Electrical engineering: passive components and circuit responses

Voltages are V, current is A, resistance is ohm, capacitance is F, inductance is H and angular frequency is rad/s. RMS phasors are assumed for power helpers. Negative voltage/current are signed quantities; passive resistances are nonnegative, and denominators used to infer current/resistance must be nonzero. Parallel-resistance helpers require strictly positive branches; represent an ideal short circuit separately rather than passing a zero branch.

RLC current convention

SeriesRLC models current for an applied voltage. The transfer function is I(s)/V(s)=s/(L*s^2+R*s+1/C), with zero initial energy. The previous numerator 1 represented charge, despite the response being named current. That error is corrected here. The response to a 1 V step approaches zero current for R>0 and has integrated charge C. User-supplied transient grids must be uniformly spaced and strictly increasing, as required by SciPy step response. Lossless R=0 is allowed and oscillates without decay.

import numpy as np
from engineeringkit.electrical_engineering import SeriesRLC
result = SeriesRLC(2, 1, .25).step_response(np.linspace(0, 20, 20001))
fig, ax = result.plot()  # current [A]

For the underdamped case the independent benchmark uses exp(-R*t/(2L))*sin(wd*t)/(L*wd), wd^2=1/(LC)-(R/(2L))^2. Tests also integrate current to obtain C, verify resonance produces purely resistive impedance, and check source energy equals capacitor storage plus resistor dissipation for RC charging. Capacitor impedance excludes zero frequency, where the ideal capacitor is an open circuit.

Thévenin/Norton helpers transform supplied equivalent parameters; to reduce a network, find its open-circuit voltage and short-circuit current with nodal_analysis. Power factor is bounded by [-1,1], allowing signed real power. Balanced sinusoidal operation is assumed for three-phase power; the result does not cover unbalanced phases or harmonic distortion (use symmetrical components for the former).

Nodal analysis

nodal_analysis solves a DC network of resistors, ideal current sources and ideal voltage sources by modified nodal analysis. Node 0 is ground and non-ground nodes are numbered 1..n. A current source (a, b, I) drives I from node a through itself into node b; a voltage source (p, m, V) holds v_p - v_m = V, and its reported current is the one leaving its + terminal into the circuit. A floating node or a loop of voltage sources makes the system singular; the function rejects it when the condition number exceeds 1e12 rather than returning a meaningless answer. The result’s kcl_residual is the largest net current leaving any node, an independent check of the solve.

Mesh analysis

mesh_analysis is the dual of nodal analysis for a planar circuit of resistors and ideal voltage sources. Meshes (the windows of the drawn circuit) are numbered 1..n and carry clockwise currents; 0 stands for the outside. A resistor (a, b, R) lies on the boundary between meshes a and b (b = 0 on the outer edge), and a voltage source (a, b, V) drives current clockwise around mesh a and therefore anticlockwise around mesh b. Resistor currents are reported in the clockwise sense of their first mesh, i_a - i_b. Current sources are not accepted (they would need supermeshes); convert them to Thévenin form or use nodal_analysis. The tests solve the same ladder both ways.

Second-order transients

rl_current is Helmholtz’s first-order rise or decay through L/R. rlc_damping returns the neper frequency alpha = R/2L, the resonant frequency omega0 = 1/sqrt(LC) [rad/s] and the regime of a series RLC circuit (for a parallel circuit alpha = 1/2RC instead). The comparison alpha = omega0 is made within a relative 1e-9, inside which the critical formula is used, so circuits a hair away from critical damping do not lose accuracy to cancellation. rlc_step_voltage is the capacitor voltage from rest; the loop current is C dv/dt, which SeriesRLC.step_response returns, and the benchmark tests check both against scipy.signal.step. R, L and C are scalar-only.

Electronics

The op-amp helpers are ideal: infinite gain and input resistance, zero output resistance, no rail limits, offset or bandwidth. op_amp_integrator integrates a sampled input by the trapezoidal rule, so its accuracy depends on the sampling. shockley_diode_current covers forward and reverse bias but not breakdown, series resistance or high injection; it refuses voltages above 40 n V_T, where the current would be about e^40 times I_S and no real diode is still described. The temperature enters only through V_T: I_S is an input, though in a real diode it grows steeply with temperature. diode_operating_point brackets the load-line crossing and solves it with Brent’s method: on the current for forward bias, where the diode voltage n V_T ln(1 + i/I_S) cannot overflow, and on the voltage for reverse bias. A Lambert-W closed form checks it.

Machines

dc_motor_torque and dc_motor_speed describe a separately excited (or permanent-magnet) DC machine with constant flux k*Phi; series and shunt field interaction, armature reaction and brush drop are left out. induction_motor_torque uses the per-phase T circuit (stator R1 + jX1, magnetizing jXm, rotor R2/s + jX2, all referred to the stator) without a core-loss resistance; the synchronous speed is the mechanical speed 2 pi f/(poles/2). The torque is computed multiplied through by s^2, so slip 0 gives zero torque instead of a division by zero, negative slip is generating and slip above 1 is plugging. Deep-bar and double-cage rotors, whose resistance changes with slip, are not modelled.

Transmission lines

transmission_line_z0 and transmission_line_velocity are lossless models using H/m and F/m. propagation_constant keeps R [ohm/m] and G [S/m] and returns gamma = alpha + j beta [1/m]; distortionless_condition tests Heaviside’s R/L = G/C. reflection_coefficient, standing_wave_ratio and input_impedance assume a lossless line of real characteristic impedance and a passive, finite load (an open circuit is the limit Gamma = 1 and cannot be passed as an impedance). The electrical length is beta*l in radians, so a quarter wavelength is pi/2. Where a line transforms the load into an open circuit the input impedance is numerically very large.

Power systems

symmetrical_components and phase_components use the Fortescue operator a = exp(j 2 pi/3) with the phase-a members of each sequence set; they accept volts, amps or per-unit values alike. admittance_matrix builds the bus admittance matrix from pi-model lines (series impedance and total line charging, per unit). gauss_seidel_load_flow treats bus 0 as the slack bus and every other bus as a PQ bus with the given consumed power (generation is negative load); PV buses, tap-changing transformers and reactive limits are not modelled. It does not raise on non-convergence: check converged and history. An acceleration factor of 1.4-1.6 usually cuts the number of sweeps, but too large a value slows or prevents convergence. Tests check the two-bus case against its closed form and the example checks a five-bus network against scipy.optimize.root.

newton_raphson_load_flow solves the same problem in polar coordinates and adds PV buses, {bus: |V|}: their real power comes from p_load (negative for a generator), their q_load entry is ignored and their reactive output is reported in power. Its history is the largest power mismatch before each step, so the last entry is below tol on convergence. The Jacobian is dense, which is fine for teaching networks of a few dozen buses. Generator reactive limits are not enforced. The tests check quadratic convergence and agreement with Gauss-Seidel.

per_unit_base returns the base impedance and current for a base power (three-phase total) and a base voltage (line to line); change_base moves a nameplate impedance to the system base. fault_currents computes the phase currents of three-phase, single-line-to-ground (phase a), line-to-line (b-c) and double-line-to-ground faults from the Thévenin sequence impedances at the bus, with optional fault impedance; prefault load current is neglected. Use one unit system, per unit or volts and ohms, throughout.

Transient stability

swing_response integrates the classical single-machine, infinite-bus swing equation (constant voltage behind transient reactance, constant mechanical power) with integrate_ode, switching from the fault-on to the post-fault power-angle curve at the clearing time. H is the inertia constant on the machine base and powers are per unit on the same base. The stable flag is a first-swing check: the angle stays below the post-fault unstable equilibrium pi - arcsin(Pm/P3). Later swings, governor and excitation action and multi-machine effects are outside the model. equal_area_clearing_angle returns the prefault angle and the critical clearing angle; when the fault-on curve keeps the rotor from ever passing the unstable equilibrium it returns that equilibrium, and when the post-fault network cannot hold the rotor at all it raises ValueError. critical_clearing_time has a closed form only for a fault that drops the electrical power to zero; otherwise bisect on swing_response, as the example does. Tests check equal areas, energy conservation after clearing and that the critical time separates stable from unstable swings.

Textbook reference problems

tests/test_reference_problems.py works six examples from Glover, Overbye and Sarma’s Power System Analysis and Design (6th ed. Examples 9.3 and 11.1; 7th ed. Examples 6.9, 12.4 and 12.5) and checks the printed answers: the five-bus Newton-Raphson case, a bolted single-line-to-ground fault, the swing angle three cycles after a terminal fault, the maximum swing after a cleared fault and the critical clearing time. All agree to the printed digit but one. In Example 12.4 the book prints delta2 = 0.7003 rad, solved iteratively from 2.4638 cos(delta2) + delta2 = 2.5843, whose root is 0.70020; swing_response gives 0.70025 from the unrounded clearing angle. The left side is flat there (its derivative is about -0.59), so an iteration stopped at a residual of 1e-4 is off by about that much in the angle; the test accepts one unit in the last printed digit.