1. Chapter 11
AC power analysis
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2. rms value
The RMS value is the effective value of a varying voltage or current. It is the equivalent steady DC (constant) value which gives the same effect.
effective value or DC-equivalent value
The rms value of a periodic function is defined as the square root of the mean value of the squared function.
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3. If the periodic function is a sinusoid, then
What do AC meters show, is it the RMS or peak voltage?
AC voltmeters and ammeters show the RMS value of the voltage or current.
What does '6V AC' really mean, is it the RMS or peak voltage?
If the peak value is meant it should be clearly stated, otherwise assume it is the RMS value.
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4. AC power analysis Instantaneous Power Suppose: i(t) v(t) N
Invariable part
Sinusoidal part
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5. E page415 figure 10.2
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6. Stored energy
WLav
In the sinusoidal steady state an inductor operates with a current iL(t)=IAcos(wt). The corresponding energy stored in the element is
Average stored energy
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7. Stored energy
In the sinusoidal steady state the voltage across a capacitor is vc(t)=VAcos(wt). The energy stored in the element is
Average stored energy
WCav
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8. Average power
The average power is the average of the instantaneous power over one period real power
Note :
There are other methods to calculate P. 1) 1) 2)
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9. Instantaneous power, real power
Instantaneous power waveforms for a voltage of 2V peak and a current of 1.5A peak
Flowing separately in a resistor, a capacitor and an inductor
Inductor case
Pav = 0
Resistor case
Average power
Pav=0.5Vm*Im
Pav=vrms*irms
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Capacitor case Pav = 0

10. Apparent power <0 or  >0 S=VrmsIrms (VA) Power factor
current leads voltage or current lags voltage
< or  >0
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11. Reactive power (VAR) Resistor: Q=0 Inductor: Q=VrmsIrms
Capacitor: Q=-VrmsIrms
To any passive single port network
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12. The power triangle S Q  P
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13. EXAMPLE
Find the average power delivered to the load to the right of the interface in Figure 8-64.
Fig. 8-64
SOLUTION:
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14. Complex power
Complex power is the complex sum of real power and reactive power
=P+jQ
So =VI*
Where V is the voltage phsor across the system and I* is the complex conjugate of the current phasor.
The magnitude of complex power is just apparent power
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15. Are these equations right?
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16. Maximum power transfer
Fig. 8-66: A source-load interface in the sinusoidal steady state.
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17. we know P is maximized when RL=RT
Let XL=-XT then
we know P is maximized when RL=RT
the maximum average power
where |VT| is the peak amplitude of the Thevenin equivalent voltage
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18. EXAMPLE
                                                                                 
(a) Calculate the average power delivered to the load in the circuit shown in Figure 8-67 for Vs(t)=5cos106t, R=200 ohm, and RL=200 ohm.
(b) Calculate the maximum average power available at the interface and specify the load required to draw the maximum power.
SOLUTION:
(a)
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20. If the load must be a resistor, how get the maximum power on it?
Question:
If the load must be a resistor, how get the maximum power on it?
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21. Maximum power transfer when ZL is restricted
RL and XL may be restricted to a limited range of values.
In this situation, the optimum condition for RL and XL is to adjust XL as near to –XT as possible and then adjust RL as close to as possible
the magnitude of ZL can be varied but its phase angle cannot.
Under this restriction, the greatest amount of power is transferred to the load when the magnitude of ZL is set equal to the magnitude of ZT
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22. Note:
If the load is a resistor, then what value of R results in maximum average-power transfer to R? what is the maximum power then?
If ZL cannot be varied but ZT can, what value of ZT results in maximum average-power transfer to ZL?
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