ISRO Scientist Electrical cheat sheet

Thevenin resistance is found by deactivating all independent sources (voltage sources shorted, current sources opened) and looking into the terminals; dependent sources must remain active.

Maximum power transfer to a load occurs when R_L = R_th for resistive networks, giving efficiency 50% at that point; for AC networks Z_L = Z_th* (conjugate match).

In a series RLC circuit at resonance, omega_0 = 1/sqrt(LC), impedance is purely resistive and minimum, current is maximum, and the quality factor Q = (1/R)sqrt(L/C) = omega_0L/R.

Superposition applies only to linear circuits and only to responses (currents/voltages), not to power; it fails for circuits containing nonlinear elements.

Nodal analysis: form GV = I with conductances; mesh analysis: form RI = V with resistances. Choose the method giving fewer simultaneous equations.

Two-port parameters: for a reciprocal network z12 = z21, y12 = y21, h12 = -h21, and ABCD determinant AD - BC = 1 for reciprocal two-ports.

Star-delta conversion: R_A = (R_ABR_AC)/(R_AB + R_BC + R_CA); delta-star: R_AB = (R_AR_B + R_BR_C + R_CR_A)/R_C.

Transient in an RC circuit: v_C(t) = V_final + (V_initial - V_final)*exp(-t/tau) with tau = RC; in RL circuits tau = L/R.

Reciprocity theorem: in a linear bilateral network, the ratio of response to excitation is unchanged when source and response positions are interchanged.

Transformer EMF equation: E = 4.44fPhi_m*N (RMS), so volts-per-turn is fixed by flux and frequency; core flux depends on applied voltage, not load.

Transformer equivalent circuit referred to primary: R_02 = R_1 + a^2R_2 and X_02 = X_1 + a^2X_2, where a = N_1/N_2; maximum efficiency occurs when copper loss equals iron loss.

DC machine EMF: E = (PPhiNZ)/(60A); torque T = (PPhiI_aZ)/(2pi*A). For a shunt motor, flux is nearly constant so speed falls roughly linearly with load current.

DC motor speed control: N proportional to (V - I_a*R_a)/Phi; armature voltage control gives constant-torque operation below base speed, field weakening gives constant-power operation above base speed.

Induction motor slip s = (N_s - N_r)/N_s; rotor frequency f_r = s*f; rotor copper loss = s * air-gap power, and mechanical power developed = (1-s) * air-gap power.

Synchronous machine: E_f = V + I_a*(R_a + jX_s) for generator; the power-angle relation P = (E_f*V/X_s)*sin(delta) shows maximum power at delta = 90 degrees.

Starting of squirrel-cage induction motors: direct-on-line for small ratings, star-delta reduces phase voltage by 1/sqrt(3) and starting torque to one-third, autotransformer and rotor-resistance starting for slip-ring machines.

Parallel operation of transformers requires equal voltage ratios, same polarity, and similar per-unit impedance; load sharing is inversely proportional to per-unit impedance.

Per-unit system: Z_pu = Z_actual/Z_base with Z_base = V_base^2/S_base; when changing bases, Z_pu_new = Z_pu_old*(S_new/S_old)*(V_old/V_new)^2.

Short transmission line (<80 km) uses lumped R and X; medium line uses nominal-T or nominal-pi; long line uses ABCD (distributed) parameters with surge impedance Z_c = sqrt(L/C).

Ferranti effect: on an open-circuited or lightly loaded long line, receiving-end voltage exceeds sending-end voltage because of line charging capacitance.

Symmetrical components: positive, negative and zero sequence; zero-sequence currents flow only through paths with a neutral return, so delta windings block them.

Protection: overcurrent relays for feeders, differential protection for transformers and generators, distance (impedance) relays for transmission lines with zones 1 (80%), 2 (120%) and 3.

Circuit breaker rating: making current (peak) and breaking current (RMS symmetrical) are specified; at rated symmetrical breaking current, the peak making current is about 2.55 times the RMS value for a 50 Hz system.

Load flow: Newton-Raphson converges quadratically and uses the Jacobian; Gauss-Seidel is simple but slow; the slack bus fixes voltage magnitude and angle.

Corona loss increases with voltage and conductor surface gradient, and is reduced by bundled conductors or larger conductor diameter; it also causes radio interference.

Closed-loop transfer function T(s) = G(s)/(1 + G(s)H(s)); the characteristic equation is 1 + G(s)H(s) = 0, and its roots are the closed-loop poles.

Routh-Hurwitz criterion: the number of sign changes in the first column of the Routh array equals the number of right-half-plane poles; a zero row is replaced by the derivative of the auxiliary polynomial.

Steady-state error: for a type-0 system with a step input, e_ss = 1/(1 + K_p); for type-1 with a ramp, e_ss = 1/K_v; for type-2 with a parabola, e_ss = 1/K_a.

Second-order system: natural frequency omega_n and damping ratio zeta; percent overshoot = exp(-pizeta/sqrt(1 - zeta^2))100, peak time = pi/(omega_nsqrt(1 - zeta^2)), settling time about 4/(zetaomega_n).

Bode plots: a pole at the origin gives -20 dB/decade slope; a simple pole contributes -20 dB/decade and -90 degrees of phase after the corner; gain margin and phase margin are read at the phase and gain crossover frequencies.

Nyquist criterion: N = P - Z, where N is encirclements of -1, P is open-loop right-half-plane poles and Z is closed-loop right-half-plane poles; stability requires Z = 0.

State-space: x_dot = Ax + Bu, y = Cx + Du; the system is controllable if the controllability matrix [B AB A^2B ...] has full rank, and observable if [C; CA; CA^2; ...] has full rank.

Lead compensation increases phase margin and bandwidth; lag compensation reduces steady-state error but slows the response.

Single-phase full-wave controlled rectifier with resistive load: average output voltage = (2*V_m/pi)*cos(alpha); with a highly inductive load and freewheeling, the same formula holds but the output can go negative for alpha > 90 degrees.

Three-phase fully controlled bridge: V_dc = (3*sqrt(3)V_m/pi)cos(alpha) = 1.35V_LLcos(alpha) for continuous conduction.

Buck converter: V_o = DV_in; boost: V_o = V_in/(1 - D); buck-boost: V_o = -V_inD/(1 - D); all in continuous conduction mode.

A step-down chopper with a resistive load gives a discontinuous-current average voltage of V_o = V_inD(D + delta)/... in the discontinuous mode; in continuous mode V_o = D*V_in.

Thyristor turn-on requires anode positive with respect to cathode plus a gate pulse; turn-off (commutation) requires anode current below the holding current for long enough to remove stored charge.

Inverters: a single-phase bridge with 180-degree conduction gives a square wave; PWM control reduces low-order harmonics and allows voltage and frequency control; the fundamental RMS of a square wave is 4V_dc/(pisqrt(2)).

Induction motor drives: V/f control keeps air-gap flux constant below base speed; above base speed, field weakening gives constant-power operation; slip compensation improves speed regulation.

Switching losses in a power device are proportional to switching frequency and to the overlap of voltage and current during transitions; snubbers reduce dv/dt and di/dt stress.

Op-amp ideal assumptions: infinite open-loop gain, infinite input impedance, zero output impedance, infinite bandwidth; the virtual-short concept gives V+ = V- in negative feedback.

Inverting amplifier gain = -R_f/R_in; non-inverting gain = 1 + R_f/R_in; integrator output = -(1/RC)integral of v_in dt; differentiator output = -RCdv_in/dt.

Slew rate limits the maximum frequency of a full-scale sine output: f_max = SR/(2piV_peak).

Boolean identities: De Morgan's laws (A+B)' = A'B' and (AB)' = A' + B'; A + A'*B = A + B; consensus theorem AB + A'C + BC = AB + A'C.

Karnaugh map grouping uses powers of two (1, 2, 4, 8) with wrap-around adjacency; don't-care terms may be used to enlarge groups.

A JK flip-flop with J = K = 1 toggles; a D flip-flop output follows input at the clock edge; a T flip-flop toggles when T = 1; a master-slave arrangement prevents race-around.

An n-bit ripple counter has n flip-flops and modulus 2^n; a MOD-N counter needs at least ceil(log2(N)) flip-flops and reset logic.

CMOS static power is negligible; dynamic power = alphaCV_dd^2*f, where alpha is the activity factor.

A system is linear if it satisfies superposition and time-invariant if a shift in input causes the same shift in output; LTI systems are fully described by their impulse response h(t).

Convolution in time corresponds to multiplication in frequency: y(t) = x(t)*h(t) and Y(f) = X(f)*H(f).

Nyquist criterion: to reconstruct a band-limited signal of maximum frequency f_m, the sampling rate must exceed 2*f_m; otherwise aliasing occurs.

Fourier series coefficients of a periodic signal of period T: c_n = (1/T)integral over T of x(t)exp(-j2pint/T) dt.

Laplace transform pairs: u(t) gives 1/s, exp(-at) gives 1/(s+a), t gives 1/s^2, sin(wt) gives w/(s^2 + w^2); initial value theorem: x(0+) = lim s*X(s) as s tends to infinity.

Z-transform: x[n] = a^n*u[n] gives z/(z - a) with ROC |z| > |a|; the ROC determines stability and causality.

Measurement: a PMMC meter reads average value and is used for DC; a moving-iron meter reads RMS and works on AC and DC; a rectifier-type meter reads average but is calibrated in RMS for sine waves.

Wattmeter method: two-wattmeter measurement of three-phase power gives total P = W1 + W2 and reactive power Q = sqrt(3)*(W1 - W2) for a balanced load.

Instrument transformer burden and ratio error: a current transformer's phase angle error and ratio error depend on the secondary burden and core magnetization.

Maxwell's equations in differential form: div D = rho, div B = 0, curl E = -dB/dt, curl H = J + dD/dt.

Poisson's equation: del^2 V = -rho/epsilon; Laplace's equation: del^2 V = 0 in charge-free regions.

Capacitance of a parallel-plate capacitor: C = epsilonA/d; energy stored = 0.5CV^2 = 0.5integral of epsilon*E^2 dV.

Biot-Savart law: dB = (mu_0I/4pi)*(dl x r_hat)/r^2; Ampere's law: closed-loop integral of H dot dl = I_enclosed.

Poynting vector S = E x H gives power flow per unit area; intrinsic impedance of free space is 377 ohms.

Breakdown in gases follows Paschen's law: breakdown voltage is a function of the product of pressure and gap distance (p*d); a minimum exists on the Paschen curve.

Standard lightning impulse is 1.2/50 microseconds (virtual front and tail times); switching impulse is 250/2500 microseconds; the 50% flashover voltage is used to characterize insulation.

High-voltage generation: Cockcroft-Walton multiplier for DC, Van de Graaff for very high DC, cascaded transformers and resonant transformers for AC, and Marx generator for impulse voltages.

A square matrix is invertible if and only if its determinant is nonzero; the eigenvalues satisfy det(A - lambda*I) = 0, and the trace equals the sum of eigenvalues while the determinant equals their product.

Cayley-Hamilton theorem: every square matrix satisfies its own characteristic equation, allowing powers of A to be reduced.

Rank of a matrix equals the number of nonzero rows in row-echelon form; a system Ax = b is consistent if rank(A) = rank([A|b]).

Taylor series: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ...; the mean value theorem states f(b) - f(a) = f'(c)(b - a) for some c in (a, b).

First-order linear ODE: dy/dx + P(x)y = Q(x) has integrating factor exp(integral P dx); second-order homogeneous solutions use the characteristic equation.

Fourier transform of a rectangular pulse is a sinc function; the Fourier transform of exp(-a|t|) is 2a/(a^2 + w^2).

Numerical methods: Newton-Raphson iteration x_{n+1} = x_n - f(x_n)/f'(x_n) has quadratic convergence; trapezoidal rule error is of order h^2, Simpson's rule error of order h^4.

Runge-Kutta fourth-order method for ODEs uses four slope estimates per step and has local error of order h^5.

Percentage change: successive changes of a% and b% give a net change of a + b + a*b/100 percent.

Compound interest: A = P*(1 + r/n)^(nt); simple interest: SI = Pr*t/100.

Time-speed-distance: distance = speed*time; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Permutations and combinations: nPr = n!/(n-r)!; nCr = n!/(r!(n-r)!); the number of ways to arrange n objects with repetitions is n!/(p!q!...).

Logical reasoning: in syllogisms, conclusions must follow from the premises regardless of real-world truth; use Venn diagrams for categorical statements.

Number series and coding-decoding questions are solved by finding constant differences, ratios, or letter-position shifts.

Verbal ability: subject-verb agreement, correct preposition use, and spotting misplaced modifiers; in reading comprehension, the answer must be supported by the passage, not by outside knowledge.

C operators: && and || short-circuit; the ternary operator (a ? b : c) evaluates only the selected branch; the sizeof operator returns size_t in bytes.

Pointers: an array name decays to a pointer to its first element; pointer arithmetic scales by the size of the pointed-to type; NULL must not be dereferenced.

Storage classes: auto (local), static (lifetime of program, retains value), extern (defined elsewhere), register (hint for a register).

Data structures: a stack is LIFO with push/pop at the top; a queue is FIFO with enqueue at the rear and dequeue at the front; a binary search tree has in-order traversal in sorted order.

Complexity: binary search is O(log n); merge sort and heap sort are O(n log n); quicksort average is O(n log n) but worst case O(n^2); hash table lookup is average O(1).

Digital logic: a half adder gives sum = A XOR B and carry = A AND B; a full adder adds three inputs; a multiplexer with n select lines handles 2^n inputs.

Number systems: hexadecimal to binary is a direct 4-bit mapping; signed two's complement of an n-bit number has range -2^(n-1) to 2^(n-1) - 1.

Memory: a ROM is non-volatile and programmed at manufacture; a RAM is volatile; a flip-flop stores 1 bit, while a register is a group of flip-flops.