Instantaneous and Average Power & Maximum Average Power Transfer

               Power is said to be the most important quantity in electric utilities, electronic and even communication systems because these systems involve transmission of power from one point to another. Using ac in power generation began after the successful contribution of Tesla. Ac allowed high voltage power transmission from different power generating plant to the consumers compared to dc.

INSTANTANEOUS POWER

               Instantaneous power- the power absorbed by an element at any instant of time. It is just the product of instantaneous voltage v(t) across the element and the instantaneous current i(t) through it.

            

We noticed in the sketch of p (t) above that the power is positive at some part and negative for the rest of the cycle. This means that when:

p (t)>0: power is absorbed by the circuit

p(t)<0: power is absorbed by the source

The conditions stated above is possible to happen because of the presence of inductors and capacitors which are considered to be a storage elements. I conclude that power is transferred from the circuit to the source. Such as if a certain circuit has large voltage and current, the circuit has large power otherwise, less power. Power is directly proportional to the voltage and current. Thus, when voltage and current increases, power also increases.

AVERAGE POWER

               Average power- the average of instantaneous power over one period.

SAMPLE PROBLEM:

If v(t) = 160 cos 50t V and i(t) = –20 sin(50t – 30°) A, calculate the instantaneous power and the average power.

SOLUTION:

v(t) =160cos(50t)

i(t) = -20sin(50t −30°) = 2cos(50t −30°+180°−90°)

i(t) = 20cos(50t +60°) 

MAXIMUM AVERAGE POWER TRANSFER

               The maximum average power can be transferred to the load if X_L=-X_TH and R_L=R_TH. The load impedance Z_L must be equal to the complex conjugate of the Thevenin impedance Z_TH. We calculate V_TH and R_TH to get the maximum average power transfer.

SAMPLE PROBLEM:

 For real life application, we also have what we called power matching. For example, you bought an electronic appliance from other country then when you used it, it doesn't work. What would be the possible reasons why this happened? Maybe one of those reasons would be, the electric power in other country doesn't match to the electric power here in Philippines since we commonly used 50-60 Hz ac power. As we noticed, the electrical devices and appliances in our household such as electric fan, television, computer and etc. have its own power rating. This indicates how much power the equipment requires. Over power rating may lead to a permanent damage to the appliance.