Fundamentals of Engineering Exam (FE) Electrical, Electronics and Computing Syllabus

Define the phasor representation of $v(t)=V_m\cos(\omega t + \phi)$.

The phasor is $\mathbf{V} = V_{m}\angle\phi$ (or RMS $\frac{V_m}{\sqrt{2}}\angle\phi$), a complex number encoding amplitude and phase at a single frequency $\omega$.

Give the impedance of a resistor, inductor, and capacitor in the phasor (frequency) domain.

Resistor: $Z_{R}=R$. Inductor: $Z_{L}=j\omega L$. Capacitor: $$Z_{C}=\frac{1}{j\omega C}=-\frac{j}{\omega C}$$ Impedance is measured in ohms.

Define impedance $Z$ and admittance $Y$ in terms of their components.

$Z = R + jX$ where $R$ is resistance and $X$ is reactance. Admittance $Y = \dfrac{1}{Z} = G + jB$ where $G$ is conductance and $B$ is susceptance.

How does inductive reactance differ from capacitive reactance regarding phase?

For an inductor, current lags voltage by $90^{\circ}$ ($X_{L}=\omega L$, positive). For a capacitor, current leads voltage by $90^{\circ}$ ($X_{C}=\frac{1}{\omega C}$, negative). Mnemonic: ELI the ICE man.

Give the resonant frequency of a series (or parallel) RLC circuit.

$$\omega_{0} = \frac{1}{\sqrt{LC}}\quad\text{or}\quad f_{0} = \frac{1}{2\pi\sqrt{LC}}$$ At resonance the inductive and capacitive reactances cancel and the impedance is purely resistive.

Relate the RMS value to the peak value for a sinusoid.

$$V_{rms} = \frac{V_{m}}{\sqrt{2}} \approx 0.707\,V_{m}$$ The same factor applies to current. RMS is the DC-equivalent heating value.

Give the formula for average (real) power delivered to an AC load and define power factor.

$$P = V_{rms}I_{rms}\cos\theta$$ where $\theta$ is the phase angle between voltage and current. The power factor is $\text{pf}=\cos\theta$; it is leading for capacitive loads and lagging for inductive loads.

Define complex power $S$ and its components.

$$S = P + jQ = V_{rms}I_{rms}^{*}$$ $P$ is real power (W), $Q$ is reactive power (VAR), and $|S|=V_{rms}I_{rms}$ is apparent power (VA). Also $|S|=\sqrt{P^{2}+Q^{2}}$.

What is the power triangle and the relationship among P, Q, and S?

A right triangle with real power $P$ (adjacent), reactive power $Q$ (opposite), and apparent power $S$ (hypotenuse): $$S^{2}=P^{2}+Q^{2},\quad \theta=\tan^{-1}\!\frac{Q}{P}$$

For a balanced three-phase system, relate line and phase quantities in a Y (wye) connection.

In a balanced wye: $V_{L}=\sqrt{3}\,V_{ph}$ (line voltage leads phase by $30^{\circ}$) and $I_{L}=I_{ph}$.

For a balanced three-phase delta connection, relate line and phase quantities.

In a balanced delta: $V_{L}=V_{ph}$ and $I_{L}=\sqrt{3}\,I_{ph}$.

Give the total real power in a balanced three-phase load.

$$P_{total} = \sqrt{3}\,V_{L}I_{L}\cos\theta = 3\,V_{ph}I_{ph}\cos\theta$$ where $\theta$ is the per-phase power-factor angle.

State the ideal transformer voltage, current, and impedance relationships in terms of turns ratio $a=N_1/N_2$.

$$\frac{V_{1}}{V_{2}}=\frac{N_{1}}{N_{2}}=a,\quad \frac{I_{1}}{I_{2}}=\frac{N_{2}}{N_{1}}=\frac{1}{a},\quad Z_{in}=a^{2}Z_{L}$$ An ideal transformer conserves power: $V_{1}I_{1}=V_{2}I_{2}$.

What is a PN junction diode and its ideal forward/reverse behavior?

A diode is a junction of p-type and n-type semiconductor. Ideally it conducts (acts as a short) when forward biased and blocks current (acts as an open) when reverse biased. Silicon diodes have a forward drop of about $0.7\ \text{V}$.

Write the Shockley diode equation.

$$I = I_{S}\left(e^{V/(nV_{T})} - 1\right)$$ where $I_{S}$ is reverse saturation current, $n$ the ideality factor, and $V_{T}=\frac{kT}{q}\approx 26\ \text{mV}$ at room temperature.

Name the three terminals of a BJT and the three regions of operation.

Terminals: base, collector, emitter. Operating regions: cutoff (off), active (amplification, $I_{C}=\beta I_{B}$), and saturation (fully on). The current gain is $\beta=\frac{I_{C}}{I_{B}}$.

What is a Zener diode primarily used for?

A Zener diode is operated in reverse breakdown at a controlled Zener voltage $V_{Z}$, providing a stable reference/regulated voltage. It is used in voltage regulator and clamping circuits.

List the ideal characteristics of an operational amplifier.

Infinite open-loop gain, infinite input impedance, zero output impedance, infinite bandwidth, and zero input offset. Under negative feedback the two golden rules apply: no current into the inputs, and $V_{+}=V_{-}$ (virtual short).

Give the gain of an inverting amplifier and a non-inverting amplifier.

Inverting: $$\frac{V_{out}}{V_{in}} = -\frac{R_{f}}{R_{1}}$$ Non-inverting: $$\frac{V_{out}}{V_{in}} = 1 + \frac{R_{f}}{R_{1}}$$

Write the output of an ideal op-amp summing (inverting adder) amplifier.

$$V_{out} = -R_{f}\left(\frac{V_{1}}{R_{1}} + \frac{V_{2}}{R_{2}} + \cdots + \frac{V_{n}}{R_{n}}\right)$$ If all input resistors are equal to $R$, then $V_{out}=-\frac{R_{f}}{R}\sum V_{k}$.

State Coulomb's law for the force between two point charges.

$$F = \frac{1}{4\pi\varepsilon}\,\frac{q_{1}q_{2}}{r^{2}} = k\frac{q_{1}q_{2}}{r^{2}}$$ where $k\approx 8.99\times 10^{9}\ \text{N·m}^{2}/\text{C}^{2}$ in free space. The force is along the line joining the charges.

State Gauss's law for the electric field in integral form.

$$\oint_{S}\vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_{0}}$$ The net electric flux through a closed surface equals the enclosed charge divided by the permittivity.

Write the magnetic force (Lorentz force) on a charge moving in electric and magnetic fields.

$$\vec{F} = q\vec{E} + q\vec{v}\times\vec{B}$$ The magnetic part is perpendicular to both velocity and field, with magnitude $qvB\sin\theta$.

Convert the binary number $1011_{2}$ to decimal.

$$1011_{2} = 1\cdot 2^{3} + 0\cdot 2^{2} + 1\cdot 2^{1} + 1\cdot 2^{0} = 8+0+2+1 = 11_{10}$$

How do you form the two's complement of a binary number, and what is it used for?

Invert all bits (one's complement) and add 1. Two's complement represents signed integers so that subtraction can be done by addition; the most significant bit indicates sign (1 = negative).

State De Morgan's theorems.

$$\overline{A \cdot B} = \overline{A} + \overline{B}$$ $$\overline{A + B} = \overline{A}\cdot\overline{B}$$ The complement of a product is the sum of complements, and vice versa.

Give the Boolean output expressions for XOR and XNOR of two inputs $A$ and $B$.

XOR: $A \oplus B = A\overline{B} + \overline{A}B$ (output 1 when inputs differ). XNOR: $\overline{A \oplus B} = AB + \overline{A}\,\overline{B}$ (output 1 when inputs are equal).

Distinguish combinational from sequential logic circuits.

Combinational logic outputs depend only on the present inputs (e.g., adders, multiplexers, decoders) and have no memory. Sequential logic outputs depend on present inputs and stored state/past history (e.g., flip-flops, registers, counters) and require a clock or feedback.

Describe the behavior of a D flip-flop and a JK flip-flop.

A D flip-flop captures the input $D$ on the active clock edge: $Q_{next}=D$. A JK flip-flop: $J=K=0$ holds, $J=1,K=0$ sets, $J=0,K=1$ resets, and $J=K=1$ toggles the output.

What is the time complexity of binary search, and what is its prerequisite?

Binary search runs in $O(\log n)$ time. It requires the array to be sorted; each step halves the remaining search interval by comparing the target to the middle element.

Compare the average-case time complexity of bubble sort and merge sort.

Bubble sort is $O(n^{2})$ (compares/swaps adjacent elements). Merge sort is $O(n\log n)$ using divide-and-conquer, making it far more efficient for large $n$, though it needs $O(n)$ extra space.