Page 1 Physics Including Human Applications 554 Chapter 25 Alternating Currents GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms and use it in an operational definition: effective values of current and voltage reactance impedance power factor resonance Q-factor AC Circuits Solve alternating-current problems involving resistance, inductance, and capacitance in a series circuit. Phasor Diagrams Draw phasor diagrams for alternating current circuits. Transformer Explain the operation of the transformer. AC Measurements Describe the use of alternating currents in physiological measurements. PREREQUISITES Before you begin this chapter you should have achieved the goals of Chapter 22, Basic Electrical Measurements, and Chapter 24, Electromagnetic Induction. Page 2 Physics Including Human Applications 554 Chapter 25 Alternating Currents GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms and use it in an operational definition: effective values of current and voltage reactance impedance power factor resonance Q-factor AC Circuits Solve alternating-current problems involving resistance, inductance, and capacitance in a series circuit. Phasor Diagrams Draw phasor diagrams for alternating current circuits. Transformer Explain the operation of the transformer. AC Measurements Describe the use of alternating currents in physiological measurements. PREREQUISITES Before you begin this chapter you should have achieved the goals of Chapter 22, Basic Electrical Measurements, and Chapter 24, Electromagnetic Induction. Physics Including Human Applications 555 Chapter 25 Alternating Currents 25.1 Introduction Most of your experience with electricity has probably been with alternating current (AC) circuits. Do you know the difference between AC and direct current (DC) electricity? At some time or another you have probably read the printing on the end of a light bulb. Usually a light bulb has printed on it that it is for use in a 120-volt AC circuit. What was the manufacturer trying to indicate? Could you use a 120-V DC bulb in your household sockets? What, if anything, would happen? Why is AC the dominant form of electrical energy in use? More than 90 percent of electrical energy is used as AC electricity. If you owned an electric train in your childhood, you may recall that you used a transformer to reduce household voltage to the voltage required by the electric train. The transformer is a device that plays a most important role in the use of electrical energy. Do you know what its functions are in transmission of electric power? What is the role of the transformer in the coupling of two AC circuits? These are a few of the questions we will discuss in this chapter. 25.2 Nomenclature Used for Alternating Currents In Chapter 24, we learned that a simple AC generator can be made by rotating a coil of conducting wire in a constant magnetic field. The voltage produced by such a generator has the form shown in Figure 25.1. This sinusoidal voltage alternates polarity + to -and produces an alternating current in a circuit connected to such an AC source. This alternating polarity is in contrast to the unindirectional nature of DC current. In the United States the standard frequency in home and industrial use is 60 cycles/second (Hz). This means that there is a reversal of the direction of the current every 1/120 second. Radio broadcast frequencies are of the order of 10 6 Hz. Some Page 3 Physics Including Human Applications 554 Chapter 25 Alternating Currents GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms and use it in an operational definition: effective values of current and voltage reactance impedance power factor resonance Q-factor AC Circuits Solve alternating-current problems involving resistance, inductance, and capacitance in a series circuit. Phasor Diagrams Draw phasor diagrams for alternating current circuits. Transformer Explain the operation of the transformer. AC Measurements Describe the use of alternating currents in physiological measurements. PREREQUISITES Before you begin this chapter you should have achieved the goals of Chapter 22, Basic Electrical Measurements, and Chapter 24, Electromagnetic Induction. Physics Including Human Applications 555 Chapter 25 Alternating Currents 25.1 Introduction Most of your experience with electricity has probably been with alternating current (AC) circuits. Do you know the difference between AC and direct current (DC) electricity? At some time or another you have probably read the printing on the end of a light bulb. Usually a light bulb has printed on it that it is for use in a 120-volt AC circuit. What was the manufacturer trying to indicate? Could you use a 120-V DC bulb in your household sockets? What, if anything, would happen? Why is AC the dominant form of electrical energy in use? More than 90 percent of electrical energy is used as AC electricity. If you owned an electric train in your childhood, you may recall that you used a transformer to reduce household voltage to the voltage required by the electric train. The transformer is a device that plays a most important role in the use of electrical energy. Do you know what its functions are in transmission of electric power? What is the role of the transformer in the coupling of two AC circuits? These are a few of the questions we will discuss in this chapter. 25.2 Nomenclature Used for Alternating Currents In Chapter 24, we learned that a simple AC generator can be made by rotating a coil of conducting wire in a constant magnetic field. The voltage produced by such a generator has the form shown in Figure 25.1. This sinusoidal voltage alternates polarity + to -and produces an alternating current in a circuit connected to such an AC source. This alternating polarity is in contrast to the unindirectional nature of DC current. In the United States the standard frequency in home and industrial use is 60 cycles/second (Hz). This means that there is a reversal of the direction of the current every 1/120 second. Radio broadcast frequencies are of the order of 10 6 Hz. Some Physics Including Human Applications 556 microwave devices have frequencies the order of 10 10 Hz. So the AC current in common use is of relatively low frequency. If you have two equal resistors and in one there is an AC current of 1 ampere maximum and in the other a DC current of one ampere, would you expect the two resistors to produce the same heat? Let us consider the situation to see what we should expect. In the resistor with the DC current, the current is constant. In the resistor with AC current, the current varies from 0 to 1 A in one direction and the same in the opposite direction. You have learned that the heat produced depends upon the square of current for a given resistor. Hence, you would have been correct if you had said that more heat is produced by the 1-A DC current. For a varying current we would expect the heat produced to be proportional to the average value of the square of the current. The square root of this quantity is called effective current. That is, it is the current that would produce the same heating effect as one ampere of DC current. Let us find the relationship between the effective AC current and maximum AC current. The AC current at any instant of time can be expressed as a sinusoidal function of time, i = i max sin ?t (25.1) where i max is the maximum value of the current, ? is the angular frequency of the AC, and t is the time in seconds. The square of the current will be proportional to the power dissipated in the resistor; so i 2 = i 2 max sin 2 ?t (25.2) We can draw a graph of the square of the current versus the time(Figure 25.2). The total heating effect will be proportional to the area under the curve. You see that the curve between p and 2p radians is exactly the same as that from 0 to p radians. So we will need to consider only half of the total cycle. The curve for the square of the DC current is a horizontal line represented by AB. The problem is then to find the altitude for the rectangle with the base of p which has the same areas under the i 2 max sin 2 ?t curve. By inspection you might say that the area under AB is about equal to the area under the i 2 max sin 2 ?t curve if A 1 = A 2 and A 3 = A 4 . This is true if OC is equal to i 2 max /2. You can show this by measuring the two areas. Page 4 Physics Including Human Applications 554 Chapter 25 Alternating Currents GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms and use it in an operational definition: effective values of current and voltage reactance impedance power factor resonance Q-factor AC Circuits Solve alternating-current problems involving resistance, inductance, and capacitance in a series circuit. Phasor Diagrams Draw phasor diagrams for alternating current circuits. Transformer Explain the operation of the transformer. AC Measurements Describe the use of alternating currents in physiological measurements. PREREQUISITES Before you begin this chapter you should have achieved the goals of Chapter 22, Basic Electrical Measurements, and Chapter 24, Electromagnetic Induction. Physics Including Human Applications 555 Chapter 25 Alternating Currents 25.1 Introduction Most of your experience with electricity has probably been with alternating current (AC) circuits. Do you know the difference between AC and direct current (DC) electricity? At some time or another you have probably read the printing on the end of a light bulb. Usually a light bulb has printed on it that it is for use in a 120-volt AC circuit. What was the manufacturer trying to indicate? Could you use a 120-V DC bulb in your household sockets? What, if anything, would happen? Why is AC the dominant form of electrical energy in use? More than 90 percent of electrical energy is used as AC electricity. If you owned an electric train in your childhood, you may recall that you used a transformer to reduce household voltage to the voltage required by the electric train. The transformer is a device that plays a most important role in the use of electrical energy. Do you know what its functions are in transmission of electric power? What is the role of the transformer in the coupling of two AC circuits? These are a few of the questions we will discuss in this chapter. 25.2 Nomenclature Used for Alternating Currents In Chapter 24, we learned that a simple AC generator can be made by rotating a coil of conducting wire in a constant magnetic field. The voltage produced by such a generator has the form shown in Figure 25.1. This sinusoidal voltage alternates polarity + to -and produces an alternating current in a circuit connected to such an AC source. This alternating polarity is in contrast to the unindirectional nature of DC current. In the United States the standard frequency in home and industrial use is 60 cycles/second (Hz). This means that there is a reversal of the direction of the current every 1/120 second. Radio broadcast frequencies are of the order of 10 6 Hz. Some Physics Including Human Applications 556 microwave devices have frequencies the order of 10 10 Hz. So the AC current in common use is of relatively low frequency. If you have two equal resistors and in one there is an AC current of 1 ampere maximum and in the other a DC current of one ampere, would you expect the two resistors to produce the same heat? Let us consider the situation to see what we should expect. In the resistor with the DC current, the current is constant. In the resistor with AC current, the current varies from 0 to 1 A in one direction and the same in the opposite direction. You have learned that the heat produced depends upon the square of current for a given resistor. Hence, you would have been correct if you had said that more heat is produced by the 1-A DC current. For a varying current we would expect the heat produced to be proportional to the average value of the square of the current. The square root of this quantity is called effective current. That is, it is the current that would produce the same heating effect as one ampere of DC current. Let us find the relationship between the effective AC current and maximum AC current. The AC current at any instant of time can be expressed as a sinusoidal function of time, i = i max sin ?t (25.1) where i max is the maximum value of the current, ? is the angular frequency of the AC, and t is the time in seconds. The square of the current will be proportional to the power dissipated in the resistor; so i 2 = i 2 max sin 2 ?t (25.2) We can draw a graph of the square of the current versus the time(Figure 25.2). The total heating effect will be proportional to the area under the curve. You see that the curve between p and 2p radians is exactly the same as that from 0 to p radians. So we will need to consider only half of the total cycle. The curve for the square of the DC current is a horizontal line represented by AB. The problem is then to find the altitude for the rectangle with the base of p which has the same areas under the i 2 max sin 2 ?t curve. By inspection you might say that the area under AB is about equal to the area under the i 2 max sin 2 ?t curve if A 1 = A 2 and A 3 = A 4 . This is true if OC is equal to i 2 max /2. You can show this by measuring the two areas. Physics Including Human Applications 557 Let us use the symbol I for the effective current. We can see from Figure 25.2 that the effective current squared is equal to one-half of i max squared, I 2 = (1/2) i 2 max (25.3) It follows, taking the square root of both sides of Equation 25.3 that the effective current is equal to the maximum AC current divided by the square root of two, I = i max /SQR RT [2] = 0.707 i max (25.4) The effective current is called the root mean square current. Similarly we will use E to represent the effective emf of an AC source and V to represent the effective voltage drop in an AC circuit: E = E rms = E max / SQR RT [2] = 0.707 E max (25.5) V = V rms = V max /SQR RT [2] = 0.707 V max (25.6) The AC meters which you use measure the V rms and I rms values of an AC circuit. If the AC line in your home is said to be 110 V, that is the effective value of the voltage. The maximum or peak voltage would be 110 x SQR RT [2] or 155 V. 25.3 Phase Relations of Current and Voltage in AC Circuits In an AC circuit containing only resistance, the instantaneous voltage and current are always in phase. This means they are both zero at the same time and reach their maximum value at the same time. This is shown in Figure 25.3. In Chapter 24 you learned that whenever the current is changing in an inductive coil, an emf is induced. This induced emf depends upon the induction of the coil and on the time rate of change of current. In a coil, continuously changing AC current produces an alternating emf from self-induction. According to Lenz's law this induced emf is opposite to the applied emf. If you neglect the resistance of the coil, the emf of the source will just be equal to the emf of self-induction in the coil. The induced emf will cause the current in the coil to lag behind the applied emf by one-quarter of a cycle. We say that the current in an inductor lags the voltage by 90 o , or that the phase of the voltage across the inductor leads the phase of the current by 90 o . See Figure 25.4. Page 5 Physics Including Human Applications 554 Chapter 25 Alternating Currents GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms and use it in an operational definition: effective values of current and voltage reactance impedance power factor resonance Q-factor AC Circuits Solve alternating-current problems involving resistance, inductance, and capacitance in a series circuit. Phasor Diagrams Draw phasor diagrams for alternating current circuits. Transformer Explain the operation of the transformer. AC Measurements Describe the use of alternating currents in physiological measurements. PREREQUISITES Before you begin this chapter you should have achieved the goals of Chapter 22, Basic Electrical Measurements, and Chapter 24, Electromagnetic Induction. Physics Including Human Applications 555 Chapter 25 Alternating Currents 25.1 Introduction Most of your experience with electricity has probably been with alternating current (AC) circuits. Do you know the difference between AC and direct current (DC) electricity? At some time or another you have probably read the printing on the end of a light bulb. Usually a light bulb has printed on it that it is for use in a 120-volt AC circuit. What was the manufacturer trying to indicate? Could you use a 120-V DC bulb in your household sockets? What, if anything, would happen? Why is AC the dominant form of electrical energy in use? More than 90 percent of electrical energy is used as AC electricity. If you owned an electric train in your childhood, you may recall that you used a transformer to reduce household voltage to the voltage required by the electric train. The transformer is a device that plays a most important role in the use of electrical energy. Do you know what its functions are in transmission of electric power? What is the role of the transformer in the coupling of two AC circuits? These are a few of the questions we will discuss in this chapter. 25.2 Nomenclature Used for Alternating Currents In Chapter 24, we learned that a simple AC generator can be made by rotating a coil of conducting wire in a constant magnetic field. The voltage produced by such a generator has the form shown in Figure 25.1. This sinusoidal voltage alternates polarity + to -and produces an alternating current in a circuit connected to such an AC source. This alternating polarity is in contrast to the unindirectional nature of DC current. In the United States the standard frequency in home and industrial use is 60 cycles/second (Hz). This means that there is a reversal of the direction of the current every 1/120 second. Radio broadcast frequencies are of the order of 10 6 Hz. Some Physics Including Human Applications 556 microwave devices have frequencies the order of 10 10 Hz. So the AC current in common use is of relatively low frequency. If you have two equal resistors and in one there is an AC current of 1 ampere maximum and in the other a DC current of one ampere, would you expect the two resistors to produce the same heat? Let us consider the situation to see what we should expect. In the resistor with the DC current, the current is constant. In the resistor with AC current, the current varies from 0 to 1 A in one direction and the same in the opposite direction. You have learned that the heat produced depends upon the square of current for a given resistor. Hence, you would have been correct if you had said that more heat is produced by the 1-A DC current. For a varying current we would expect the heat produced to be proportional to the average value of the square of the current. The square root of this quantity is called effective current. That is, it is the current that would produce the same heating effect as one ampere of DC current. Let us find the relationship between the effective AC current and maximum AC current. The AC current at any instant of time can be expressed as a sinusoidal function of time, i = i max sin ?t (25.1) where i max is the maximum value of the current, ? is the angular frequency of the AC, and t is the time in seconds. The square of the current will be proportional to the power dissipated in the resistor; so i 2 = i 2 max sin 2 ?t (25.2) We can draw a graph of the square of the current versus the time(Figure 25.2). The total heating effect will be proportional to the area under the curve. You see that the curve between p and 2p radians is exactly the same as that from 0 to p radians. So we will need to consider only half of the total cycle. The curve for the square of the DC current is a horizontal line represented by AB. The problem is then to find the altitude for the rectangle with the base of p which has the same areas under the i 2 max sin 2 ?t curve. By inspection you might say that the area under AB is about equal to the area under the i 2 max sin 2 ?t curve if A 1 = A 2 and A 3 = A 4 . This is true if OC is equal to i 2 max /2. You can show this by measuring the two areas. Physics Including Human Applications 557 Let us use the symbol I for the effective current. We can see from Figure 25.2 that the effective current squared is equal to one-half of i max squared, I 2 = (1/2) i 2 max (25.3) It follows, taking the square root of both sides of Equation 25.3 that the effective current is equal to the maximum AC current divided by the square root of two, I = i max /SQR RT [2] = 0.707 i max (25.4) The effective current is called the root mean square current. Similarly we will use E to represent the effective emf of an AC source and V to represent the effective voltage drop in an AC circuit: E = E rms = E max / SQR RT [2] = 0.707 E max (25.5) V = V rms = V max /SQR RT [2] = 0.707 V max (25.6) The AC meters which you use measure the V rms and I rms values of an AC circuit. If the AC line in your home is said to be 110 V, that is the effective value of the voltage. The maximum or peak voltage would be 110 x SQR RT [2] or 155 V. 25.3 Phase Relations of Current and Voltage in AC Circuits In an AC circuit containing only resistance, the instantaneous voltage and current are always in phase. This means they are both zero at the same time and reach their maximum value at the same time. This is shown in Figure 25.3. In Chapter 24 you learned that whenever the current is changing in an inductive coil, an emf is induced. This induced emf depends upon the induction of the coil and on the time rate of change of current. In a coil, continuously changing AC current produces an alternating emf from self-induction. According to Lenz's law this induced emf is opposite to the applied emf. If you neglect the resistance of the coil, the emf of the source will just be equal to the emf of self-induction in the coil. The induced emf will cause the current in the coil to lag behind the applied emf by one-quarter of a cycle. We say that the current in an inductor lags the voltage by 90 o , or that the phase of the voltage across the inductor leads the phase of the current by 90 o . See Figure 25.4. Physics Including Human Applications 558 The inductance in the circuit not only causes the current to lag the emf but it also reduces the current to a smaller value than it would have if there were no inductance present. The voltage drop V across an inductance L is given by V = I?L = 2pÆ’LI (25.7) where Æ’ is the frequency in cycles per second and L is the inductance in the circuit in henries. It follows that the AC current in an inductance is given by the voltage drop across the inductance divided by 2pÆ’ times the inductance, I = V/(2 Æ’L) (25.8) The factor 2pfL is called the inductive reactance of the circuit, represented by X L , and is measured in ohms L is in henries and f is the frequency (in Hz), X L = 2pÆ’L (25.9) An inductive element in an AC circuit acts as an inertia element and impedes the alternating current. We also note that this effect depends directly upon the frequency. Hence for high frequency an inductor exhibits a large inertia and thus greatly impedes a high-frequency alternating current. If you connect a capacitor to an AC source the plates of the capacitor become charged alternatively positive and negative, according to the surge of charges back and forth in the connecting wires. So, even though there can be no constant DC current through a capacitor, we can say that there can be an alternating current through a capacitor. A capacitor does present an impedance to an alternating current which is called capacitive reactance. We shall represent the capacitive reactance by X c . In a way analogous to the definition of capacitance as the ratio of voltage to charge, we shall define the capacitive reactance as the ratio of the voltage drop across the capacitor to the current through the capacitor, X c = V/I (25.10) where X c = 1/(2pÆ’C) ohms (25.11) and C is the capacitance in farads and Æ’ is the frequency in cycles per second (Hz). Note that the capacitive reactance decreases as the frequency increases. A capacitor has infinite impedance for DC sources. The inertia effect of a capacitor in an AC circuit decreases as the frequency increases.Read More

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