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Chapter 23 Alternating Current Circuits Topics Covered in Chapter 23 23-1: AC Circuits with Resistance but No Reactance 23-2: Circuits with XL Alone 23-3: Circuits with XC Alone 23-4: Opposite Reactances Cancel 23-5: Series Reactance and Resistance © 2007 The McGraw-Hill Companies, Inc. All rights reserved. Topics Covered in Chapter 23 23-6: Parallel Reactance and Resistance 23-7: Series-Parallel Reactance and Resistance 23-8: Real Power 23-9: AC Meters 23-10: Wattmeters 23-11: Summary of Types of Ohms in AC Circuits 23-12: Summary of Types of Phasors in AC Circuits McGraw-Hill © 2007 The McGraw-Hill Companies, Inc. All rights reserved. 23-1: AC Circuits with Resistance but No Reactance Fig. 23-1 In this figure, combinations of series and parallel resistances are shown. All voltages and currents throughout the resistive circuits are in phase. There is no reactance to cause a lead or lag in either current or voltage. 23-2: Circuits with XL Alone Fig. 23-2 A series inductive circuit is shown in Fig. 23-2. The ohms of XL are just as effective as ohms of R in limiting the current or producing a voltage drop. XL has a phasor quantity with a 90° phase angle. 23-2: Circuits with XL Alone Fig. 23-3 A parallel inductive circuit is shown in Fig. 23-3. The ohms of XL are just as effective as ohms of R in limiting the current or producing a voltage drop. XL has a phasor quantity with a 90° phase angle. 23-3: Circuits with XC Alone Fig. 23-4 Series XC Capacitive reactances are shown in Fig. 23-4 Since there is no R or XL, the series ohms of XC can be combined directly. 23-3: Circuits with XC Alone Fig. 23-5 Parallel XC Capacitive reactances are shown in Fig. 23-5. Since there is no R or XL, parallel IC currents can be added. 23-3: Circuits with XC Alone XC and XL are phasor opposites. R XL XC R When analyzing series circuits: Opposite reactances in series must be subtracted. If XL is larger, the net reactance is inductive. If XC is larger, the net reactance is capacitive. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 23-4: Opposite Reactances Cancel In a circuit with both XL and XC, the opposite phase angles enable one to offset the effect of the other. For XL and XC in series, the net reactance is the difference between the two series reactances. In parallel circuits, the net reactive current is the difference between the IL and IC branch currents. 23-4: Opposite Reactances Cancel Fig. 23-6: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 23-4: Opposite Reactances Cancel Fig. 23-7: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 23-5: Series Reactance and Resistance The resistive and reactive effects of series reactance and resistance must be combined by phasors. For series circuits, all the ohms of opposition are added to find ZT. First, add all series resistances for one total R. Combine all series reactances, adding all XLs and all XCs and finding X by subtraction. The total R and net X can be added by phasors to find the total ohms of opposition in the entire series circuit. 23-5: Series Reactance and Resistance Magnitude of ZT After the total R and net reactance X are found, they can be combined by the formula ZT = R2 + X2 23-6: Parallel Reactance and Resistance In parallel circuits, the branch currents for resistance and reactance are added by phasors. Then the total line current is found by IT = IR2 + IX2 23-6: Parallel Reactance and Resistance Parallel IC and IL are phasor opposites. IC IR IR IL Opposite currents in parallel branches are subtracted. If IL is larger, the circuit is inductive. If IC is larger, the circuit is capacitive. 23-6: Parallel Reactance and Resistance Parallel RCL Circuit Analysis VA = 120 R = 30 Ω XC = 60 Ω XL = 24 Ω IT = 5 A IT = IR2 + IX2 = 42 + 32 = 5A 2A 4A The circuit is inductive. 5A 4A 3A IT = 5 A 23-6: Parallel Reactance and Resistance Parallel RCL Circuit Impedance R = 30 W VA = 120 XC = 60 W XL = 24 W IT = 5 A 4A ZEQ = 3A IT = 5 A VA IT = 120 5 = 24 Ω 23-6: Parallel Reactance and Resistance Parallel RCL Circuit Phase Angle VA = 120 R = 30 W XC = 60 W XL = 24 W IT = 5 A Θ= Tan-1 − IX = Tan−1 − IR −37° 3A 3 4 4A IT = 5 A = −37° 23-7: Series-Parallel Reactance and Resistance Figure 23-12 shows how a series-parallel circuit can be reduced to a series circuit with just one reactance and one resistance. The triangle diagram in (d) shows total impedance Z (141 Ω). Fig. 23-12: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 23-7: Series-Parallel Reactance and Resistance Waveforms and Phasors for a Series RCL Circuit Θ = 0 VR I L Θ = −90 I VR R C I VC VC VL Θ = 90 VL I Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 23-7: Series-Parallel Reactance and Resistance Series RCL Circuit Analysis R=4W 4A 20 V XC = 12 W L XL = 9 W Z= R XNET 3W The net reactance is 3 W, capacitive. R2 + X2 = 42 + 32 = 5Ω 4W Z=5W Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. I= V Z = 20 5 =4A 23-7: Series-Parallel Reactance and Resistance Series RCL Circuit Phase Angle I 20 V L XL = 9 W ΘZ= Tan-1 − XNET 3W R −37° 4W 5W R=4W Z = 5Ω XC = 12 W ΘZ= Tan-1 ± X / R The net reactance is 3 W, capacitive. X = Tan−1 − R 3 = −37 4 Note: Since the circuit is capacitive, the source voltage lags the source current by 37 degrees. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 23-7: Series-Parallel Reactance and Resistance Series RCL Voltage Drops 20 V 4A L R=4Ω VR = IR = 4 × 4 = 16 V XC = 12 Ω VC = IXC = 4 × 12 = 48 V VL = IXL = 4 × 9 = 36 V XL = 9 Ω VC and VL are phasor opposites, so the net reactive voltage is the difference between the two or 12 V. R 16 V XNET 12 V Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. VT = 162 + 122 = 20 V 23-8: Real Power In an ac circuit with reactance, the current I supplied by the generator either leads or lags the generator voltage V. The product VI is not the real power produced by the generator, since the instantaneous voltage may have a high value while at the same time the current is near zero, or vice versa. 23-8: Real Power The real power in watts can always be calculated as I2R, where R is the total resistive component of the circuit. To find the corresponding value of power as VI, this product must be multiplied by the cosine of the phase angle Θ. Then Real power = P = I2R or Real power = P = VI cos Θ 23-8: Real Power Series RCL Circuit Power Dissipation R=4W 4A 20 V XC = 12 W L XL = 9 W Note: the power dissipation is zero in ideal capacitors and ideal inductors. All of the dissipation takes place in the circuit’s resistance. P = V × I × Cos Θ = 20 × 4 × 0.8 = 64 W P = I2R = 42× 4 = 64 W −37 4A 20 V The source voltage and source current are not in phase and the true power is not equal to VI. It is equal to VI × power factor. 23-8: Real Power Parallel RCL Circuit Power Dissipation VA = 120 R = 30 W XC = 60 W XL = 24 W IT = 5 A P = V × I × Cos Θ = 120 × 5 × 0.8 = 480 W P= −37° 3A 4A IT = 5 A V2 R = 1202 30 = 480 W The source voltage and source current are not in phase and the true power is not equal to VI. It is equal to VI × power factor. 23-9: AC Meters An ac meter must produce deflection of the meter pointer up-scale regardless of polarity. This deflection is accomplished by one of the following methods for nonelectronic ac meters. Thermal type Electromagnetic type Rectifier type All analog ac meters (meters with scales and pointers) have scales calibrated in rms values, unless noted otherwise on the meter. 23-10: Wattmeters The wattmeter shown in Fig. 23-14, uses fixed coils to measure current in a circuit, and the moving coil measures voltage. The deflection is proportional to power. Either dc power or real ac power can be read directly by the wattmeter. Fig. 23-14: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 23-11: Summary of Types of Ohms in AC Circuits Ohms of opposition limit the amount of current in dc circuits or ac circuits. Resistance is the same for either case. Ac circuits can have ohms of reactance because of the variations in alternating current or voltage. Reactance XL is the reactance of an inductor with sine-wave changes in current. Reactance XC is the reactance of a capacitor with sine-wave changes in voltage. 23-12: Summary of Types of Phasors in AC Circuits Fig. 23-15 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.