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# Brief information from the theory

The three-phase system was first created to solve the problem of converting electrical energy into mechanical energy by creating a circular rotating magnetic field. It is widespread due to the simplicity of the three-phase asynchronous motor.

The main elements of three-phase circuits are three-phase generator, three-phase transformer, single-phase or three-phase load and connecting wires.

A three-phase generating system can be created from three single-phase sources, the EMF of which can be represented by expressions:

eA = EAm ∙ sin ω ∙ t;

eB = EBm ∙ sin (ω ∙ t + 2400);

eC = ECm ∙ sin (ω ∙ t + 1200).

In complex form these EDS can be recorded:

EA = EAm ∙ e j∙ω t;

EB = EBm ∙ e(ϳ∙ω∙t-120˚) ;

EC = ECme(ϳ∙ω∙t-240 ˚).

The time diagram of the three-phase voltage generator is shown in fig. 5.1. Fig. 5.1. The timing diagram of voltages of three-phase generator

Most often, the energy systems of industrial, administrative and residential complexes will be implemented according to the star-star scheme with a zero wire, as shown in fig. 5. 2 a, sometimes under the scheme "star" - "star" shown in fig. 5..2 b.

There are the following types of three-phase loads.

1. A symmetrical load, which is equal to the complex impedance voltages phase of the receiver:

ZA = ZB = ZC = ZA . a b

Fig. 5. 2. Connection scheme of three-phase systems: a ? ?star? - ?star? with zero wire, b ? ?star? - ?star?

1. Uniform load at which the resistance modules of the receiver phases are equal:

ZA = ZB = ZC.

2. Uniform load, which is equal to the arguments of the resistance and the phase of the receiver:

A = ⱷB = ⱷC.

3, Asymmetric load at which resistance complexes are not equal in all phases of the receiver:

ZAZBZC.

For the ?star? - ?star? scheme with zero wire under arbitrary load, the following relations are valid for the phase current complexes

Ia = Ua / Za ; Ib = Ub / Zb ; Ic = Uc / Zc.

For neutral current (zero wire):

I0 = Ia + Ib + Ic .

For linear and phase voltages of the same circuit

Ul = ∙ Uph.

At symmetric load, the phase current modules are equal to each other

Ia = Ib = Ic = Uph / Zph,

and the current in the neutral is zero

I0 = 0.

For the "star" - "star" schema without neutral for arbitrary load neutral voltage

UN = (UA * Ya+ UBYb + UCYC) / (Y0 + Ya + Yb + Yc).

The voltage on the load phases,,

U?a = UA̶ UN ; U?b = UB̶ UN ; U?c = UC̶ UN.

The currents in the phases of the load:

Ia = U?a / Z? ; Ib= U?b/ Zb ; Ic= U?c/ Zc..

At the same time, in each phase of the circuit, a complex linear currents is equal to the phases current.

Il = Iph .

Phase shift angles between phase currents and voltages

a = arctg (Xa / Ra); ⱷb = arctg (Xb / Rb); ⱷc = arctg (Xc / Rc),

where Ra, Rb, Rc ? resistance of the load phases;

Xa, Xb, Xc ? their reaction resistances.

Active power of three-phase circuit under arbitrary load

P = Pa + Pb + Pc ;

reactive power

Q = Qa + Qb + Qc ;

full power

S = .

P = 3 ∙ Uph ∙ Iph cos l = ∙ Ul ∙ Il cosl;
Q = 3 ∙ Uph ∙ Iph sinl = ∙ Ul ∙ Il sinl;
S = 3 ∙ Uph ∙ Iph = ∙ Ul ∙ Il .