Published On: Sat, Mar 14th, 2020

Three-phase system connected in a triangle – All formulas

Author: Radoje Jankovic.

Calculated and drawn by Jankovic

Original work

The following will be shown here:
1. Formation of line voltages in a three-phase winding connected in a triangle
2. Voltage, current and power in a three-phase system with impedances connected in a triangle
3. Power of a three-phase symmetric system with a resistive load connected in a triangle
4. Power of a three-phase symmetric system with active loads connected in a triangle
5. Calculation of three-phase resistive circuits with a consumer connection in a triangle

Therefore, everything is so clear that anyone can solve every three-phase system in a triangle and find the answers to many questions and problems you encounter in your daily practice and your work on the repair of three-phase electrical appliances.

1. FORMATION OF LINE VOLTAGES IN THREE-PHASE COIL CONNECTED

The scheme of connection and designation in FIG. 1.

FIG. 1. A three-phase windings connected to a triangle
FIG. 2. In the case of a three-phase windings connected to a triangle, the connecting conductors obtain the potential of a two-phase connection point

Line voltages for a three-phase winding connected to a triangle:

FIG. 3. Vector representation of the line voltage of one phase of the winding connected to a triangle
FIG. 4. Vector relations of voltage for a three-phase winding connected to a triangle

2. Voltage, current and power in a three-phase system with impedances connected to a triangle

Scheme of connection and markings shown in Fig. 1. The system is symmetrical.


Instantaneous voltage values:

Generator voltages, FIG. 1.

Phase voltages

Instantaneous values ​​of currents
Load phase currents

Load line currents

Phi, complex load impedance argument
Vm, maximum voltage value
V = V1, phase voltages = line voltages

FIG. 1. Three-phase system connected in a triangle

Generator = Generator,
Linija = Line
Potrošač = consumer, load

Vector diagram of currents and voltages of a three-phase system of FIG. 1, is shown in FIG. 2.

FIG. 2. Voltages and currents in a three-phase system with a triangle connection

Effective voltages and currents
Voltages:

Line and phase currents

Complex voltages

Complex currents

Load line currents

Iz  –  struja kroz potrošač = The current through load 

For a symmetric system, a vector diagram in two forms is shown in Figs. 3 and FIG. 4.

FIG. 3. Vector diagram of voltage and current – first form
FIG. 4. Vector diagram of current and voltage in phase start – second form
 

Currents for symmetrically pure ohmic load

Line currents

For

The currents are:

Phase vector currents of consumers-loads and line currents are shown in Figs. 5. while in FIG. 6 shows the line and phase currents of a three phase consumer of FIG. 1 connected to a triangle.

Sl. 5. Vektors of phase currents of loads and the line voltages
FIG. 6. Line currents of a three-phase consumer connected to the triangle of FIG. 1. The line current system is 90⁰ phase delayed relative to the phase current system.

UNSYMMTERICAL LOADED THREE-PHASE SYSTEM WITH CONSUMERS CONNECTED IN THE TRIANGLE

Impedance

Voltgages

Phase currents:

Complex power:

where

Consumer-load phase angles in individual phases.

Phase currents:

Line currents:

Example of calculation 1.

Resistors of values ​​R1 = 20 Ω, R2 = 10 Ω, R3 = 5 Ω are connected to a triangle and connected to three-phase voltage 460/260 VAC. Calculate all phase and line currents.

Callcullations:

The line voltage is Vl = 460 V, so the currents through the resistors are:

Phase currents,

Complex phase currents now:

Line currents are:

That is:

Calculation Example 2.

Three windings whose reactive resistances XL1 = 5 Ω, XL2 = 4 Ω, XL3 = 3 Ω are connected to a triangle and connected to a line voltage of 270 V, 60 Hz. Calculate phase and line currents in this electrical circuit.

Calculation:

Phase currents with voltage

Are

That is

Line currents are:

Calculation example 3.

Three phase symmetric line voltage system Vl = 200 V, powered by a consumer whose impedances are:

And

connected in a triangle. Calculate phase line currents in a circuit.

Calculation

Consumer phase currents are:

Currents in power lines:

3. THE POWER OF A THREE-PHASE SYMMETRIC SYSTEM WITH A RESISTANCE LOAD CONNECTED IN A TRIANGLE

The scheme of the connection and the markings in FIG. 1. The system is symmetrical.

FIG. 1. A resistive three-phase consumer-load connected to a triangle

From (a) it follows:

FIG. 2. Diagram of phase and line currents for a symmetrical resistive consumer connected in a triangle

From (b) it follows:

From (c) is:

Pdelta – total power of load in W

Vf1, Vl – phase and line voltage in V

If, Il – phase and line current in A

Rf – the resistance of load in the branch of triangle in ohms.

Pdelta – total load power in W
Vf1, Vl – phase and line voltage in V
If, Il – phase and line current in A
Rf – load resistance in the branch of triangle in ohms.

4. POWER OF A THREE-PHASE SYMMETRIC SYSTEM WITH ACTIVE CONDUCTIVES CONNECTED IN A TRIANGLE

The scheme of the connection and the markings in FIG. 1. The system is symmetrical.

Three-phase power

Active conductivity

Phase current

Line current

Power of the consumer-load

FIG. 1. A three-phase consumer-load connected in a triangle

From (a) it follows:

From (b) is:

Phase conductivity:

Line current:

5. CALCULATION OF THREE-PHASE RESISTANCE ELECTRIC CIRCUITS WITH CONNECTOR CONNECTION

Example of calculation 1.

Using the form for a resistive three-phase electric circuit with a connection of resistors in a triangle, calculate all parameters for the resistive consumer in Figs. 1.

Calculation

According to FIG. 1. there is:

Line voltage: Vl = 400 V
Phase voltage: Vf = Vl


Phase current:

Line current:

FIG. 1. Resistive consumer-load connected to a triangle and connected to a three-phase network

Power per phase:

Total three-phase power:

Calculation example 2.

The power of the water heater in Example 1 is 1,500 W per phase. Calculate all the parameters of this three-phase electric circuit using all power forms and check the bills.

Calculation

Power per phase and voltage:

Phase current:

Line currents:

The resistance of heater per phase:

Total three-phase power:

Phase current:

Line current:

Line voltage check:

Example of calculation 3.

Using the forms for active conductivities in a three-phase electrical circuit, calculate all the parameter of the three-phase consumer in the electrical circuit in Figs. 2.

Calculation

Phase / line voltage:

Phase current:

Line current:

FIG. 2. Three active conductances connected in the triangle

Power per phase:

Total power of three-phase consumer:

Example of calculation 4.

The three-phase heaters of FIG. 2 have a power per phase of 3,000 W. Using the active electrical conductivity forms, calculate all the parameters of this electrical circuit.

Calculation

Phase / line voltage:

Phase power:

Total power:

Line currents:

Phase currents:

Phase conductivity of consumer or load in phase

Phase current, check:

Line current, check:

That is all.

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Three-phase system connected in a triangle – All formulas