\approx 2W$ for narrowband FM ($D\ll 1$).

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Amplitude Modulation, Angle Modulation, Frequency Modulation (FM) · Haykin, Communication Systems · Lathi & Ding, Modern Digital and Analog Communication Systems

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Computer Networks
4% of the exam

Routing & switching, topologies, LAN/WAN, OSI & TCP/IP models, network security.

OSI and TCP/IP Models (with Network Topologies)

The seven-layer OSI stack, the four-layer TCP/IP stack, encapsulation into PDUs, and protocols per layer (FE subarea D), plus the star/bus/ring/mesh physical topologies that belong to FE subarea B.

OSI layer order (top to bottom): $\text{App} \to \text{Pres} \to \text{Sess} \to \text{Trans} \to \text{Net} \to \text{Link} \to \text{Phys}$; The seven OSI layers, layer 7 down to layer 1. Mnemonic: All People Seem To Need Data Processing.
TCP/IP layer order: $\text{Application} \to \text{Transport} \to \text{Internet} \to \text{Network Interface}$; The four TCP/IP layers, top to bottom; the practical model used on the internet.
OSI to TCP/IP mapping: $\{7,6,5\}\!\to\!\text{App},\;4\!\to\!\text{Trans},\;3\!\to\!\text{Internet},\;\{2,1\}\!\to\!\text{Net.\ Interface}$; TCP/IP Application = OSI App+Pres+Sess; Network Interface = OSI Data Link+Physical; Transport and Internet map one-to-one.
PDU by layer: $\text{message} \to \text{segment} \to \text{packet} \to \text{frame} \to \text{bits}$; Protocol Data Unit at Application, Transport, Network, Data Link, and Physical respectively. Packet = datagram.
Encapsulation rule: $\text{PDU}_{n} = \text{Header}_{n} \,\|\, \text{PDU}_{n+1}$; Each layer n prepends its header to the PDU from the layer above; $\|$ denotes concatenation. Reversed on receipt (decapsulation).
Device-to-layer map: $\text{hub}\!=\!L1,\quad \text{switch/bridge}\!=\!L2,\quad \text{router}\!=\!L3$; Physical, Data Link, and Network layers respectively; the layer a device reads to make forwarding decisions.
Full-mesh link count (subarea B): $L = \dfrac{n(n-1)}{2}$; Topology metric (FE subarea B, Network topologies): number of direct links in a full mesh of n nodes; quantifies the cost of full redundancy.
TCP vs UDP: $\text{TCP: connection-oriented, reliable};\;\; \text{UDP: connectionless, best-effort}$; TCP detects loss/duplication/reordering and corrects; UDP has less overhead but no delivery guarantee.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Computer Networks (OSI and TCP/IP models, encapsulation) · NCEES FE Electrical and Computer Exam Specification — Computer Networks subareas B (Network topologies: mesh, ring, star) and D (Network models: OSI, TCP/IP) · ISO/IEC 7498-1 — OSI Basic Reference Model · Tanenbaum & Wetherall, Computer Networks

Read the Computer Networks chapter

Digital Systems
8% of the exam

Number systems, Boolean logic, K-maps, flip-flops & counters, state machines, timing.

Boolean Algebra and Logic Gates

Master the Boolean axioms, De Morgan's theorems, the seven core gates, and the SOP/POS forms that turn any truth table into a circuit.

Identity laws: $A + 0 = A,\quad A\cdot 1 = A$; OR with 0 and AND with 1 leave a variable unchanged. The starting point for most simplifications.
Null (dominance) laws: $A + 1 = 1,\quad A\cdot 0 = 0$; OR with 1 forces 1; AND with 0 forces 0. Use to collapse a term to a constant.
Complement laws: $A + \overline{A} = 1,\quad A\cdot\overline{A} = 0$; A variable ORed with its complement is 1; ANDed with its complement is 0. Key to eliminating variables.
Idempotent and involution: $A + A = A,\quad A\cdot A = A,\quad \overline{\overline{A}} = A$; Repeated operands collapse; double inversion cancels. Lets you duplicate or remove terms freely.
Distributive laws: $A(B+C) = AB + AC,\quad A + BC = (A+B)(A+C)$; Both AND-over-OR and OR-over-AND hold. The second form has no arithmetic analog — used to convert between SOP and POS.
Absorption: $A + AB = A,\quad A(A+B) = A$; Deletes a term subsumed by a simpler one. The single most useful simplification on the exam.
Redundancy / simplification: $A + \overline{A}B = A + B,\quad A(\overline{A}+B) = AB$; Removes a complemented literal that adds nothing. Easy to miss; check whenever a variable and its complement coexist.
Consensus theorem: $AB + \overline{A}C + BC = AB + \overline{A}C$; The consensus term $BC$ is redundant when $A$ appears true in one product and complemented in another. Drop it.
De Morgan's theorems: $\overline{A\cdot B} = \overline{A} + \overline{B},\quad \overline{A+B} = \overline{A}\cdot\overline{B}$; Complement of a product/sum. Use to invert expressions and to map between AND/OR and NAND/NOR.
XOR and XNOR: $A\oplus B = A\overline{B} + \overline{A}B,\quad \overline{A\oplus B} = AB + \overline{A}\,\overline{B}$; XOR is true when inputs differ (mod-2 sum); XNOR is the equality/coincidence detector.
Canonical SOP / POS: $F = \textstyle\sum m_i = \prod M_j$; Sum of minterms for the 1-rows of the truth table, or product of maxterms for the 0-rows. Both specify the same function.
NAND as inverter: $\overline{A\cdot A} = \overline{A}$; Tie both NAND inputs together to make a NOT. Foundation of NAND-universal logic.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Logic Operations and Boolean Algebra · NCEES FE Reference Handbook — Electrical and Computer Engineering: De Morgan's Theorems · IEEE Std 91/91a — Graphic Symbols for Logic Functions · Mano & Ciletti, Digital Design

Read the Digital Systems chapter

Computer Systems
5% of the exam

Microprocessors, memory technology and hierarchy, interfacing.

Memory Technology and Hierarchy

SRAM/DRAM/ROM/flash, the cache-driven memory hierarchy, address decoding and cache mapping, virtual memory, and the effective-access-time math the FE rewards.

Average memory access time (AMAT): $AMAT = t_{hit} + (1-h)\,t_{penalty}$; $h$ = hit rate, $t_{hit}$ = fast-level access time, $t_{penalty}$ = extra time to service a miss. Hit time is always paid; the penalty only on misses.
Two-level cache AMAT: $AMAT = t_{L1} + m_{L1}\big(t_{L2} + m_{L2}\,t_{mem}\big)$; $m_{L1},m_{L2}$ are local miss rates. The L2 AMAT is nested as the L1 miss penalty; extend the same way for L3.
Cache capacity: $C = S\times A\times B$; $C$ = data capacity (bytes), $S$ = number of sets, $A$ = associativity (ways), $B$ = block size (bytes). From the FE handbook.
Cache address field widths: $\#offset=\log_2 B,\;\; \#index=\log_2 S,\;\; \#tag=n-\#index-\#offset$; $n$ = total address bits. Offset selects a byte in the block, index selects a set, tag is stored and compared.
Number of cache sets: $S = \dfrac{C}{A\times B}$; Solve the capacity relation for sets. Direct-mapped has $A=1$; fully associative has $S=1$.
Hit and miss rate: $h + m = 1$; $h$ = fraction of accesses found in the level, $m=1-h$ = fraction that miss. Dimensionless.
Address lines for a memory size: $\#\text{addr lines} = \log_2(\#\text{addressable words})$; A $W$-word chip needs $\log_2 W$ lines; a $T$-word bank needs $\log_2 T$ total lines.
Chip count and select lines: $N_{chips}=\dfrac{T}{W},\qquad \#\text{select} = \log_2\!\frac{T}{W}$; $T$ = total words, $W$ = words per chip. High-order lines decode to chip-selects; low-order $\log_2 W$ lines address within a chip.
Effective access time with page faults: $EAT = (1-p)\,t_{mem} + p\,t_{fault}$; $p$ = page-fault rate, $t_{mem}$ = in-memory access time, $t_{fault}$ = fault service time (disk). Same form as AMAT for virtual memory.
DRAM vs SRAM: $\text{SRAM: 6T latch, no refresh};\;\;\text{DRAM: 1T1C, periodic refresh}$; SRAM is faster/lower-density (cache); DRAM is denser/cheaper but must be refreshed (main memory). Both volatile.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Computer Systems (Memory/Storage Types, Cache, Tag/Index/Offset, C = S·A·B) · Patterson & Hennessy, Computer Organization and Design — The Memory Hierarchy · Hennessy & Patterson, Computer Architecture: A Quantitative Approach — Appendix B (Memory Hierarchy)

Read the Computer Systems chapter

Software Engineering
4% of the exam

Algorithms and complexity, data structures, iteration, recursion, control flow, testing.

Algorithms and Complexity

Read Big-O/Theta/Omega, compare searching and sorting costs, and solve recurrences so you can pick the right algorithm and predict how it scales.

Big-O (upper bound): $f(n)=O(g(n))\iff \exists\,c,n_0:\ f(n)\le c\,g(n)\ \forall n\ge n_0$; Asymptotic ceiling on growth; drop constants and lower-order terms. Used to state worst-case scaling of an algorithm.
Big-Omega (lower bound): $f(n)=\Omega(g(n))\iff \exists\,c,n_0:\ f(n)\ge c\,g(n)\ \forall n\ge n_0$; Asymptotic floor on growth; states the best the algorithm can do for large $n$.
Big-Theta (tight bound): $f(n)=\Theta(g(n))\iff f=O(g)\ \text{and}\ f=\Omega(g)$; Exact growth class. Merge/heap sort are $\Theta(n\log n)$ in all cases; use when both bounds match.
Growth hierarchy: $O(1)<O(\log n)<O(n)<O(n\log n)<O(n^2)<O(n^3)<O(2^n)<O(n!)$; Ordering for large $n$. Resolve any 'which is faster' question by placing each candidate on this ladder.
Binary search cost: $T(n)=T(n/2)+O(1)\Rightarrow O(\log_2 n)$; Comparisons to find/reject a key in a SORTED array of $n$ elements; at most $\lceil\log_2 n\rceil$.
Linear search cost: $T_{\text{worst}}=n,\qquad T_{\text{avg}}=\tfrac{n}{2}\ \to\ O(n)$; Scan of an unsorted array of $n$ elements; no preprocessing required.
Merge / heap sort: $T(n)=2T(n/2)+O(n)\Rightarrow \Theta(n\log n)$; Guaranteed linearithmic in every case. Merge sort needs $O(n)$ aux space; heap sort needs $O(1)$.
Quick sort: $T_{\text{avg}}=O(n\log n),\quad T_{\text{worst}}=O(n^2)$; Fast average pivot-partition sort; degrades to $O(n^2)$ on a poor pivot (e.g., sorted input, naive pivot).
Bubble / insertion sort: $T_{\text{avg}}=T_{\text{worst}}=O(n^2),\quad T_{\text{best,insertion}}=O(n)$; Simple $O(n^2)$ sorts; insertion sort is $O(n)$ on already-sorted data, the others' best case is still $O(n^2)$.
Master Theorem form: $T(n)=a\,T(n/b)+f(n)$; Solve divide-and-conquer recurrences by comparing $f(n)$ with $n^{\log_b a}$; equal magnitude adds a $\log n$ factor.
Naive recursive Fibonacci: $T(n)=T(n-1)+T(n-2)+O(1)=\Theta(\varphi^{\,n})$; Branching recursion without memoization is exponential; the iterative version is $O(n)$ time, $O(1)$ space.
Polynomial-time runtime scaling: $\frac{T(kn)}{T(n)}=k^{\,p}\quad\text{for } T(n)=\Theta(n^p)$; How runtime scales when input multiplies by $k$. For $O(n^2)$, doubling input quadruples time.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering: Algorithms and Algorithm Efficiency (Big-O) · NCEES FE Reference Handbook — Electrical and Computer Engineering: Software Syntax Guidelines · Cormen, Leiserson, Rivest, Stein — Introduction to Algorithms

Read the Software Engineering chapter

FE Electrical and Computer formula sheet

Mathematics
11% of the exam

Complex Numbers and Phasors

Probability and Statistics
4% of the exam

Central Tendency and Dispersion

Ethics and Professional Practice
4% of the exam

Codes of Ethics and Professional Obligations

Engineering Economics
5% of the exam

Time Value of Money

Properties of Electrical Materials
4% of the exam

Electrical Properties of Materials

Circuit Analysis
11% of the exam

Phasors, Impedance, and AC Power

Linear Systems
5% of the exam

Laplace Transforms and Transfer Functions

Signal Processing
5% of the exam

Sampling and the Nyquist Theorem

Electronics
7% of the exam

Operational Amplifiers

Power Systems
8% of the exam

AC Power and Three-Phase Systems

Electromagnetics
4% of the exam

Electrostatics and Magnetostatics

Control Systems
6% of the exam

Closed-Loop Response and Stability

Communications
5% of the exam

Analog Modulation

Computer Networks
4% of the exam

OSI and TCP/IP Models (with Network Topologies)

Digital Systems
8% of the exam

Boolean Algebra and Logic Gates

Computer Systems
5% of the exam

Memory Technology and Hierarchy

Software Engineering
4% of the exam

Algorithms and Complexity

Now use them on real questions

Mathematics11% of the exam

Complex Numbers and Phasors

Probability and Statistics4% of the exam

Central Tendency and Dispersion

Ethics and Professional Practice4% of the exam

Codes of Ethics and Professional Obligations

Engineering Economics5% of the exam

Time Value of Money

Properties of Electrical Materials4% of the exam

Electrical Properties of Materials

Circuit Analysis11% of the exam

Phasors, Impedance, and AC Power

Linear Systems5% of the exam

Laplace Transforms and Transfer Functions

Signal Processing5% of the exam

Sampling and the Nyquist Theorem

Electronics7% of the exam

Operational Amplifiers

Power Systems8% of the exam

AC Power and Three-Phase Systems

Electromagnetics4% of the exam

Electrostatics and Magnetostatics

Control Systems6% of the exam

Closed-Loop Response and Stability

Communications5% of the exam

Analog Modulation

Computer Networks4% of the exam

OSI and TCP/IP Models (with Network Topologies)

Digital Systems8% of the exam

Boolean Algebra and Logic Gates

Computer Systems5% of the exam

Memory Technology and Hierarchy

Software Engineering4% of the exam

Algorithms and Complexity

Now use them on real questions

Mathematics
11% of the exam

Probability and Statistics
4% of the exam

Ethics and Professional Practice
4% of the exam

Engineering Economics
5% of the exam

Properties of Electrical Materials
4% of the exam

Circuit Analysis
11% of the exam

Linear Systems
5% of the exam

Signal Processing
5% of the exam

Electronics
7% of the exam

Power Systems
8% of the exam

Electromagnetics
4% of the exam

Control Systems
6% of the exam

Communications
5% of the exam

Computer Networks
4% of the exam

Digital Systems
8% of the exam

Computer Systems
5% of the exam

Software Engineering
4% of the exam