1 Lecture #9 EGR 272 – Circuit Theory II Read: Chapter 10 in Electric Circuits, 6 th Edition by Nilsson Chapter 10 - Power Calculations in AC Circuits Instantaneous Power: p(t) = v(t)  i(t) = instantaneous power Average Power: Instantaneous power is not commonly used. Average power, P, is more useful. P = P AVG = P DC = average or real power (in watts, W) P can be defined for any waveform as: For periodic waveforms with period T (including sinusoids), P can be expressed as:

2 Lecture #9 EGR 272 – Circuit Theory II Average power can be calculated: 1)by inspection (for simple cases) 2)by integration (for more complex cases) Example: Find the average power absorbed to the resistor below.

3 Lecture #9 EGR 272 – Circuit Theory II Example: Find the average power absorbed to the resistor below. 10 v(t) + _ i(t) t [s] v(t) [V]

4 Lecture #9 EGR 272 – Circuit Theory II RMS Voltage and Current V RMS = root-mean-square voltage (also sometimes called V eff = effective voltage) V RMS = the square root of the average value of the function squared Discuss how the name RMS essentially gives the definition:

5 Lecture #9 EGR 272 – Circuit Theory II Calculating average power for a resistive load using RMS values A key reason that RMS values are used commonly in AC circuits is that they are used in power calculations that are very similar to those used in DC circuits. Recall that power in DC circuits can be calculated using: Similarly, instantaneous power to a resistor can be calculated using Show that power can be also be calculated using RMS values as follows: Development:

6 Lecture #9 EGR 272 – Circuit Theory II Example: Find I RMS for the waveform below and use I RMS to calculate the average power absorbed to the resistor. Recall that P was calculated earlier using instantaneous power. RMS values can be calculated: 1)by inspection (for simple cases) 2)by integration (for more complex cases)

7 Lecture #9 EGR 272 – Circuit Theory II 10 v(t) + _ i(t) t [s] v(t) [V] Example: Find V RMS for the waveform below and use V RMS to calculate the average power absorbed to the resistor. Recall that P was calculated earlier using instantaneous power.

8 Lecture #9 EGR 272 – Circuit Theory II Development: Derive the expression for V RMS above. RMS value of sinusoidal voltages and currents Using the definition of RMS voltage and current, it can be shown that sinusoidal voltage v(t) = V p cos(wt) and a sinusoidal current i(t) = I p cos(wt) that: Example: Find the power absorbed by a 10 ohm resistor with v(t) = 20cos(377t) V.

9 Lecture #9 EGR 272 – Circuit Theory II Superposition of Power Recall that, in general, superposition applies to voltage and current, but not to power. Applying superposition to find the current i: i = i 1 + i 2 or i = current due to V 1 + current due to V 2 Show that this leads to: P  P 1 + P 2

10 Lecture #9 EGR 272 – Circuit Theory II Power to a Complex Load If v(t) = V m cos(wt) then I(t) = I m cos(wt -  ) Useful identities: cos(wt -  ) = cos(  )cos(wt) + sin(  )sin(wt) cos 2 (wt) = ½ + ½ cos(2wt) sin(wt)cos(wt) = ½ sin(2wt) Show that:

11 Lecture #9 EGR 272 – Circuit Theory II Key definitions: Recall that real or average power, P, was just defined as: Since cos is an even function (i.e., cos(  ) = cos(-  )), calculating cos(  ) does not reveal the sign of the angle, so the terms leading and lagging are used with power factor. The power factor is leading if  < 0, (i.e., I leads V or the load is capacitive). The power factor is lagging if  > 0, (i.e., I lags V or the load is inductive). Apparent power = (V RMS )(I RMS ) measured in volt-amperes, VA Other important terms are apparent power and power factor, as defined below:

12 Lecture #9 EGR 272 – Circuit Theory II Examples: Determine P and the p.f. for each case below: Z 10V RMS + _