**Single Phase and Three Phase Transmission**

**Power Flow Analysis**

**1-Phase Transmission**

Let
us consider an inductive circuit and let the instantaneous voltage be

v = V_{m}
sin wt

Then the current will be i = I_{m} sin (wt-É¸) where É¸ is the angle by which the current
lags the voltage.

Assume
that the source is perfect sinusoidal with fundamental frequency component
only.

The instantaneous power is given by

p =
vi = V_{m }sin wt . I_{m} sin(wt-É¸)

= V_{m }I_{m}_{ }sin wt . sin(wt-É¸)

= V_{m }I_{m}/2_{ }[cos É¸ - cos(2wt -É¸)]

The value of p is
positive when both v and i are either positive or negative and represents the
rate at which the energy is being consumed by the load. In this case the
current flows in the direction of voltage drop. On the other hand power is
negative when the current flows in the direction of voltage rise which means
that the energy is being transferred from the load into the network to which it
is connected.

We have decomposed the instantaneous
power into two components.

The component
marked *I *defined the real
power or active power P which
physically means the useful power being transmitted. The power marked *II *contains
sin É¸ which is negative for capacitive circuit and positive for inductive
circuit. This component is known as reactive power.

p = P(1- cos 2wt)
- Q sin 2wt.

**3
Phase Transmission**

Assume that the system is balanced
which means that the 3 phase voltages and currents are balanced.

V_{a} = V_{m}
sin wt

V_{b} = V_{m}
sin(wt–120)

V_{c} = V_{m}
sin(wt+120)

I_{a} = I_{m} sin (wt-É¸)

I_{b} = I_{m} sin (wt-É¸-120)

I_{c} = I_{m} sin (wt-É¸+120)

The total power transmitted equals to
sum of the individual powers in each phase

p
= V_{a}I_{a} + V_{b}I_{b}
+ V_{c}I_{c}

_{}

_{ }= 3VI cos É¸

This shows that
the total instantaneous 3-phase power is constant and is equal to three times
the active power per phase i.e. p = 3P, where P is the power per phase.

In case of single phase transmission we noted that the instantaneous
power expression contained both the active power and reactive power expression
but here in case of 3-phase we find that the instantaneous power is constant.
This does not mean that the reactive power is of no importance in a 3-phase
system

For a 3-phase system the sum of three currents
at any instant is zero, this does not mean that the current in each phase is
zero. Similarly, even though the sum of reactive power instantaneously in
3-phase system is zero but in each phase it does exist and is equal to VI sin É¸
and, therefore, for 3-phase the reactive power is equal to Q3É¸ = 3VI sin É¸ =
3Q, here Q is the reactive power in each phase

**Balance
three phase system**

A poly-phase system is said to be
balanced if

The magnitude of corresponding
quantity are equal in each phase and

The phase difference between the
corresponding quantity is

Éµ
= 360/n

n is the number of phases

n
= 4 for 2-phase system

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