11.10 Summary

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11.10 Summary

1. The instantaneous power absorbed by an element is the product of the element’s terminal voltage and the current through the element:

$$p = vi.$$

2. Average or real po wer P (in w atts) is the a verage of instantaneous power p:

$$P = \frac{1}{T} \int_0^T p \, dt$$

If v(t) = Vmcos(ωt + θv) and i(t) = Im cos (ωt + θⁱ ), then Vrms = Vm∕√ 2 , Irms = Im∕ √ __ 2 , and

$$P = \frac{1}{2} V_m I_m \cos(\theta_v - \theta_i) = V_{\text{rms}} I_{\text{rms}} \cos(\theta_v - \theta_i)$$

Inductors and capacitors absorb no a verage power, while the a verage power absorbed by a resistor is (1∕2)Im ²R = Irms ² R.

• 3. Maximum average power is transferred to a load when the load impedance is the comple x conjug ate of the Thevenin impedance as seen from the load terminals, ZL = ZTh * .
• 4. The effective value of a periodic signal x(t) is its root-mean-square (rms) value.

$$X_{\rm eff} = X_{\rm rms} = \sqrt{\frac{1}{T} \int_0^T x^2 dt}$$

For a sinusoid, the ef fective or rms v alue is its amplitude di vided by √ __ 2 .

5. The power factor is the cosine of the phase difference between voltage and current:

$$\mathrm{pf} = \cos(\theta_v - \theta_i)$$

It is also the cosine of the angle of the load impedance or the ratio of real power to apparent power. The pf is lagging if the current lags voltage (inductive load) and is leading when the current leads v oltage (capacitive load).

6. Apparent power S (in VA) is the product of the rms values of voltage and current:

$$S = V_{\rm rms} I_{\rm rms}$$

It is also given by S = ∣S∣ = √ _______ P2 + Q² , where P is the real power and Q is reactive power.

7. Reactive power (in VAR) is:

$$Q = \frac{1}{2} V_m I_m \sin(\theta_v - \theta_i) = V_{\text{rms}} I_{\text{rms}} \sin(\theta_v - \theta_i)$$

8. Complex power S (in VA) is the product of the rms v oltage phasor and the complex conjugate of the rms current phasor . It is also the complex sum of real power P and reactive power Q.

$$\mathbf{S} = \mathbf{V}_{\rm rms} \, \mathbf{I}_{\rm rms}^* = V_{\rm rms} I_{\rm rms} / \theta_{\rm v} - \theta_{\rm i} = P + jQ$$

Also,

$$\mathbf{S} = I_{\text{rms}}^2 \, \mathbf{Z} = \frac{V_{\text{rms}}^2}{\mathbf{Z}^*}$$

• 9. The total comple x power in a netw ork is the sum of the comple x powers of the individual components. Total real power and reactive power are also, respectively, the sums of the individual real powers and the reactive powers, but the total apparent po wer is not calcu lated by the process.
• 10. Power factor correction is necessary for economic reasons; it is the process of impro ving the po wer f actor of a load by reducing the overall reactive power.
• 11. The wattmeter is the instrument for measuring the average power. Energy consumed is measured with a kilowatt-hour meter.

Review Questions

11.1 The average power absorbed by an inductor is zero.

(a) True (b) False

11.2 The Thevenin impedance of a network seen from the load terminals is 80 + j55 Ω. For maximum power transfer, the load impedance must be:

(a) −80 + j55 Ω	(b) −80 − j55 Ω
(c) 80 − j55 Ω	(d) 80 + j55 Ω

11.3 The amplitude of the voltage available in the 60-Hz, 120-V power outlet in your home is:

(a) 110 V	(b) 120 V
(c) 170 V	(d) 210 V

• 11.4 If the load impedance is 20 − j20, the power factor is
  • (a) ⧸−45*°* (b) 0 (c) 1
  • (d) 0.7071 (e) none of these
• 11.5 A quantity that contains all the power information in a given load is the
  • (a) power factor (b) apparent power (c) average power (d) reactive power
  • (e) complex power
• 11.6 Reactive power is measured in:
  • (a) watts (b) VA
  • (c) VAR (d) none of these
• 11.7 In the power triangle shown in Fig. 11.34(a), the reactive power is:
  • (a) 1000 VAR leading (b) 1000 VAR lagging (c) 866 VAR leading (d) 866 VAR lagging

Figure 11.34

For Review Questions 11.7 and 11.8.

11.8 For the power triangle in Fig. 11.34(b), the apparent power is: (a) 2000 VA (b) 1000 VAR

(c) 866 VAR	(d) 500 VAR

11.9 A source is connected to three loads Z1, Z2, and Z³ in parallel. Which of these is not true?

(a) P = P1 + P2 + P3	(b) Q = Q1 + Q2 + Q3
(c) S = S1 + S2 + S3	(d) S = S1 + S2 + S3

11.10 The instrument for measuring average power is the: