Question

Power System : Fundamentals

Answer 1

Let voltage be
\[v(t)=V_m cos(\omega t+\theta_v)\]
Let the current be
\[i(t)=I_m cos(\omega t+\theta_i)\]

Power would be multiplication of the two
\[p(t)=v(t)*i(t)=V_m cos(\omega t+\theta_v)*I_m cos(\omega t+\theta_i) \]

Answer 2

\[1/2 *V_m*I_m[cos(\theta_v-\theta_i)+cos2(\omega*t+\theta_v)*cos(\theta_v-\theta_i)+ \]
\[sin2(\omega*t+\theta_v)*sin(\theta_v-\theta_i)\]

Answer 3

now using RMS value
\[|V||I|*cos(\theta)(1+cos2(\omega*t+\theta_v)+|V||I|sin\theta*sin(\omega*t+\theta_v)\]

\[\theta\] is \[\theta_v-\theta_i\]

Answer 4

now the two term above represent the fundamental of power system

Answer 5

\[|V||I|*cos(\theta)(1+cos2(\omega*t+\theta_v)\]
is called real power, it is the power consumed by resistive load. There are three fundamental type of load( resistive , inductive, capacitive) .

Answer 6

The real power can be further classified,
\[|V||I|*cos(\theta)(1+cos2(\omega*t+\theta_v)\]
If you distribute it out , you will get two term
the first term
\[|V||I|*cos(\theta)\] is average power delivered to the load.

which furthur ,

\[|V||I|\] is called apparent power usally denoted as \[S\]

and \[cos \theta \] is power factor

Answer 7

now let's move on to second component of original power equation
\[|V||I|sin\theta*sin(\omega*t+\theta_v)\]

This is called reactive power, associated with inductive and capacitive component of load

In this case our amplitude of reactive power is given by
\[|V||I|sin\theta\].

Answer 8

which is denoted by Q

Answer 9

Another notation I like to introduce is complex power
\[S=P+jQ\]
S=Sqrt[P^2+Q^2]

Answer 10

//in progess, need to add more

Answer 11

Next , we will study Unit System

Answer 12

awesome tutorial ^_^

one thing to mention :)

(1) aparent power is also called PHANTOM or FICTICIOUS power this type of power is mainly consumed by ac machines like transformers
that is why ratiing of a transformer is in KVA
(2) and true or real power is consumed by machines like IM(induction motors) and it gives their rating in KW