FE Electrical and Computer formula sheet

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

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

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

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. Not printed in the FE Reference Handbook 10.6, which gives only non-annual compounding.

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

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

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 $P$. The FE Reference Handbook 10.6 prints the power triangle ($P$, $Q$, pf) but not this correction formula.

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

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

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

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

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 $P$. The FE Reference Handbook 10.6 prints the power triangle ($P$, $Q$, pf) but not this correction formula.

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

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

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

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 ($D\ll 1$).