Unbalanced three-phase systems and symmetrical components

Lesson 7 analysed balanced three-phase systems by solving one phase and shifting the answers by 120°. Real systems are never perfectly balanced: single-phase loads are spread unevenly across phases, a fuse blows on one phase, a conductor breaks, or a fault connects one phase to ground. Unbalance raises neutral currents, distorts voltages, overheats motors and, during faults, produces the currents that protection systems must detect. This lesson studies unbalanced wye and delta loads directly, shows what happens when a neutral is lost, explains how unbalance is measured and why it harms motors, and introduces symmetrical components, the method engineers use to analyse unbalanced conditions and faults.

Unbalanced wye loads with a neutral

If an unbalanced wye load is connected to a four-wire supply with a solid neutral, each phase's load sees its own phase voltage, regardless of the others. Each phase current is simply \(I_a = V_{an}/Z_a\), and so on, and the neutral carries the phasor sum:

Because the currents are 120° apart, the neutral current is much less than their arithmetic sum, and zero when they are balanced. It is not zero for unbalanced loads, and with non-linear loads it can exceed the phase currents, because triplen harmonics (third, ninth and so on) add in the neutral rather than cancelling (Lesson 11).

Losing the neutral

If the neutral conductor of a wye-connected unbalanced load opens, the load's star point is no longer held at the source neutral. It floats to a voltage that makes the three phase currents sum to zero, and the load voltages become unequal: phases with small loads (high impedance) get more than their rated voltage, and heavily loaded phases get less. The floating neutral voltage follows from Millman's theorem:

import numpy as np

# 120/208 V wye supply; unbalanced resistive loads per phase (ohms)
Vp = 120.0
Va, Vb, Vc = (Vp * np.exp(1j * np.radians(ang)) for ang in (0, -120, 120))
Za, Zb, Zc = 4.0, 12.0, 24.0

def show(label, Vn):
    vols = [abs(V - Vn) for V in (Va, Vb, Vc)]
    print(f"{label}: neutral shift {abs(Vn):5.1f} V; load voltages a {vols[0]:5.1f}, b {vols[1]:5.1f}, c {vols[2]:5.1f} V")

# With a solid neutral, the load neutral stays at 0 V
show("neutral connected   ", 0)

# With the neutral open, the load star point floats (Millman theorem)
Ya, Yb, Yc = 1 / Za, 1 / Zb, 1 / Zc
Vn = (Va * Ya + Vb * Yb + Vc * Yc) / (Ya + Yb + Yc)
show("neutral disconnected", Vn)

neutral connected   : neutral shift   0.0 V; load voltages a 120.0, b 120.0, c 120.0 V
neutral disconnected: neutral shift  61.1 V; load voltages a  61.1, b 151.4, c 166.5 V

With the neutral open, the lightly loaded phase c sees 167 V, 39% above rating, which can destroy electronics and lamps within seconds, while the heavily loaded phase a sees only 61 V. A lost neutral on a 120/240 V residential service has the same effect between the two halves of a house. This is why neutrals are never fused or switched alone, why their connections must be secure, and why surge and overvoltage complaints in one part of a building often trace to a loose neutral.

Unbalanced delta loads

In a delta load, each impedance is connected directly across a line-to-line voltage, so the phase currents are \(I_{ab} = V_{ab}/Z_{ab}\), etc., independent of each other even when unbalanced. The line currents follow from KCL at each corner:

With unbalanced loads, the line currents are no longer \(\sqrt{3}\) times the phase currents, and must be calculated phasor by phase.

Power in unbalanced systems

The total complex power is the sum of the complex powers of the three phases, \(S = V_{an}I_a^* + V_{bn}I_b^* + V_{cn}I_c^*\), and the convenient balanced formula \(\sqrt{3}V_LI_L\cos\theta\) no longer applies. Measurement follows from Blondel's theorem: in an \(n\)-wire system, \(n - 1\) wattmeters measure total power exactly, whatever the unbalance. A three-wire system therefore needs two wattmeters (the two-wattmeter method of Lesson 7), and a four-wire system three. Revenue meters for three-phase services measure each phase separately for the same reason.

Measuring unbalance and its effects

Voltage unbalance is quantified in two ways:

• The NEMA definition: the maximum deviation of a line voltage from the average of the three, divided by the average, as a percentage.
• The IEC definition: the ratio of negative-sequence to positive-sequence voltage, \(|V_2|/|V_1|\), defined below.

They give similar values for small unbalance. Utilities and standards generally aim to keep voltage unbalance at a service below about 2% to 3%.

Motors are especially sensitive because a small voltage unbalance produces a much larger current unbalance: the negative-sequence voltage drives currents through the motor's low negative-sequence impedance (close to its locked-rotor impedance), creating a reverse-rotating field that brakes the rotor and heats it. As a rule of thumb, current unbalance is six to ten times the voltage unbalance, and temperature rise in the hottest winding increases by roughly twice the square of the percent voltage unbalance.

Symmetrical components

In 1918 Charles Fortescue showed that any set of three unbalanced phasors can be written as the sum of three balanced sets:

• Positive sequence (\(V_1\)): three equal phasors 120° apart in the normal order a, b, c, the component a balanced system has.
• Negative sequence (\(V_2\)): three equal phasors 120° apart in the reverse order a, c, b.
• Zero sequence (\(V_0\)): three equal phasors in phase with each other.

With the operator \(a = 1\angle 120°\), the transformation is

import numpy as np

a = np.exp(2j * np.pi / 3)            # the 120-degree operator
A = np.array([[1, 1, 1], [1, a ** 2, a], [1, a, a ** 2]])

def polar(z):
    return f"{abs(z):7.2f} at {np.degrees(np.angle(z)):7.1f} deg"

# Unbalanced phase currents measured on a feeder (A)
Iabc = np.array([100 * np.exp(1j * np.radians(-20)),
                 80 * np.exp(1j * np.radians(-150)),
                 90 * np.exp(1j * np.radians(95))])
I012 = np.linalg.solve(A, Iabc)
for name, I in zip(("zero", "positive", "negative"), I012):
    print(f"{name:8s} sequence: {polar(I)} A")
print(f"neutral current 3*I0 = {polar(3 * I012[0])} A; direct sum = {polar(Iabc.sum())} A")
print(f"current unbalance I2/I1 = {100 * abs(I012[2]) / abs(I012[1]):.1f}%")

# Check: rebuild the phase currents from the components
print(f"reconstruction error: {np.max(np.abs(A @ I012 - Iabc)):.1e} A")

zero     sequence:    7.62 at    42.5 deg A
positive sequence:   89.77 at   -24.6 deg A
negative sequence:    7.02 at   -16.0 deg A
neutral current 3*I0 =   22.86 at    42.5 deg A; direct sum =   22.86 at    42.5 deg A
current unbalance I2/I1 = 7.8%
reconstruction error: 2.5e-14 A

The components have physical meaning, which is why they are so useful:

• Only the positive sequence does useful work in a motor; the negative sequence brakes and heats it.
• The zero sequence flows only where there is a return path: in the neutral of a grounded wye system, or in the earth. The neutral current is exactly \(3I_0\). Zero-sequence current cannot flow in the lines of a three-wire system, and circulates inside a delta winding without leaving it.
• Balanced, symmetrical equipment such as transmission lines, generators and transformers acts on each sequence independently, with its own positive-, negative- and zero-sequence impedances. An unbalanced problem in a balanced network becomes three uncoupled balanced problems, each solvable per phase.

Faults and sequence networks

The most important application is short-circuit (fault) analysis. The most common fault on overhead lines is a single line-to-ground fault, which is highly unbalanced. Connecting the positive-, negative- and zero-sequence networks of the system in series at the fault point gives the fault current:

where \(V_f\) is the pre-fault phase voltage and \(Z_f\) the fault impedance. A three-phase fault, by contrast, is balanced and involves only the positive-sequence network: \(I_f = V_f/Z_1\).