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.

About the Author

- Radoje - Rade Jankovic Electrician, Electrical Technician, Electrical Engineer, PhD, Ecologist, Environmentalist, Designer, Educator, Investigator... Today PROFESSIONAL TECHNICAL WRITER AND DOCUMENTATOR Massive practical experience in almost all fields of electrical engineering over 40 years. My International education group in Facebook: EEEW - ELECTRICAL & ELECTRONIC EDUCATION WORLD where published more than 8000 small or bigger articles, lessons, technical advice, projects, technical calculations, test questions with and without answers, illustrated test questions with and without answers; all my original works and few thousands of may original photos from practice, drawings, circuit diagrams, environmental lessons and examples from everyday practice etc., etc.

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Three-phase system connected in a triangle – All formulas