Three-Phase Circuits

Three-Phase Circuits =)

       Initially we explored the idea of three-phase power systems by connecting three voltage sources together in what is commonly known as the “Y” (or “star”) configuration. This configuration of voltage sources is characterized by a common connection point joining one side of each source. (Figure below)

Three-phase “Y” connection has three voltage sources connected to a common point.

If we draw a circuit showing each voltage source to be a coil of wire (alternator or transformer winding) and do some slight rearranging, the “Y” configuration becomes more obvious in Figure below.

Three-phase, four-wire “Y” connection uses a "common" fourth wire.

The three conductors leading away from the voltage sources (winding's) toward a load are typically called lines, while the windings themselves are typically called phases. In a Y-connected system, there may or may not (Figure below) be a neutral wire attached at the junction point in the middle, although it certainly helps alleviate potential problems should one element of a three-phase load fail open, as discussed earlier.

Three-phase, three-wire “Y” connection does not use the neutral wire.

Overview and Insights

-The conductors connected to the three points of a three-phase source or load are called lines.

-The three components comprising a three-phase source or load are called phases.

-Line voltage is the voltage measured between any two lines in a three-phase circuit.

-Phase voltage is the voltage measured across a single component in a three-phase source or load.

-Line current is the current through any one line between a three-phase source and load.

-Phase current is the current through any one component comprising a three-phase source or load.

-In balanced “Y” circuits, line voltage is equal to phase voltage times the square root of 3, while line current is equal to phase current.

 

-In balanced Δ circuits, line voltage is equal to phase voltage, while line current is equal to phase current times the square root of 3.

-Δ-connected three-phase voltage sources give greater reliability in the event of winding failure than Y-connected sources. However, Y-connected sources can deliver the same amount of power with less line current than Δ-connected sources.

Power Factor & Complex Power

Power Factor & Complex Power =)

Power Factor 

v(t) = Vm cos(ωt + θv)

and

i(t) = Im cos(ωt + θi)

The average power is a product of two terms. The product Vrms Irms is known as the apparent power S. The factor cos(θv − θi) is called the power factor (pf).

S = Vrms Irms

The apparent power (in VA) is the product ofthe rms values ofvoltage and current.

The power factor is dimensionless, since it is the ratio of the average power to the apparent power,

pf =P/S= cos(θv − θi)

The angle θv − θi is called the power factor angle, since it is the angle whose cosine is the power factor.

The power factor is the cosine ofthe phase difference between voltage and current. It is also the cosine ofthe angle ofthe load impedance.

Complex Power

        Complex power (in VA) is the product ofthe rms voltage phasor and the complex conjugate ofthe rms current phasor. As a complex quantity, its real part is real power P and its imaginary part is reactive power Q.

It is a standard practice to represent S, P, and Q in the form of

a triangle, known as the power triangle, shown below,

Overview and Insights

- Real Power (P) in measured in W, Reactive Power (Q) in VAR, and Apparent Power (S) in VA.

- For Power Factor (PF), when theta increases PF decreases, and when theta decreases PF increases.

Reactive Power:

Q = 0 for resistive loads (unity pf).

Q < 0 for capacitive loads (leading pf).

Q > 0 for inductive loads (lagging pf).

AC Power Analysis

AC Power Analysis =)

Instantaneous and Average Power

The instantaneously power, p(t) 

Instantaneous power is the power of a any object at an instant. if you differentiate work done w.r.t time it will be the instantaneous power. If the given velocity is instantaneous the power=F*v

Average Power

The average power is the average of the instantaneous power over one period.

Maximum Average Power Transfer

We have already seen that an AC circuit can (at one frequency) be replaced by a Thévenin or Norton equivalent circuit. Based on this technique, and with the Maximum Power Transfer Theorem for DC circuits, we can determine the conditions for an AC load to absorb maximum power in an AC circuit. For an AC circuit, both the Thévenin 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 reactance cancels the reactance of the Thévenin impedance. Consequently, for maximum power transfer, the Thévenin and load reactances must be equal in magnitude but opposite in sign; furthermore, the resistive parts -according to the DC maximum power theorem- must be equal. In another words the load impedance must be the conjugate of the equivalent Thévenin impedance. The same rule applies for the load and Norton admittances.

RL= Re{ZTh} and XL = - Im{ZTh}

The maximum power in this case:

Where V2Th and I2N represent the square of the sinusoidal peak values.

For maximum average power transfer, the load impedance ZL must be equal to the complex conjugate ofthe Thevenin impedance ZTh.

Overview and Insights

              Power analysis is another chapter and view of understanding in our class since it has lesser circuit analysis. Power is the most important quantity in electric utilities, electronic, and communication systems, because such systems involve transmission of power from one point to another.