AC Power Analysis

Instantaneous Power

The instantaneous power p(t) absorbed by an element
is the product of the instantaneous voltage v(t) across the element and the instantaneous current i(t) through it. Assuming the passive sign convention, 

p(t)= v(t)i(t)

 Let the voltage and current at the terminals of the circuit be 

v(t)= Vm cos(ωt +θv)

i(t)= Im cos(ωt +θi)

where Vm and Im are the amplitudes (or peak values), and θv and θi are the phase angles of the voltage and current, respectively. The instantaneous power absorbed by the circuit is 

p(t)= v(t)i(t)= VmIm cos(ωt +θv)cos(ωt +θi)

We apply the trigonometric identity 

cosAcosB =1/2[cos(A−B)+cos(A+B)]

and express Eq. (11.3) as

p(t)=1/2VmIm cos(θv −θi)+1/2VmIm cos(2ωt +θv +θi)

Average Power

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

MAXIMUM AVERAGE POWER TRANSFER

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

This result is known as the maximum average power transfer theorem for the sinusoidal steady state. Setting RL = RTh and XL =−XTh

Pmax = |VTh|^2/8RTh

EFFECTIVE OR RMS VALUE

The effective value of a periodic current is the dc current that delivers the same average power to a resistor as the periodic current.

The idea of effective value arises from the need to measure the effectiveness of a voltage or current source in delivering power to a resistive load.

The effective value of a periodic signal is its root mean square(rms)value.

LEARNINGS:

In this topic I learned that the instanstanues power is valid for signals of any waveform. The unit of Instantaneous power is VA. Complex power is the product of the complex effective voltage and the complex effective conjugate current. The unit of complex power is VA.