10.1 COMPLEX POWER IN CIRCUITS WITH AC SIGNALS

Transcription

1 HAPER 10 Power in A ircuits HAPER OUINE 10.1 omplex Power in ircuits with A ignals 10. How to alculate omplex Power 10.3 omplex Power alculations in eries Parallel ircuits 10.4 Power Factor and pf orrection 10.1 OMPEX POWER IN IRUI WIH A IGNA What concept is illustrated in the plots in Figure 4.1? Explain the concept in the following equation and relate it to the plots: VI p p Pave VRM IRM Veff Ieff (10.1) Note: he RM and eff subscripts have identical meaning (RM will be used in this chapter, as it was in h. 4). What is true power? Real power? Figure 4.1 (reproduced) he previous power development is valid for what type of component in a circuit? Why? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 1

2 What is the phase relationship between the voltage and the current for inductors? For capacitors? Why for each? he graphical multiplication of v(t) and i(t) when they are ±90 out of phase is shown in Figure Which is leading, the voltage or the current leading? What is the average power in this case? Why? Is this result the same as the p(t) result for the resistive load? Why or why not? What is the significance of positive and negative power? First, explain the passive sign convention: What is positive power for inductors (Fig. 10.a)? Figure 10.1 What is positive power for capacitors (Fig. 10.a)? Figure 10. ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page

3 What is negative power for inductors (Fig. 10.b)? What is negative power for capacitors (Fig. 10.b)? hus, what is the net average power for ideal inductors and capacitors? How is energy storage in inductors and capacitors for A signals quantified? Need a power-like quantity that corresponds to energy storage for inductors and capacitors Need a method to express both electrical energy conversion and electrical energy storage with A signals Identify the following equations: E V E I (10.) 1 1 Note: calculus is needed to derive the stored energy expressions for A signals. Alternatively, an explanation: alculus average value of energy stored in a capacitor or an inductor expressed with RM values: E V E I (10.3) (ave) 1 1 RM (ave) RM Notation: he capital letter E shall be used to indicate D energy or average energy in the A case. Explain how the following equations were obtained: 1 V I E V V I X V I RM RM RM RM RM RM RM (10.4) V V V I E I I I 1 1 RM 1 RM RM RM RM RM RM X (10.5) What is striking between these two general results? What is the average energy stored in an inductor or a capacitor directly proportional to? What is the average energy stored in an inductor or a capacitor inversely proportional to? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 3

4 Does (V RM I RM )/() in this energy expression contain a power-like quantity? If so, what is it? If not, why not? What does the power-like quantity represent for inductors and capacitors? How can this power-like quantity be utilized for inductors and capacitors? Recall impedance: Z R jx What is the phase relationship between the voltage and the current for the real part of Z? Why? What is the phase relationship between the voltage and the current for the imaginary part of Z? Why? What is the phase relationship between the A voltage and current when the power is real? What happens to electrical energy for real power? What is the phase relationship between A voltage and current that represents stored energy? By analogy to impedance, what type of number is the power that represents energy storage? Based on the previous discussion, explain the following expression for complex power : P jq (10.6) where P = power in watts (W), Q = power in volt-amperes reactive (VARs), and = power in volt-amperes (VA). Note: do not confuse reactive power Q with electric charge Q know from context which one is appropriate Now relate V RM I RM to complex power: (in-phase part of V I ) 0 (90 out-of -phase part of V I ) 90 RM RM RM RM (in-phase part of V I ) j(90 out-of -phase part of V I ) (10.7) RM RM RM RM How does one separate the V RM I RM product into real and imaginary parts? tart with: V I ( V )( I ) V I ( )??? RM RM RM V RM I RM RM V I What is wrong with this equation? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 4

5 Has the total phase angle ( V + I ) ever appeared up to now? What is the phase shift between voltage and current that is physically significant for impedance? Hence, the phase angle should be the not the sum: VRM IRM ( V I) (10.8) How is the negative (opposite sign) of an angle obtained with complex numbers? Explain Figure I ( I ) I (10.9) RM RM I RM I Figure 10.3 hus, the complex conjugate is Explain each step that follows: V I ( V )( I ) V ( I ) (10.10) RM RM RM V RM I RM V RM I V I ( ) V I (10.11) RM RM V I RM RM What is? Refer to Figure 10.4: V I cos jv I sin P jq (10.1) RM RM RM RM P Re ( ) V I cos (10.13) RM RM Figure 10.4 where Re means Q Im ( ) V I sin RM RM (10.14) where Im means ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 5

6 V I cos jv I sin RM RM RM RM ( V I cos ) 0 ( V I sin ) 90 RM RM RM RM P jq (10.15) P jq (10.16) where: = in units of P = in units of = of complex power in rectangular form, Q = in units of = of complex power in rectangular form, = in units of = of complex power in polar form, and = in units of = of complex power in polar form. Note: he unit watt is reserved for power that represents energy conversion. he key complex power expression is (10.17) VI VI( V I) P jq Are the phasor voltage and current effective (RM) or peak values? Explain. Why is the VI product called apparent power for A signals? What is = V I? (10.18) What is Q = Q Q? (10.19) Why is the sign for Q positive? Why is the sign for Q negative? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 6

7 10. HOW O AUAE OMPEX POWER here are two ways to calculate the total complex power provided by a source to a circuit: a. or b. Determine the total phasor voltage and the total phasor current supplied by the source to the circuit (single source circuits only) hen use Equation (10.17): VI VI( V I) P jq Determine the real or reactive power of each component in the circuit Add all the real powers to obtain the total power P OA Add all the reactive powers to obtain the total reactive power Q OA (positive Q for inductors, negative Q for capacitors) hen form the total complex power: P jq (Multiple source circuits will be covered in the next section). Example (Explain each step.) Determine the total complex power provided by the source to the circuit shown in Figure : : V 100 V R = 10 X = 15 RM trategy: olution: Z R jx 10 j15 Figure 10.5 V 10 0 I A Z 10 j15 VI (10 0 )( ) (10 0 )( ) j VA (3.08 j4.6) VA 3.08 W j4.6 VAR ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 7

8 Why are both real power and reactive power present? Which complex power calculation method was demonstrated in this example? Explain each step as it relates to determining complex power using the second method: Note: All phasors are assumed to have RM magnitudes throughout this discussion. For a resistance, V I V I ( ) V I 0 0 P j0 P (10.0) R R R R R V I R R R R R VR PR VR IR IRR QR 0 (resistances only) (10.1) R Which of the numbers is complex in the previous equation? Explain: For a capacitor, V I V I ( ) V I jq jq (10.) V I V P Q V I I X (capacitors only) (10.3) 0 X For an inductance, V I V I ( ) V I jq jq (10.4) V I ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 8

9 V P Q V I I X (inductors only) (10.5) 0 X Explain how complex power is determined using the next equation: [ ] (10.6) VI P j Q Q where the summation sign is designated by the uppercase Greek letter sigma (). P is Q is Q is Example 10.. (Fill in the steps.) Determine the complex power in the circuit shown in Figure 10.5 repeated below. Given: Desired: trategy: Figure 10.5 olution: Answer: P jq 3.08 W j4.6 VAR (3.08 j4.6) VA VA ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 9

10 he complex power can be visualized using the power triangle. A plot in the complex number plane Defined by three quantities: - the origin - the real power on the real axis - the reactive power on the imaginary axis Example ketch and label the power triangle for the circuit in Figure Given: 3.08 W j4.6 VAR VA P = 3.08 W Q = 4.6 VAR = 5.55 VA = 56.3 Desired: power triangle trategy: Plot P and Q in approximate proportion ketch the triangle abel P, Q,, olution: Explain why the power triangle flips between the inductive and capacitive cases (Figure 10.7). Figure 10.7 ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 10

11 Determine the equivalent aspect of the following two equations: VI VI ( V I) (10.7) V V V V Z ( V I) Z I I I I (10.8) For example, check out the results in Examples and 10..: his fact is a useful check in complex power calculations. Form a summary statement for each relation that follows: 3.08 W j4.6 VAR VA Z R jx 10 j15 Z (10.9) VI VI( V I) P jq VR PR VR IR IRR QR 0 (10.30) R V P Q V I I X (10.31) 0 X V P Q V I I X (10.3) 0 X [ ] (10.33) VI P j Q Q ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 11

12 10.3 OMPEX POWER AUAION IN ERIE PARAE IRUI Example (Explain each step.) Determine the complex power provided by the source to the circuit shown in Figure 10.8 by (a) determining and summing the individual powers of the components, and (b) VI. Given: circuit in Figure 10.8 Desired: trategy: a. Z, I, V x using series parallel analysis Q = I X Q V X olution: a. Z x b. x P Vx R P j( Q Q ) V I ( Z R)( Z ) (13)( j13) Z Z 13 j13 R Z Z Z j x V I A Z x V IZ ( )( ) V x RM RM Figure 10.8 Q I X (.7343) (65) VAR Q Vx VAR X 13 Vx P W R 13 P j( Q Q ) j( ) W j7.06 VAR 44 W j7 VAR VI ( )( ) b. ( )( ) VA j W j7 VAR Is the circuit inductive or capacitive from an impedance viewpoint? Why? Is the circuit inductive or capacitive from a complex power viewpoint? Why? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 1

13 Multiple source circuits: onsider a circuit with more than one source. an the powers in each component due to each source be summed? Why or why not? an the voltages (or currents) for each component be summed? If so, under what condition? he complex power for each component is determined from that total phasor voltage (or current) for that component. he total complex power of the circuit is determined by summing the complex powers of the individual components. Example Determine the total complex power provided by the source to the circuit in Example he circuit schematic is repeated below (Figure 10.9). Figure 10.9 Given: Desired: trategy: he circuit from Figure 9.18 is repeated in Figure 10.9 with the current in each branch labeled. olution: (Perform all steps on separate paper.) ub-answers and answer to check as you proceed: For the source: I A, I A, I A b a c For the 150 source: I A, I A, I A c a b uperposition: I A, I A, I A a b c Answer: = ( ) + j( ) = j = 18.4 W j18.0 VAR ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 13

14 10.4 POWER FAOR AND PF ORREION What is the power factor angle? Definition of power factor (pf): How does the P/ ratio relate to in the power triangle (see Figure 10.7, repeated to the right)? Figure 10.7 onsider able 10.1 for the physical significance of pf. Explain the trend in pf versus power factor angle. ABE 10.1 power factor power factor angle When the power factor angle is ±90, what is the pf? Is the complex power real, reactive, or a mixture? Why? When the power factor angle is 0, what is the pf? Is the complex power real, reactive, or a mixture? Why? When the power factor is in the middle region between 1 and 0, is the complex power primarily real, reactive, or a mixture? Why? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 14

15 Given the power factor, can one tell whether the power factor angle is positive or negative? Why or why not? erminology applied to pf: leading or lagging. It is applied to the current relative to the voltage: If the power factor is leading, then the current leads the voltage. Is the circuit capacitive or inductive? Why? If the power factor is lagging, then the current lags the voltage. Is the circuit capacitive or inductive? Why? Major application of power factor: power factor correction Example: reat the circuit in Examples and 10.. as a block with complex power (Figure 10.1): Figure 10.1 Is the Q zero? Why or why not? Explain the following calculation: pf P leading j Is real, reactive, or both powers significantly present in the circuit? Explain. ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 15

16 What was added to the circuit in Fig. 10.1(b) as shown in Fig ? Figure Explain the mathematical statements that follow: P jq j( ) ( j0) VA 3.08 W What is the total complex power? What is the total real power? What is the total reactive power? pf P j j0 VI I A A V 10 0 Note: he original current in the circuit without pf correction (from Ex or 10..) is A. What are the primary effects of power factor correction with regards to each of the following aspects? (1) he power factor () he phase between the total circuit voltage and current (3) he circuit current What is the impact of (3) on wire sizes, power dissipation in wires, and costs? How should the component for pf correction be connected into the circuit (series or parallel)? Why? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 16

17 What type of component is added in parallel with the load in this example? Why? What is the value of the component to add in parallel in Figure 10.13? Identify what is known and what is unknown in the following equations: Q V X (10.35) X (10.36) Assume frequency is known. In this example, it is 60 Hz. Explain the following calculations: X V Q X mh 60 Example (Explain each step.) Determine (a) the reactance for power factor correction in the circuit shown in Figure 10.8 (repeated to the right). (b) Determine the component value if the frequency is 60 Hz. (c) Determine the pf both before and after pf correction. : W j7.08 VAR (from Example ) V V RM f = 60 Hz : a. X for pf correction Figure 10.8 b. value of or for pf correction c. pf before pf correction pf after pf correction : Qpf correction Im ( ) (inductive or capacitive, as appropriate) X V Q 1 X P pf ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 17

18 olution: (Explain each step.) Qpf correction Im ( ) 7.08 VAR (capacitive) X V Q X 60(145.39) F P pfbefore j pf after What is the difference and the significance of Figure with respect to Figure 10.8? Figure hus, what is power factor correction? What is the general approach to power factor correction? ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 18

19 Example (Explain or fill in each step, as appropriate.) Determine the reactance for power factor correction in the circuit shown in Figure Given: V V RM load 1: load : Figure Desired: : Determine for each load from pf: = cos 1 (pf). Determine for each load from V, I,, and leading/lagging status VI 1 Q Im ( ) (inductive or capacitive, as appropriate) olution: pf correction V X Q (Perform the calculations per the strategy. heck against the answers provided on the next page.) ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 19

20 45.00 (positive due to lagging pf) (negative due to leading pf) kva kva Q X (.97 j6.578) kva pf correction kvar (capacitive) earning Objectives Discussion: an you perform each learning objective for this chapter? (Examine each one.) As a result of successfully completing this chapter, you should be able to: 1. Describe why complex power is needed to express power in A circuits.. Describe complex power, apparent power, real power, reactive power, power factor angle, and power factor and the differences between them. 3. alculate complex power, apparent power, real power, reactive power, power factor angle, and power factor for components, groups of components, and entire circuits using two approaches: a. complex power equation in terms of phasor voltage and phasor current, and b. summing real or reactive powers of individual components. 4. Describe what power factor correction is and why it is important. 5. Determine the parallel reactance and component value required for power factor correction. ontemporary Electric ircuits, nd ed., Prentice-Hall, 008 lass Notes h. 10 Page 0