3 Phase Power

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{3}A_{32} Base Sequence = 12735 String Sequence = 12735 - 3 - 32 **

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3 Phase Power

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Figure 8.56 is a *Wye* (or *Y*) configuration of a Balanced three phase AC circuit. Show that:

(a) The magnitude of the line voltages is equal to √3 times the magnitude of the phase voltages.

(b) No conducting wire is needed to connect nodes *n* and *n ^{'}*.

(c) If the 3 balanced load impedances are replaced with 3 equal resistances

**The string**:

S_{7}P_{3}A_{32} (Force - Push).
**The math**:

Pj Problem of interest is of type *force*. Power and energy problems are *force problems*

Three-phase power is a configuration of three sinusoidal voltages that are generated out of phase with each other. It came about as a result of the need to improve the efficiency of single phase power delivery.
some of its advantages include:

(1) Delivery of steady constant supply of power. A single phase delivery is pulsating.

(2) Reduced transmission losses over long distances.

(3) Efficient use of conductors and circuit components

(4) Nonzero starting torque (moment of a force) for three-phase motors.

Balanced voltages implies that the voltages have equal amplitude and frequency and are out of phase by 120 degrees.

So, the phase voltages of V_{an}, V_{bn} and V_{cn} in phasor form are:

V_{an} = V_{an}<0^{o}

V_{bn} = V_{bn}<-(120)^{o}

V_{cn} = V_{cn}<-(240)^{o} =
V_{cn}<120^{o}

Sum of phase voltages equal zero. That is:

V_{an} + V_{bn} + V_{cn} = 0.

Voltage amplitudes, V are rms values,

(a) The *line voltages* (or line-line-voltages) in figure 8.56 are:

Voltages between lines *aa ^{'}* and

Voltages between lines

Voltages between lines

Line voltage between

V

V

So, V

Line voltages between

V

= V

Line voltages between

V

= V

So, amplitudes of line voltages equal √3 times amplitude of phase voltages.

(b) I

= I

= (V

So, no need for a conducting wire between the nodes since I

(c) General expression for instantaneous power

where V is rms value.

So, instantaneous power due to source V

p

Instantaneous power due to source V

p

Instantaneous power due to source V

p

Total power P(t) = p

= 3V

So, P(t) is a constant.

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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