1. Sinsuoidal steady state power
Instantaneous and average (real) power
Reactive power
Complex power
Power factor
Related educational modules:
Section 2.9.0, 2.9.1 1

2. AC power Power is still the product of voltage and current:
We are now interested in the case in which the voltage and current are sinusoids: 2

3. v(t) and i(t) are related3

4. Instantaneous AC power
Our previous (time domain) definition of power is called the instantaneous power
In terms of our sinusoidal voltage & current:
After some trigonometry and algebra:
The power consists of a DC (constant) part and an AC (sinusoidal part) 4

5. Graphical representation of p(t)5

6. Alternate representation for p(t)
Which can be decomposed into two plots:
Average (real) power and reactive power 6

7. Average Power
We are generally more interested in the average power delivered to a load:
Average power is:
This is also called the real power (it’s the power that’s provided to the resistive part of the load over time)
Units are watts 7

8. RMS values
We want to assess the power delivered by different types of time-varying signals
The power delivered to a resistive load:
Find a DC (constant) value which delivers the same average power as the time-varying signal
Called the effective or RMS value of the signal
Used to “compare” different time-varying signals 8

9. Note why it’s called “RMS”
Note: we want our average power to look like an “average” current squared times resistance or an “average” voltage squared divided by resistance
We want to define these “effective” values
Note why it’s called “RMS” 9

10. RMS values – continued Average power: Effective DC value:
Equating to time-average value: , 10

11. Annotate previous slide to show VRMS, IRMS notation (RMS = “effective”)11

12. Definition of RMS values
The effective (or RMS) value of a signal is equal to the DC value which provides the same average power to a resistor
For sinusoidal signal with no DC offset: ,
Average power in terms of RMS values: 12

13. Apparent power and power factor
Power in terms of RMS values:
The average (real) power is the product of apparent power and the power factor
Apparent power: (units = volt-amps = VA)
Power factor (pf): (unitless)
Power factor is leading or lagging, to denote whether current leads or lags voltage 13

14. Interpretation of apparent power and pf
Power factor is a property of the load
For a complex load, the power delivered to the load is not exactly the power supplied by the generator
If ZL is real  pf = 1
If ZL is imaginary  pf = 0, and no average power is delivered to the load 14

15. On previous slide, mention reactive power again.15

16. Complex Power
Complex power is a way to conveniently expressing the various power parameters and their relationships 
or: 16

17. Annotate previous slide to show real (average) power and reactive power17

18. Power relationships Complex power:
Magnitude of S is the apparent power (units = VA)
The real part of S is the average power (units = watts)
The imaginary part of S is the reactive power (units = VAR) 18

19. Power Triangle 19

20. Example For the circuit below,
(a) find the average power delivered by the source
(b) find the powers absorbed by the resistor and capacitor
(c) find the apparent and reactive powers delivered by the source
(d) sketch a power triangle for the source 20

21. (a) find the average power delivered by the source21

22. (b) find the powers absorbed by the resistor and capacitor22

23. (c) find the apparent and reactive powers delivered by the source23

24. (d) sketch a power triangle for the source
Apparent power: 391VA
Average power: 305W
Reactive power: -244VAR 24

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