Unbalanced Three Phase System

A special technique for handling unbalanced three phase systems is the method of symmetrical components, which is beyond the scope of this text. We will learn this in the near future.

Make sure to read “what a three phase circuit is” first.

After we learn about three phase circuit, we will learn:

1. Balanced three phase voltage
2. Balanced three phase power
3. Unbalanced three phase power
4. Three phase power measurement

Unbalanced Three Phase Systems

Learning and understanding three phase systems would be incomplete without learning and analyzing unbalanced three phase systems.

An unbalanced three phase system is when our three phase AC circuit has sinusoidal waveforms of its voltage or current not equal or/and not shifted exactly by 120o from each other.

An unbalanced three-phase system is not a rare thing in power transmission and distribution.

When we are dealing with either:

1. Balanced three-phase system
2. Unbalanced three-phase system

We need to know what caused them to go “balanced” or “unbalanced”.

There are few causes of this unbalanced system:

1. The voltage sources are not equal in magnitude and/or have differences in phase angle from each other phase.
2. The loads are unequal from each other.
3. The impedances are not equal from each other, caused by different wire sizes, wire lengths, source impedances.
4. There is a system fault happening in our electrical circuit, such as damaged components, blown fuses, or excessive power loss caused by the imbalance.
5. Harmonic distortion caused by nonlinear components such as inductors and capacitors. These components are able to distort our sinusoidal waveforms either voltage or current.

In balanced system, we always have:

1. Equal voltage sources in magnitude and phase angle. For example, a three-phase system with a voltage source at 120V and 50Hz frequency for each phase.
2. Equal load impedance. For example, a three-phase system with only resistance loads or inductive loads or capacitive loads with the same value for all lines.

Thus,

An unbalanced system is due to unbalanced voltage sources or an unbalanced load.

There is also one thing to remember that an unbalanced three-phase voltage source is a very rare phenomenon.

Analyzing unbalanced three-phase systems will take a lot of time. Hence, in this post we will assume that every circuit we use has balanced voltage sources and unbalanced load impedances.

Unbalanced Load in Three Phase System

A three-phase system is balanced if all the line loads are equal to each other. If one of the loads is increased, then it will be an unbalanced system.

Why?

Because that line will draw more current than the other two.

What are the effects of an unbalanced three phase system? There will be:

• Increased heat by three phase motors.
• Reduced lifetime of machine by increased heat and stress.
• Increased operational costs since the reactive power is generated more.
• Power losses I²R increased since active power is less than reactive power.
• Motor drives become unreliable and reduced efficiency because the generated torque is lower.
• Significant neutral current flow in the circuit where it won’t be happening in a balanced circuit.

Properties of Unbalanced Three-Phase System

• The three-phase waveform is disturbed.
• The line currents are not equal to each other.
• Neutral wire is needed.
• Higher power losses.

How to Solve Unbalanced Three Phase System

Is there a way to fix an unbalanced three phase system? Before we are dealing with how to make an unbalanced circuit to balanced circuit, there are few things we can do to solve an unbalanced three phase system into balanced three phase system:

• Load balancing by distributing the load evenly to each phase.
• Fault detector can help us to address any faults and repair them as soon as possible.
• Power factor correction will boost our power factor to generate and consume more efficient power.
• Apply a filter to diminish the harmonic distortion that exists in our circuit.
• Impedance matching to ensure every impedance in each phase is adjusted.

Even if we are dealing with three-phase systems, we can still use mesh and nodal analysis. Of course KCL and KVL are also useful.

Observe the circuit below. Here we have an star-connected four-wire unbalanced three-phase system consists of:

• Balanced three-phase voltage source (not drawn in the circuit)
• Unbalanced star-connected load impedances (Z₁, Z₂, and Z₃).

Since we already set the load impedances are unbalanced, all the Z_A, Z_B, and Z_C are unequal.

Using Ohm’s law, we get the line currents as:

In a balanced three-phase system, current produced in the neutral line should be zero. But here, the current in the neutral line is not zero.

Applying KCL at node N gives the neutral line current as

If we are dealing with a three-phase system which doesn’t have a neutral line, we can still use mesh analysis to find line currents I_a, I_b, and I_c. This includes the delta-star, star-delta, and delta-delta three-phase (and three-wire) system.

If we are talking about long distance power distribution and transmission, we are dealing with multiple three-wire systems with earth as the neutral conductor.

To calculate power in an unbalanced three-phase system requires that we find the power in each phase.

The total power is not simply three times the power in one phase but the sum of the powers in the three phases.

Unbalanced Three-Phase Systems Problem Examples

1. The unbalanced star-load of the circuit before has balanced voltage sources 100 V and the a c b sequence. Calculate the line currents and the neutral current, if known

Z_A = 15 Ω
Z_B = 10 + j5 Ω
Z_C = 6 − j8 Ω.

Solution:

Using Equation.(1), the line currents are

Using Equation.(2), the current in the neutral line is

2. Observe the circuit in Figure.(2) below. Here we have an unbalanced circuit with star-star connection three-wire. Find:
(a) the line currents
(b) the total complex power absorbed by the load
(c) the total complex power supplied by the source.

Solution:

(a) We use mesh analysis to find the required currents. For mesh 1,

Or

For mesh 2,

Or

Equations.(2.1) and (2.2) form a matrix equation:

The determinants are

The mesh currents are

The line currents are

(b) We can now calculate the complex power absorbed by the load.

For phase A,

For phase B,

For phase C,

The total complex power absorbed by the load is

(c) We check the result above by finding the power supplied by the source. For the voltage source in phase a,

For the source in phase b,

For the source in phase c,

The total complex power supplied by the three-phase source is

showing that S_S + S_L = 0 and confirming the conservation principle of ac power.

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