Power Systems

The three EMFs of a balanced three-phase set are 120 degrees apart and sum to zero. Gather them at a point for Y (a neutral); close them into a loop for Δ (no circulating current). Tune the phase spacing and watch the sum vanish.

The line voltage is √3 times the phase voltage not as a number to memorize but as the geometry of a difference of two phasors 120 degrees apart. Watch the difference grow to √3 and learn the duality: √3 lands on the voltage in Y and on the current in Δ.

Three-phase power is just three times one phase, and in line values it becomes √3 V_L I_L cosφ regardless of connection. Drag the power-factor gauge to see the active power move against the apparent power, and learn that the Y and Δ substitutions arrive at the same formula.

The instantaneous power of a balanced three-phase set is constant in time, because each phase pulsation (2ω) is 120 degrees apart and the three sum to zero (this is A1). Add phases 1, 2, 3 to watch the pulsation vanish, and learn that constant power means smooth, constant torque.

Under balance (A1) the three currents sum to zero and the neutral is empty. When one phase drifts the sum leaves zero, and the leftover flows as the Y neutral current I_N = I_a+I_b+I_c = 3 I_0 (zero sequence). Drag the tip of the I_a phasor to make unbalance yourself and learn that balance is one special point.

When a load pulls its current out of step with the voltage by φ, power splits into a right triangle. Active P = S cosφ (real work), reactive Q = S sinφ (only sloshing), apparent S = VI (the hypotenuse). Drag φ to reshape the triangle and learn S² = P² + Q² and the power factor cosφ = P/S.

Cancel the lagging reactive Q_L of a coil load with the leading reactive Q_C of a capacitor, and the net reactive Q_net = Q_L − Q_C falls, shrinking the apparent S and the line current together. Drag the capacitor to shrink the triangle and learn the needed Q_C = P(tanφ1 − tanφ2) and overcompensation.

Equipment is sized to the composite maximum demand, not the sum of ratings. The diversity factor (sum of maxima / composite ≥ 1), demand factor (maximum / installed ≤ 1) and load factor (average / maximum) measure that gap. Drag two load peaks out of step to see the composite fall below the sum and learn the three ratios.

Three-phase three-wire power is measured with two single-phase wattmeters (Blondel: n wires → n−1 meters). Each reads V_L I_L cos(30°∓φ), so the sum W1+W2 = √3 V_L I_L cosφ = P (A3) and the difference √3(W2−W1) = Q. Drag the current phase to see W1 flip negative below a power factor of 0.5.

A transmission line is not a perfect wire but a series impedance Z = R + jX. R is conductor resistance, X the reactance from the magnetic field (inductance), and at high voltage usually X > R, so the impedance angle is near 80 degrees. Drag the line length to watch the impedance triangle grow and learn why a wire behaves more like a reactor than a resistor.

Express every quantity as a ratio to a base (per-unit), and with V_base set in the transformer turns ratio the per-unit value of one impedance is the same on both sides. Z_base = V_base²/S_base, Z_pu = Z/Z_base. Drag the ratio a to see ohms swing with a² while per-unit does not budge.

Current through the line Z=R+jX lowers the receiving voltage (e ≈ I(R cosφ + X sinφ)) and the resistance soaks up loss 3I²R. Since I = P/(√3 V cosφ) for the same power, raising the voltage cuts the drop as 1/V and the loss as 1/V². Drag the transmission voltage to watch the loss fall steeply.

As the load changes the I·Z drop sags the receiving voltage (heavy) or raises it (light, Ferranti). Since the X sinφ term dominates, reactive plant (capacitor = voltage up, reactor = voltage down) and transformer tap changing hold it in band. Drag the capacitor to lift a sagging voltage into the band (0.95 to 1.05).

The distribution topology sets how far a single fault cuts power. Radial is cheap but cuts everything downstream, the loop back-feeds so only the faulted section drops, the network reroutes with no outage. For the same fault, switch the topology with tabs to watch the interrupted customers shrink, and learn the bargain of buying reliability with the cost of wire and switchgear.

Between two buses, active power is pushed not by voltage magnitude but by the phase-angle difference δ. On a reactance-X line P = (V_s V_r / X) sin δ, the transfer peaks at δ=90° and beyond it crosses the steady-state stability limit. Reactive power is pushed by the magnitude difference. Drag the phase angle along the power-angle curve to see P rise then turn over.

Any unbalanced three-phase set decomposes uniquely into three balanced sets — positive, negative and zero sequence. I_a = I_1 + I_2 + I_0 and the zero I_0 = (I_a+I_b+I_c)/3 is the neutral residual of A5. Balanced gives only positive; an unbalanced fault brings out negative and zero. Drag the tip of I_a to watch negative and zero grow from zero.

The fault type sets the connection of the three sequence circuits (Z1, Z2, Z0) and their combined impedance sets the fault current. A three-phase short uses positive only (E/Z1, usually largest), a line-to-ground puts all three in series (3E/(Z1+Z2+Z0)), a line-to-line uses positive and negative (√3E/(Z1+Z2)). Switch the fault type with tabs to see the connection and current, and learn that with no zero-sequence path the ground current is zero.

A relay measures current and voltage to judge a fault and sends a trip to the breaker, which interrupts the arc to isolate the faulted section. An overcurrent relay is inverse-time (faster for bigger current), and stacking the curves apart by a grading margin gives selectivity, where the relay nearest the fault trips first. Drag the fault current to see the operating-time gap between the near relay and the backup.

Renewables connect through an inverter (power electronics), not a synchronous generator, actively synchronizing their output to grid frequency and phase. The key difference is inertia: a synchronous machine slows frequency change (RoCoF = ΔP/2H) but an inverter has inertia ≈0. Drag the renewable share to see that as inertia falls the frequency drops faster and deeper after an event.

Frequency reflects supply-demand balance and inertia (primary), governor (secondary) and AGC (tertiary) restore 60 Hz. Transient stability is whether the angle opened by a fault keeps synchronism. Equal-area criterion: if the decelerating area A2 after clearing absorbs the accelerating area A1 during the fault (A1≤A2), synchronism holds, else it slips. Drag the clearing angle to see the critical-angle boundary and learn that fast protection is stability.