BALANCED WYE-WYE CONNECTION

The voltages in the three phase power system are produced by a synchronous generator. In a balanced system, each of the three instantaneous voltages have equal amplitudes but separated from other voltages by a phase angle of 120 degrees. The three voltages are typically labeled a, b and c. The common reference point for the three phase voltages is designated as the neutral connection and is labeled as n. We may defined either a positive phase sequence or a negative sequence. The three sources Van, Vbn, and Vcn are designated as the line to neutral voltages in the three phase system.

ABC sequence
line to line voltage


-the voltage differences between the phases.


Vab = Van-Vbn


Vbc = Vbn-Vcn


Vca = Vcn-Van


Line to neutral voltage


Van   =  Vp∠0

Vbn  =  Vp∠-120
Vcn    =  Vp∠+120

MAXIMUM AVERAGE POWER TRANSFER

The maximum power transfer theorem for DC circuit, we can determine the condition for an AC load to absorb maximum power in an AC circuit. For an AC circuit, both the thevenin impedance and the load can have a reactive component. Although these reactances do not absorb any average power, they will limit the circuit current unless the load reactances cancels the reactance of the thevenin impedance. For the maximum power transfer, the thevenin and load reactances must be equal in magnitude but opposite in sign.

Maximum Average Power is transferred from a source to a load, the load impedance should be chosen equal to the conjugate of the Thevenin equivalent impedance representing the reminder of the network.

Maximum power transfer theorem is explained by the figure. Let a network consisting of one or more independent sources and other resistive elements be represented by its Thevenin's equivalent circuit.  We find out now the value of load resistance for which the power transferred to it is the maximum.

The average power dissipated in the load is the square of the current multiplied by the resistive portion (the real part)   of the load impedance.

POWER FACTOR

Power Factor is the ratio of true power or watts to apparent power or volts amps. They are identical only when current and voltage are in phase than the power factor is 1.0 The power in an Ac circuit is very seldom equal equal to the direct product of volts and amperes. In order to find the power of a single phase ac circuit the products of volts and amperes mus be multiplied by the power factor. Low power factor is usually not that much of a problem in residential homes.

The angle of this power triangle graphically indicates the ratio between the amount of dissipated power and the amount absorbed/returned power. It also happens to be the same angle  as that of the circuit's impedance in polar form.

A low power factor is the result of inductive loads such as transformers and electric motors. Unlike resistive loads creating heat by consuming kilowatts, inductive loads require a current flow to create magnetic field to produce the desired work.




LEADING and LAGGING power factors

• Inductive loads - transformers, motors and wound coils - consumes reactive power with current waveform lagging the voltage.
• Capacitive loads - capacitor banks or buried cables - generates reactive power with current phase leading the voltage. 

THREE PHASE SEQUENCE

Three phase electric power systems have at least three conductors carrying voltage waveforms that are 2π/3 radians offset in time.

A Three Phase Wave

The sinusoidal functions or in waves has two different, but closely related meanings. One is the initial angle of a siusoidal function at its origin and is sometimes called phase offset.

Three Phase AC generators

The principles of a three phase generator are basically the same as that of a single phase generator, except that there are three equally spaced windings and three  output voltages that are all 120 degrees  out of phase with one another.

Three Phase Ac Motor

The Ac induction motor is a rotating electric machine designed to operate from a three phase source of alternating voltage. The stator is a classic three phase stator with the winding displaced by 120 degrees.

INSTANTANEOUS POWER AND AVERAGE POWER

Power is the most important quantity in electric utilities, electronics . . . because such system involve transmission of power from one point to another.


Instantaneous power is the power at any instant of time. In Ac circuit the instantaneous electric power is given by 

P= VI

but these quantities are continuously varying. Instantaneous Power p(t) is the power, p(t)= u(t)*i(t). It is the product of the time functions of the voltage and current. This definition of instantaneous  power is valid for signals of any waveform. The unit for instantaneous power is VA.

REAL and ACTIVE POWER

P can be define in two ways: as the real part of the complex power or as the simple average of the instantaneous power. The second definition is more general because with it we can define the instantaneous power for any signal waveform, not just sinusoids.

The instantaneous power at any time t can be expressed as:

Pinstantaneous= VmImsinωtsin(ωt-ɸ)

Average power is more convenient to measure. It is the average of the instantaneous power over one period. And it is given by:

Pave= VIcosɸ

where ɸ is the phase angle between the current and the voltage and where V and I are understood to be the effective or rms of the voltage and current. The term cos ɸ is called the power factor for the circuit.