Single-phase half-wave uncontrolled rectifier

Subscripts s and o identify the source and the complete output load; dc denotes the mean, rms the total RMS, and ac the RMS of the zero-mean ripple. Load types are R, RL, RE, RLE, and R‖C. In R‖C, the resistor and filter capacitor are in parallel; the opposing series voltage is not included. E is the effective series DC voltage (zero when disabled); v_E = E, p_E = Ei_o, P_E = EI_o,dc. The symbol E_returned elsewhere denotes returned energy in joules, not the DC voltage E in volts. Unless explicitly labeled as a reference, readouts use the selected circuit.

v_s = V_m sin θ, θ = 2πft. The amplitude is peak, not RMS. v_o spans the complete series load, top minus bottom: v_o = Ri_o + v_L + E. Positive i_o flows down and enters the + terminal of E. D_F is antiparallel to the complete load including E. Diode voltages are anode minus cathode. The ideal AC source can accept returned instantaneous power.

R > 0, L ≥ 0 and E ≥ 0. Selecting the noninductive load bypasses the saved L; enabling E changes R/RL into RE/RLE. Disabling E bypasses it without deleting its saved setting. Both diodes have the same constant V_F. With fixed magnetic flux and motor speed, E approximates back-EMF and Ei_o electromechanical conversion power; as a battery it is an ideal constant-voltage charging model. Speed, torque dynamics, state of charge, battery chemistry, magnetic saturation, switching losses, reverse recovery and source impedance are not modeled. Negative E and battery discharge are excluded. Each edit selects periodic steady state; it is not a live switching event.

With an ideal source, D₁ conducts while v_s − V_F can supply both resistor R and capacitor C. During conduction, v_o = V_m sin θ − V_F, and the diode current is the sum of the resistor and capacitor currents:

D₁ turns off at β, the first angle after the voltage peak where i_D1(β) = 0. From then on, only the capacitor supplies R:

Conduction restarts at α in the next period, when the rising source catches up with the capacitor voltage. The solver finds α numerically from:

Then ΔV_o = V_o,max − V_o,min = V_m(1 − sin α). For ωRC ≫ 1 the discharge is almost linear and lasts almost a whole period, which gives the usual estimate ΔV_o ≈ (V_m − V_F)/(fRC); it overestimates the ripple, because the real discharge is shorter and curves upwards. The capacitor current has zero mean, so I_D1,avg = I_o,dc. The peak diode current occurs at α, or at θ = atan(1/(ωRC)) if that angle lies inside the conduction interval. While D₁ blocks, v_D1 = v_s − v_o, which approaches −(2V_m − V_F) when the ripple is small.

For a purely resistive load (L = 0 and E = 0), i_o = v_o/R, so I_o,rms = V_o,rms/R and P_out = V_o,rmsI_o,rms. These shortcuts require instantaneous voltage/current proportionality. Do not use them as general rules with inductance or a nonzero DC voltage; use the average instantaneous power and show the RMS product separately:

In periodic steady state, ⟨v_L⟩ = ⟨p_L⟩ = 0, so V_o,dc = RI_o,dc + E and P_out = RI_o,rms² + EI_o,dc. The DC-source term is absorbed power, not loss. Instantaneously p_s = p_R + p_L + p_E + p_D1 + p_DF; its average gives P_in = P_out + P_D1 + P_DF. Setting V_F = 0 removes diode losses, not the load-power term Ei_o, which is positive when E > 0 and current flows. With all currents zero and E > 0, v_o is constant, FF_v = 1 and RF_v = 0; current ratios, PF, THD and power-efficiency ratios are undefined. A flat battery-voltage trace therefore does not establish useful power transfer.

Here the analyzed current is the single-phase source current. In the equations, I_rms, I_dc, and I_1,rms refer to its full-period RMS, signed mean, and fundamental RMS. Both ratios use the fundamental-current RMS as denominator, never total RMS. DC excluded removes the mean-current contribution only from the numerator; DC included retains it. The readouts are percentages (100 × the ratio); exports retain dimensionless ratios unless explicitly labeled %. Both are undefined when the fundamental is zero. The rectifier values use the complete source-current waveform, including capacitor charging current, independently of the number of spectrum bars displayed.

φ₁ is the lag of the source-current fundamental relative to the sinusoidal source voltage; it is not generally atan(ωL/R). CF = I_s,pk/I_s,rms. TUF is defined here as P_dc/(V_s,rmsI_s,rms). No primary winding or transformer magnetization is modeled.

Diode means and RMS values use a full period. Reverse-voltage peaks include both sides of switching jumps. With no current, v_D1 = v_s − E; when D_F is present, v_DF = −E. If the circuit is off at the negative source peak, D₁ blocks V_m + E. It can block less if current still flows then; use the actual waveform maxima. β is the last D₁ turn-off, not necessarily the end of load current. The interval list retains any brief initial conduction, gap and restart. These peaks are operating stresses, not component ratings.

The solution is analytic on each conduction interval. Device switching instants are located explicitly. Event-split 16-point Gauss–Legendre quadrature, with additional resolution near exponential tails, evaluates current and power metrics and Fourier coefficients. Charts use a separate drawing grid with the switching events inserted. Values at an ideal switching discontinuity follow the right-hand interval. The live consistency checks below compare independently integrated and analytic quantities; displayed values are rounded.