Chapter 24

Three-Phase Systems

Three-Phase Voltage Generation

Three-phase generators

Three sets of windings and produce three ac voltages

Windings are placed 120° apart

Voltages are three identical sinusoidal voltages 120° apart

Three-Phase Voltage Generation

Set of voltages such as these are balanced

If you know one of the voltages

The other two are easily determined

Four-Wire Systems

Three loads have common return wire called neutral

If load is balanced

Current in the neutral is zero

Current is small

Wire can be smaller or removed

Current may not be zero, but it is very small

Four-Wire Systems

Outgoing lines are called line or phase conductors

Three-Phase Relationships

Line voltages

Voltages between lines either at the generator (E_AB) or at the load (V_AB)

Three-Phase Relationships

Phase voltages

Voltages across phases

For a Y load, phases are from line to neutral

For D load, the phases are from line to line

Three-Phase Relationships

Line currents

Currents in line conductors

Phase currents

Currents through phases

For a Y load two currents are the same

Voltages in a Wye Circuit

For a balanced Y system

Magnitude of line-to-line voltage is times the magnitude of phase voltage

Each line-to-line voltage

Leads corresponding phase voltage by 30°

Line-to-line voltages form a balanced set

Voltages for a Wye Circuit

Nominal voltages

120/208-V

277/480-V

347/600-V systems

Voltages for a Wye Circuit

Given any voltage at a point in a balanced, three-phase Y system

Determine remaining five voltages using the formulas

Currents for a Wye Circuit

Line currents

Same as phase currents

I_a = V_an/Z_an

Line currents form a balanced set

If you know one current

Determine the other five currents by inspection

Currents for a Delta Load

In a balanced delta

The magnitude of the line current is times the magnitude of the phase current

Currents for a Delta Load

Each line current lags its corresponding phase current by 30°

For any current in a balanced, three-phase delta load

Determine remaining currents by inspection

Power in a Balanced System

To find total power in a balanced system

Determine power in one phase

Multiply by three

Use ac power formulas previously developed

Power in a Balanced System

Since magnitudes are the same for all three phases, simplified notation may be used

Active Power to a Balanced Wye Load

P_f = V_fI_fcos q_f

P_T = 3P_f = 3V_fI_fcos q_f

P_T = V_LI_Lcos q_f

P_f = I_f²R_f

P_T = 3I_f²R_f

Reactive Power to a Balanced Wye Load

Q_f = V_fI_fsin q_f

Q_T = V_LI_Lsin q_f

Q_f = I_f²X_f

Units are VARs

Apparent Power to a Balanced Wye Load

S_f = V_fI_f

S_T = V_LI_L

S_f = I_f²Z_f

Apparent Power to a Balanced Wye Load

Units are VAs

Power factor is

F_p = cos q_f = P_T/S_T = P_f/S_f

Power to a Balanced Delta Load

Power formulas for D load are identical to those for Y load

In all these formulas

Angle q_f is phase angle of the load impedance

Power to a Balanced Delta Load

You can also use single-phase equivalent in power calculations

Power will be power for just one phase

Measuring Power in Three-Phase Circuits

Measuring power to a 4-wire Y load requires three wattmeters (one meter per phase)

Loads may be balanced or unbalanced

Total power is sum of individual powers

Measuring Power in Three-Phase Circuits

If load could be guaranteed to be balanced

Only one meter would be required

Its value multiplied by 3

Measuring Power in Three-Phase Circuits

For a three-wire system

Only two meters are needed

Loads may be Y or D

Loads may be balanced or unbalanced

Total power is algebraic sum of meter readings

Measuring Power in Three-Phase Circuits

Power factor for a balanced load

Obtain from wattmeter readings using a watts ratio curve

Measuring Power in Three-Phase Circuits

From this, q can be determined

Power factor can then be determined from cos q

Unbalanced Loads

Use Ohm’s law

For unbalanced four-wire Y systems without line impedance

Three-wire and four-wire systems with line and neutral impedance

Require use of mesh analysis

Unbalanced Loads

One of the problems with unbalanced loads

Different voltages are obtained across each phase of the load and between neutral points

Unbalanced Loads

Unbalanced four-wire D systems without line impedance are easily handled

Source voltage is applied directly to load

Three-wire and four-wire systems with line and neutral impedance

Require use of mesh analysis

Power System Loads

Single-phase power

Residential and business customers

Single-phase and three-phase systems

Industrial customers

Therefore, there is a need to connect both single-phase and three-phase loads to three-phase systems

Power System Loads

Utility tries to connect one third of its single-phase loads to each phase

Three-phase loads are generally balanced

Power System Loads

Real loads

Seldom expressed in terms of resistance, capacitance, and inductance

Rather, real loads are described in terms of power, power factors, etc.


	Three-phase generators
		Three sets of windings and produce three ac voltages
	Windings are placed 120° apart
		Voltages are three identical sinusoidal voltages 120° apart


	Set of voltages such as these are balanced
	If you know one of the voltages
		The other two are easily determined


	Three loads have common return wire called neutral
	If load is balanced
		Current in the neutral is zero
	Current is small
		Wire can be smaller or removed
		Current may not be zero, but it is very small


	Line voltages
		Voltages between lines either at the generator (E_AB) or at the load (V_AB)


Phase voltages
	Voltages across phases
		For a Y load, phases are from line to neutral
		For D load, the phases are from line to line


	Line currents
		Currents in line conductors
	Phase currents
		Currents through phases
		For a Y load two currents are the same


	For a balanced Y system
		Magnitude of line-to-line voltage is times the magnitude of phase voltage
	Each line-to-line voltage
		Leads corresponding phase voltage by 30°
	Line-to-line voltages form a balanced set


	Given any voltage at a point in a balanced, three-phase Y system
		Determine remaining five voltages using the formulas


Line currents
	Same as phase currents
	I_a = V_an/Z_an
Line currents form a balanced set
	If you know one current
		Determine the other five currents by inspection


	In a balanced delta
		The magnitude of the line current is times the magnitude of the phase current


	Each line current lags its corresponding phase current by 30°
	For any current in a balanced, three-phase delta load
		Determine remaining currents by inspection


	To find total power in a balanced system
		Determine power in one phase
		Multiply by three
	Use ac power formulas previously developed


	Since magnitudes are the same for all three phases, simplified notation may be used


	P_f = V_fI_fcos q_f
	P_T = 3P_f = 3V_fI_fcos q_f
	P_T = V_LI_Lcos q_f
	P_f = I_f²R_f
	P_T = 3I_f²R_f


	Q_f = V_fI_fsin q_f
	Q_T = V_LI_Lsin q_f
	Q_f = I_f²X_f
	Units are VARs


	Units are VAs
	Power factor is
	F_p = cos q_f = P_T/S_T = P_f/S_f


	Power formulas for D load are identical to those for Y load
	In all these formulas
		Angle q_f is phase angle of the load impedance


	You can also use single-phase equivalent in power calculations
		Power will be power for just one phase


	Measuring power to a 4-wire Y load requires three wattmeters (one meter per phase)
	Loads may be balanced or unbalanced
	Total power is sum of individual powers


	If load could be guaranteed to be balanced
		Only one meter would be required
		Its value multiplied by 3


	For a three-wire system
		Only two meters are needed
	Loads may be Y or D
	Loads may be balanced or unbalanced
	Total power is algebraic sum of meter readings


	Power factor for a balanced load
		Obtain from wattmeter readings using a watts ratio curve


	From this, q can be determined
	Power factor can then be determined from cos q


	Use Ohm’s law
		For unbalanced four-wire Y systems without line impedance
	Three-wire and four-wire systems with line and neutral impedance
		Require use of mesh analysis


	One of the problems with unbalanced loads
		Different voltages are obtained across each phase of the load and between neutral points


	Unbalanced four-wire D systems without line impedance are easily handled
		Source voltage is applied directly to load
	Three-wire and four-wire systems with line and neutral impedance
		Require use of mesh analysis


	Single-phase power
		Residential and business customers
	Single-phase and three-phase systems
		Industrial customers
		Therefore, there is a need to connect both single-phase and three-phase loads to three-phase systems


	Utility tries to connect one third of its single-phase loads to each phase
	Three-phase loads are generally balanced


	Real loads
		Seldom expressed in terms of resistance, capacitance, and inductance
		Rather, real loads are described in terms of power, power factors, etc.