1. Engineering Circuit Analysis
CH5 AC circuit power analysis
5.1 Instantaneous Power
5.2 Average Power
5.3 Effectives values of Current ＆ Voltage
5.4 Apparent Power and Power Factor

2. Ch5 AC circuit power analysis
5.1 Instantaneous Power
Given instantaneous power current ＆ voltage, the instantaneous power
delivered to any devices is :
If it is a resistor
If it is a inductor
If it is a capacitor

3. Ch5 AC circuit power analysis
5.1 Instantaneous Power
The total amount of power supplied by the source = total amount of power
delivered to the circuit elements
Known
Power supplied by the source
Power delivered to R:
Power delivered to L: PL(t) = VL(t) ∙ iL(t) R + - V0 - +

4. Ch5 AC circuit power analysis
5.1 Instantaneous Power
Since

5. Ch5 AC circuit power analysis
5.1 Instantaneous Power
How if both ＆ sinusoidal signals?
It is known
, leading to possible to be the
averaged power?
twice of
Time invariant

6. ) Ch5 AC circuit power analysis 5.1 Instantaneous Power
Example 8.1 Given a voltage source of μ(t) V, a 5 μF capacitor and a 200 Ω resistor being connected in series. Find at t = 1.2ms, the powers being absorbed by the resistor and capacitor respectively. If PS = PC + PR?
When t<0 ,Vs=40V
When t=0 , Vs=100V , and + ) - ~
Hence
In this 1st order RC circuit
Hence ,

7. Ch5 AC circuit power analysis
5.1 Instantaneous Power
The average power (over time) for the instantaneous power
The average power over time intervals to
--If is periodic as , where is the period
Validate the previous suggestion !

8. Ch5 AC circuit power analysis
5.1 Instantaneous Power
Example : the averaged power for
It is a periodic signal with a period of
Phase different between ＆ , or
Validate the previous suggestion !

9. Ch5 AC circuit power analysis
5.2 Average Power
Two observations:
-- power absorbed by an ideal resistor, ,
-- power absorbed by a purely reactive element,
Example 8.2 (p.213)
Given and
Determine P and p(t).

10. ) Ch5 AC circuit power analysis 5.2 Average Power
Example 8.4 Determine the average power absorbed by each of the passive elements and supplied by each of the sources. + ) - ~
Apply KVL, we can determine the currents are:
The current that passes through the resistor is:
Hence ,
Both L & C are reactive elements , and
For the source on the left mesh
For the source on the right mesh
The power of the left supplies 50W power and it is absorbed by the resistor and the source on the right

11. Ch5 AC circuit power analysis
5.2 Average Power P

12. Ch5 AC circuit power analysis
5.2 Average Power
Remark : superposition is applicable for evaluating the averaged power
of a non periodic function which can be decomposed into
two periodic functions with different periods,

13. i1 = 2 cos10t – 3 cos20t A Ch5 AC circuit power analysis
5.2 Average Power
Example 8.6 (p218)
Determine the average power delivered to the 4 Ω resistor by the current i1 = 2 cos10t – 3 cos20t A.
The two different frequency components of the current can be considered separately.
i1 = 2 cos10t – 3 cos20t A

14. Ch5 AC circuit power analysis
5.3 Effectives values of Current ＆
-- represent the actual effect of a periodic function (current voltage )
in term of delivering / absorbing power
-- transforms an AC circuit into an equivalent DC circuit
-- square root of a mean square (RMS)

15. Ch5 AC circuit power analysis
5.3 Effectives values of Current ＆
RMS of the Sinusoidal wave.
Given ,
For a sinusoidal function , its RMS is
of its magnitude

16. Ch5 AC circuit power analysis
5.3 Effectives values of Current ＆
Effective value of multiple-frequency circuit
Power can be determined using super position
of 50Hz
of 100Hz
In general, given a circuit with
frequencies,

17. Ch5 AC circuit power analysis
5.3 Effectives values of Current ＆
Example …

18. Ch5 AC circuit power analysis
5.4 Apparent Power and Power Factor
Given
One can determine the averaged power delivered to the network is or
However, the effective power delivered to the network would be
is defined as the apparent power –
The maximal value of the average power can be.
Note that the unit of the Papp is VA.

19. Ch5 AC circuit power analysis
5.4 Apparent Power and Power Factor
Power factor ( )
In the above case,
If passive elements are all resistive,
If passive elements are all reactive,
If passive elements are neither purely resistive, nor purely reactive ,
, capacitive load, since lags behind
, inductive load, since leads ,
a leading PF of “—current is leading”
a lagging PF of “—current is lagging”

20. Ch5 AC circuit power analysis
5.4 Apparent Power and Power Factor
Example 8.8(P222)
Determine the apparent power supplied by the source3, and the power factor
of the combined loads. + - + -
Note that for the impedance 3+j4Ω,only the resistive component will absorb power delivered by the source. ad