ECE Board Exam flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells whether the roots are real, repeated, or complex.

Derivative of sine

The derivative of sin x is cos x, and the derivative of cos x is −sin x; the pattern repeats every fourth derivative and underlies phasor analysis of sinusoidal signals.

Integration by parts

∫u dv = uv − ∫v du. Choose u as the factor that simplifies when differentiated and dv as the factor that is easy to integrate; repeated application handles products such as x·e^x.

Law of sines

In any triangle, a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius; it applies when one side and its opposite angle are known.

Geometric series sum

For first term a₁ and ratio r ≠ 1, Sₙ = a₁(1 − rⁿ)/(1 − r); the infinite sum equals a₁/(1 − r) only when |r| is less than 1.

Limit of sin x over x

As x approaches zero measured in radians, sin x divided by x approaches 1; the limit fails in degrees and is used to derive the derivative of sine.

Permutation versus combination

nPr = n!/(n − r)! counts ordered arrangements of r items chosen from n; nCr = n!/[r!(n − r)!] counts unordered selections, so nCr equals nPr divided by r!.

Determinant of 2x2 matrix

For the matrix with rows (a, b) and (c, d), the determinant is ad − bc; a nonzero value means the matrix is invertible and the linear system has a unique solution.

Ideal op-amp assumptions

Infinite open-loop gain and input impedance, zero output impedance, and zero input bias current; with negative feedback these force both input terminals to the same voltage.

Inverting amplifier gain

With input resistor Rin to the virtual-ground inverting input and feedback resistor Rf, the closed-loop gain is −Rf/Rin; the input impedance seen by the source equals Rin.

Non-inverting amplifier gain

The signal drives the non-inverting input while R1 and Rf form a divider back to the inverting input; closed-loop gain is 1 + Rf/R1 with very high input impedance.

Silicon diode forward drop

A forward-conducting silicon junction drops about 0.7 V and a germanium junction about 0.3 V; the drop rises only slightly with current because it is logarithmic.

BJT current relationships

Collector current Ic = βIb, emitter current Ie = Ic + Ib = (β + 1)Ib, and α = Ic/Ie = β/(β + 1), where β is the common-emitter forward current gain.

LC resonance frequency

A series or parallel LC network resonates at f = 1/(2π√(LC)); at that frequency the reactances cancel and the impedance is purely resistive, minimum in series and maximum in parallel.

Thevenin equivalent circuit

Any linear two-terminal network reduces to a source Vth equal to the open-circuit voltage in series with Rth = Voc/Isc; maximum power transfers when the load equals Rth.

Full-wave rectifier ripple

A full-wave rectifier delivers ripple at twice the source frequency; with a capacitor filter the ripple factor falls as capacitance, load resistance, and frequency increase.

Transformer turns ratio

For an ideal transformer Vp/Vs = Np/Ns and Ip/Is = Ns/Np; power is conserved and impedance reflected to the primary scales as the square of the turns ratio.

Newton's second law

The net force on a body equals its mass times acceleration, F = ma, directed along the net force; weight is the gravitational case W = mg.

Ideal gas law

PV = nRT with R = 8.314 J/(mol·K); one mole of an ideal gas occupies about 22.4 liters at 0 °C and 101.3 kPa.

Carnot efficiency

No engine operating between a hot reservoir at Th and a cold reservoir at Tc, both in kelvin, can exceed η = 1 − Tc/Th; only a reversible cycle attains it.

Young's modulus

E = stress/strain = (F/A)/(ΔL/L), expressed in pascals; it measures stiffness in the linear elastic region where Hooke's law applies.

Bernoulli equation

For steady, incompressible, frictionless flow along a streamline, P + ½ρv² + ρgh is constant; where the fluid speed rises, the static pressure falls.

First law of thermodynamics

ΔU = Q − W, where Q is heat added to the system and W is work done by the system; energy is conserved and only transferred or converted between forms.

Coulomb's law

The force between two point charges is F = kq₁q₂/r² with k about 8.99 × 10⁹ N·m²/C²; like charges repel and opposite charges attract along the joining line.

Nyquist sampling theorem

A signal bandlimited to fmax must be sampled at fs ≥ 2fmax so the samples uniquely determine it; lower rates fold high frequencies back into the band as aliasing.

Shannon capacity theorem

The maximum error-free data rate over a channel of bandwidth B and signal-to-noise ratio S/N is C = B log₂(1 + S/N) bits per second.

Carson's rule

For FM, the transmitted bandwidth is approximately 2(Δf + fm), where Δf is the peak frequency deviation and fm is the highest modulating frequency.

AM modulation index

The AM index μ = Vm/Vc; a value above 1 causes overmodulation and envelope distortion, and each modulating tone creates sidebands at fc ± fm.

Quantization signal-to-noise ratio

For an ideal n-bit uniform quantizer with a full-scale sine input, SQNR is approximately 6.02n + 1.76 dB; each extra bit improves it by about 6 dB.

Geostationary orbit

A satellite about 35,786 km above the equator with a 24-hour period stays fixed over one point; the round-trip path adds roughly 240 ms of propagation delay.

IPv4 versus IPv6

IPv4 uses 32-bit addresses, about 4.3 billion in total, while IPv6 uses 128-bit addresses; IPv6 also simplifies the header and allows stateless address autoconfiguration.

Radar range equation

Received power varies as the inverse fourth power of range, Pr = Pt G² λ² σ/[(4π)³R⁴], so doubling target range requires sixteen times the transmit power.