Power Electronics and Drives: cram pack

DC-DC converters: a switch S turns on for the fraction D of each period T = 1/f. In steady state the average inductor voltage is zero (volt-second balance), which gives the ideal conversion ratios in continuous conduction: buck V_o = D·V_in, boost V_o = V_in/(1 − D), inverting buck-boost |V_o| = V_in·D/(1 − D). An ideal converter keeps the power: V_in·I_in = V_o·I_o. While S is on the inductor current rises by ΔI_L = V_L·D·T/L, with V_L = V_in − V_o for the buck and V_in for the boost and buck-boost. The output ripple is ΔV_o = ΔI_L/(8fC) for the buck and I_o·D/(fC) for the boost and buck-boost. The inductor current just touches zero (the CCM boundary) when the average inductor current equals ΔI_L/2.

DC-DC converters: a switch S turns on for the fraction D of each period T = 1/f. In steady state the average inductor voltage is zero (volt-second balance), which gives the ideal conversion ratios in continuous conduction: buck V_o = D·V_in, boost V_o = V_in/(1 − D), inverting buck-boost |V_o| = V_in·D/(1 − D). An ideal converter keeps the power: V_in·I_in = V_o·I_o. While S is on the inductor current rises by ΔI_L = V_L·D·T/L, with V_L = V_in − V_o for the buck and V_in for the boost and buck-boost. The output ripple is ΔV_o = ΔI_L/(8fC) for the buck and I_o·D/(fC) for the boost and buck-boost. The inductor current just touches zero (the CCM boundary) when the average inductor current equals ΔI_L/2.

DC motor: the armature is a resistance R_a in series with the back-EMF E, so V = E + I_a·R_a. The back-EMF grows with speed, E = k·ω, and the torque with current, T = k·I_a, with the same motor constant k (V·s/rad = N·m/A) in SI units. Convert speeds with ω = 2π·n/60. The developed power is E·I_a = T·ω; the input power is V·I_a, and the copper loss I_a²R_a and the rotational loss make the difference. The speed falls in a straight line with torque, ω = V/k − R_a·T/k², from the no-load speed V/k to zero at the stall torque k·V/R_a. At standstill E = 0, so the starting current V/R_a is large unless a resistor limits it.

DC motor: the armature is a resistance R_a in series with the back-EMF E, so V = E + I_a·R_a. The back-EMF grows with speed, E = k·ω, and the torque with current, T = k·I_a, with the same motor constant k (V·s/rad = N·m/A) in SI units. Convert speeds with ω = 2π·n/60. The developed power is E·I_a = T·ω; the input power is V·I_a, and the copper loss I_a²R_a and the rotational loss make the difference. The speed falls in a straight line with torque, ω = V/k − R_a·T/k², from the no-load speed V/k to zero at the stall torque k·V/R_a. At standstill E = 0, so the starting current V/R_a is large unless a resistor limits it.

AC power: apparent power S = V·I (VA) splits into real power P = S·cos φ (W), which does useful work, and reactive power Q = S·sin φ (var), which moves back and forth between source and load. The power factor is cos φ. A capacitor in parallel supplies reactive power locally, so the line current drops for the same real power.

AC power: apparent power S = V·I (VA) splits into real power P = S·cos φ (W), which does useful work, and reactive power Q = S·sin φ (var), which moves back and forth between source and load. The power factor is cos φ. A capacitor in parallel supplies reactive power locally, so the line current drops for the same real power.

Balanced three-phase systems: in a Y (star) connection the line-to-line voltage is √3 times the phase voltage, V_L = √3·V_ph, and the line current equals the phase current. In a Δ (delta) connection the line-to-line voltage equals the phase voltage and the line current is √3 times the phase current, I_L = √3·I_ph. The total power is three times the power of one phase: P = 3·V_ph·I_ph·cos φ = √3·V_L·I_L·cos φ.

Balanced three-phase systems: in a Y (star) connection the line-to-line voltage is √3 times the phase voltage, V_L = √3·V_ph, and the line current equals the phase current. In a Δ (delta) connection the line-to-line voltage equals the phase voltage and the line current is √3 times the phase current, I_L = √3·I_ph. The total power is three times the power of one phase: P = 3·V_ph·I_ph·cos φ = √3·V_L·I_L·cos φ. Power factor correction: capacitors supply reactive power locally, Q_C = P·(tan φ₁ − tan φ₂). Each of the three capacitors supplies Q_C/3 = V²·ω·C, where V is the voltage across that capacitor: V_L in a Δ bank, V_L/√3 in a Y bank.