## 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.

Line voltages 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

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

**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.

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.

**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.

From (a) it follows:

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

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:

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:

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.