GATE EE flashcards

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

Rank of a matrix

The number of linearly independent rows or columns, equal to the count of nonzero rows in row-echelon form; it determines whether a linear system has a unique, no, or infinite solution.

Cauchy–Riemann equations

For f(z) = u + jv to be analytic, the partial derivatives must satisfy u_x = v_y and u_y = −v_x, with continuity of the partials in a neighborhood.

Residue theorem

The contour integral of an analytic function around a closed path equals 2πj times the sum of residues of the poles enclosed; a simple pole at z0 has residue lim (z − z0)f(z).

Percentage change composition

Two successive percentage changes a and b produce a net change of a + b + ab/100 percent, so a 20% increase followed by a 20% decrease yields a 4% net decrease.

Combination formula

The number of ways to choose r objects from n distinct objects without regard to order is nCr = n!/(r!(n − r)!); permutations count ordered arrangements and equal nPr = n!/(n − r)!.

Thevenin equivalent circuit

Any linear two-terminal network can be replaced by a voltage source V_th in series with R_th, where V_th is the open-circuit voltage and R_th is the resistance seen with sources killed.

Maximum power transfer

A resistive load receives maximum power when R_L equals the Thevenin resistance R_th, giving P_max = V_th²/(4R_th); the efficiency at that operating point is only 50%.

Series RLC resonance

At resonance ω0 = 1/√(LC), the inductive and capacitive reactances cancel, impedance is purely resistive and minimum, and the quality factor Q = ω0L/R sets the bandwidth ω0/Q.

Gauss's law

The total electric flux leaving a closed surface equals the enclosed charge: ∮D·dS = Q_enclosed; for a point charge this gives D = q/(4πr²) directed radially outward.

Intrinsic impedance of free space

η0 = √(μ0/ε0) ≈ 377 Ω relates the electric and magnetic field magnitudes of a plane wave in vacuum, where E/H = η0 and the wave travels at c = 3 × 10⁸ m/s.

Skin depth

For a good conductor the current decays exponentially with depth, and the skin depth δ = √(2/(ωμσ)) is the distance at which the field falls to 1/e of its surface value.

Nyquist sampling theorem

A signal bandlimited to f_m can be reconstructed exactly only if sampled at f_s > 2f_m; sampling below that rate causes aliasing, where high frequencies appear as lower ones.

BIBO stability condition

A linear time-invariant system is bounded-input bounded-output stable exactly when its impulse response is absolutely integrable, and for a causal system when all poles lie in the left half-plane.

Final value theorem

For a signal whose sX(s) has all poles in the left half-plane, the steady-state value equals lim_{s→0} sX(s); the theorem fails if any pole lies on or right of the imaginary axis.

Transformer emf equation

The induced emf per winding is E = 4.44 f N Φ_m, so the volts per turn are the same on primary and secondary and set by the core flux and frequency.

Induction motor slip

Slip s = (N_s − N_r)/N_s, where synchronous speed N_s = 120f/P; rotor frequency equals s times supply frequency, and slip is 1 at standstill and 0 at synchronous speed.

Transformer maximum efficiency

A transformer reaches maximum efficiency at the load where variable copper loss equals constant iron loss, which typically occurs below full load at a load fraction of √(P_iron/P_copper).

Per-unit impedance

Impedance in per unit is Z_actual/Z_base with Z_base = V_base²/S_base; converting between bases uses Z_pu,new = Z_pu,old(V_old/V_new)²(S_new/S_old).

Surge impedance loading

SIL = V²/Z_c, where the surge impedance Z_c = √(L/C) of an overhead line is typically 300–400 Ω; at SIL the line neither absorbs nor supplies reactive power.

Equal-area criterion

Transient stability is judged by comparing the accelerating area with the maximum possible decelerating area on the power–angle curve; stability holds if the decelerating area can absorb the acceleration.

Routh–Hurwitz stability criterion

A necessary condition is that all coefficients of the characteristic polynomial be positive; the number of sign changes in the first column of the Routh array equals the number of right-half-plane roots.

Second-order peak overshoot

For a standard second-order system with damping ratio ζ below one, the peak overshoot is e^{−πζ/√(1 − ζ²)}, which decreases as ζ increases toward critical damping.

Steady-state error and system type

A type-0 system has finite step error 1/(1 + K_p), type-1 has zero step error and finite ramp error 1/K_v, and type-2 has zero ramp error and finite parabolic error.

PMMC instrument deflection

In a permanent-magnet moving-coil instrument the deflection is proportional to the coil current, so it reads the average value and responds only to direct current unless a rectifier is added.

Wheatstone bridge balance

The bridge is balanced when the ratio of adjacent arms is equal, R1/R2 = R3/R4, giving zero detector current; it suits medium-resistance measurement, while low resistance uses a Kelvin bridge.

Current transformer safety rule

The secondary of a current transformer must never be open-circuited while primary current flows, because the collapsing flux induces a dangerously high secondary voltage.

BJT active region

A bipolar junction transistor is in forward-active mode when the base–emitter junction is forward biased and the collector–base junction is reverse biased; then I_C = βI_B and the device amplifies.

Op-amp virtual ground

With negative feedback and ideal infinite open-loop gain, the differential input voltage is driven to zero, so the inverting input sits at the same potential as the non-inverting input.

Barkhausen criterion

An oscillator sustains oscillation when the loop gain magnitude equals one and the total loop phase shift is zero or a multiple of 360 degrees at the oscillation frequency.

De Morgan's laws

The complement of a sum equals the product of the complements, (A + B)′ = A′B′, and the complement of a product equals the sum of the complements, (AB)′ = A′ + B′.

JK flip-flop toggle condition

When both J and K inputs are high, the JK flip-flop toggles its output on each active clock edge; unlike the SR latch, this input combination is valid and useful for counters.

Buck converter voltage ratio

In continuous conduction a buck converter steps voltage down according to V_o = D·V_in, where D is the duty cycle of the switch; the inductor current ripple depends on switching frequency.

SCR latching and holding current

An SCR stays on after the gate pulse is removed until anode current falls below the holding current; the latching current, required to turn it on, is higher than the holding current.