# Eðlisfræði 2, vor 2007

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2 Again consider an inductor with inductance connected to an AC source. If the AC source provides a voltage, what is the current through the inductor as a function of time? Hint C.1 Using You can obtain the answer almost immediately if you consider the results of : The amplitude of the voltage ( ) is ; the frequency is the same as in, and the phase difference is. Do you remember what leads and what lags? Express your answer in terms of,,, and. Use the cosine function, not the sine function, in your answer. = For the amplitudes (magnitudes) of voltage and current, one can write (for the resistor) and (for the inductor). If one compares these expressions, it should not come as a surprise that the quantity, measured in ohms, is called inductive reactance; it is denoted by (sometimes ). It is called reactance rather than resistance to emphasize that there is no dissipation of energy. Using this notation, we can write (for a resistor) and (for an inductor). Also, notice that the current is in phase with voltage when a resistor is connected to an AC source; in the case of an inductor, the current lags the voltage by. What will happen if we replace the inductor with a capacitor? We will soon see. Part D Consider the potentials of points a and b on the inductor in diagram 2. If the voltage at point b is greater than that at point a, which of the following statements is true? Hint D.1 How to approach the problem Try drawing graphs of the current through the inductor and voltage across the inductor as functions of time. The current must be positive (clockwise). The current must be directed counterclockwise. The derivative of the current must be negative. The derivative of the current must be positive. It may help to think of the current as having inertia and the voltage as exerting a force that overcomes this inertia. This viewpoint also explains the lag of the current relative to the voltage. Part E Assume that at time, the current in the inductor is at a maximum; at that time, the current flows from point b to point a. At time, which of the following statements is true? Hint E.1 How to approach the problem Try drawing graphs of the current through the inductor and voltage across the inductor as functions of time. The voltage across the inductor must be zero and increasing. The voltage across the inductor must be zero and decreasing. The voltage across the inductor must be positive and momentarily constant. The voltage across the inductor must be negative and momentarily constant. Part F Now consider a capacitor with capacitance connected to an AC source (diagram 3). If the AC source provides a voltage, what is the current through the capacitor as a function of time? Hint F.1 Hint F.2 Hint F.3 The relationship between charge and voltage for a capacitor The relationship between charge and current Mathematical details Express your answer in terms of,, and. Use the cosine function with a phase shift, not the sine function, in your answer. Page 2 of 17

3 = For the amplitude values of voltage and current, one can write. If one compares this expression with a similar one for the resistor, it should come as no surprise that the quantity, measured in ohms, is called capacitive reactance; it is denoted by (sometimes ). It is called reactance rather than resistance to emphasize that there is no dissipation of energy. Using this notation, we can write, and voltage lags current by radians (or 90 degrees). The notation is analogous to for a resistor, where voltage and current are in phase, and for an inductor, where voltage leads current by radians (or 90 degrees). We see, then, that in a capacitor, the voltage lags the current by, while in the case of an inductor, the current lags the voltage by the same quantity. In a capacitor, where voltage lags the current, you may think of the current as driving the change in the voltage. Part G Consider the capacitor in diagram 3. Which of the following statements is true at the moment the alternating voltage across the capacitor is zero? Hint G.1 How to approach the problem Hint G.2 Graphs of and The current must be directed clockwise. The current must be directed counterclockwise. The current must be at a maximum. The current must be zero. Part H Consider the capacitor in diagram 3. Which of the following statements is true at the moment the charge of the capacitor is at a maximum? Hint H.1 How to approach this problem Since the voltage is directly proportional to the charge, when the charge is maximum, so is the voltage. Try drawing graphs of the (displacement) current through the capacitor and voltage across the capacitor as functions of time. Find the current when the voltage drop is maximum. Hint H.2 Graphs of and The current must be directed clockwise. The current must be directed counterclockwise. The current must be at a maximum. The current must be zero. Part I Consider the capacitor in diagram 3. Which of the following statements is true if the voltage at point b is greater than that at point a? Hint I.1 How to approach the problem Hint I.2 Graphs of and The current must be directed clockwise. The current must be directed counterclockwise. The current may be directed either clockwise or counterclockwise. Part J Consider a circuit in which a capacitor and an inductor are connected in parallel to an AC source. Which of the following statements about the magnitude of the current through the voltage source is true? Hint J.1 Driven AC parallel circuits The voltage across each element is the same at every moment in time. However, the magnitudes of the currents in an AC circuit cannot be added without consideration of the phase angle between the Page 3 of 17

5 Answer not displayed Find the amplitude of the current through the voltage source. Hint C.1 Hint C.2 The parallel connection The current through the voltage source as a vector sum Express your answer in terms of the magnitudes of the individual currents and. = Answer not displayed Part D What is the tangent of the phase angle between the voltage and the current through the voltage source? Hint D.1 The current through the voltage source as a vector sum Express in terms of and. = Answer not displayed Phasors Explained Learning Goal: To understand the concept of phasor diagrams and be able to use them to analyze AC circuits (those with sinusoidally varying current and voltage). Phasor diagrams provide a convenient graphical way of representing the quantities that change with time along with, which makes such diagrams useful for analyzing AC circuits with their inherent phase shifts between voltage and current. You have studied the behavior of an isolated resistor, inductor, and capacitor connected to an AC source. However, when a circuit contains more than one element (for instance, a resistor and a capacitor or a resistor and an inductor or all three elements), phasors become a useful tool that allows us to calculate currents and voltages rather easily and also to visualize some important processes taking place in the AC circuit, such as resonance. Let us assume that a certain quantity changes over time as. A phasor is a vector whose length represents the amplitude (see the diagram ).This vector is assumed to rotate counterclockwise with angular frequency ; that way, the horizontal component of the vector represents the actual value at any given moment. In this problem, you will answer some basic questions about phasors and prepare to use them in the analysis of various AC circuits. In parts A - C consider the four phasors shown in the diagram. Assume that all four phasors have the same angular frequency. At the moment depicted in the diagram, which of the following statements is true? leads by. leads by. Page 5 of 17

6 leads by. leads by. At the moment shown in the diagram, which of the following statements is true? lags by. lags by. lags by. lags by. At the moment shown in the diagram, which of the following statements is true? leads by. lags by. leads by. lags by. Let us now consider some basic applications of phasors to AC circuits. For a resistor, the current and the voltage are always in phase. For an inductor, the current lags the voltage by. For a capacitor, the current leads the voltage by. Part D Consider this diagram. Let us assume that it describes a series circuit containing a resistor, a capacitor, and an inductor. The current in the circuit has amplitude, as indicated in the figure. Which of the following choices gives the correct respective labels of the voltages across the resistor, the capacitor, and the inductor? Part E Now consider a diagram describing a parallel AC circuit containing a resistor, a capacitor, and an inductor. This time, the voltage across each of these elements of the circuit is the same; on the diagram, it is represented by the vector labeled. The currents in the resistor, the capacitor, and the inductor are represented respectively by which vectors? Page 6 of 17

7 A Resistor and a Capacitor in a Series AC Circuit A resistor with resistance and a capacitor with capacitance are connected in series to an AC voltage source. The time-dependent voltage across the capacitor is given by. What is the amplitude of the total current in the circuit? Hint A.1 Hint A.2 Hint A.3 How to approach the problem Applying Ohm's law to a capacitor The reactance of a capacitor Express your answer in terms of any or all of,,, and. = Answer not displayed Determining Inductance from Voltage and Current An inductor is hooked up to an AC voltage source. The voltage source has EMF and frequency. The current amplitude in the inductor is. What is the reactance of the inductor? Hint A.1 Definition of reactance Express your answer in terms of and. = What is the inductance of the inductor? Hint B.1 Reactance in terms of and Express your answer in terms of,, and. = What would happen to the amplitude of the current in the inductor if the inductance were doubled? Hint C.1 How to approach the problem There would be no change in the amplitude of the current. The amplitude of the current would be doubled. The amplitude of the current would be halved. The amplitude of the current would be quadrupled. Driven L-C Circuits Page 7 of 17

8 A Driven Series L-C Circuit Learning Goal: To understand why a series L-C circuit acts like a short circuit at resonance. An AC source drives a sinusoidal current of amplitude and frequency into an inductor having inductance and a capacitor having capacitance that are connected in series. The current as a function of time is given by. Recall that the voltages and across the inductor and capacitor are not in phase with the respective currents and. In particular, which of the following statements is true for a sinusoidal current driver? and both lag their respective currents. and both lead their respective currents. lags and leads. leads and lags The phase angle between voltage and current for inductors and capacitors is 90 degrees, or radians. Among other things, this means that no power is dissipated in either the inductor or the capacitor, since the time average of current times voltage,, is zero. What is, the voltage delivered by the current source? Hint B.1 Current in a series circuit Note that the current through the current source, the capacitor, and the inductor are all equal at all times..2 What is Find the voltage across the capacitor, the voltage across the capacitor as a function of time?.2.a Find the charge on the capacitor What is the charge on the capacitor? Express your answer in terms of the capacitance and the voltage across it. =.2.b What is the current in terms of the charge on the capacitor? With the conventions in the circuit diagram, what is the current on the capacitor in terms of its voltage? Express your answer in terms of the capacitance and the voltage across it and/or its derivative. = Integrate both sides of the equation (once you substitute in the appropriate expression for ). The constant of integration is zero because there is no average (DC) current or voltage in a pure AC circuit. Express your answer in terms of,,, and. =.3 What is What is the voltage across the inductor?, the voltage across the inductor as a function of time? Hint B.3.a Voltage and current for an inductor Page 8 of 17

9 Express in terms of,, and. = Hint B.4 Total voltage You should have defined the voltages across the capacitor and inductor in such a way that the total voltage is equal to, that is, so that total voltage is obtained by adding the voltages at time, with careful attention paid to the signs. Express in terms of some or all of the variables, and, or and,, and, of course, time. = With and the amplitudes of the voltages across the inductor and capacitor, which of the following statements is true? At very high frequencies and at very low frequencies. At very high frequencies and at very low frequencies. for all frequencies. for all frequencies. and are about the same at all frequencies. Part D The behavior of the L-C circuit provides one example of the phenomenon of resonance. The resonant frequency is. At this frequency, what is the amplitude of the voltage supplied by the current source? Express your answer using any or all of the constants given in the problem introduction. = 0 Part E Which of the following statements best explains this fact that at the resonant frequency, there is zero voltage across the capacitor and inductor? The voltage is zero at all times because. The voltages ; ; and are zero at all times. The voltage The voltage is zero only when the current is zero. is zero only at times when the current is stationary (at a max or min). At the resonant frequency of the circuit, the current source can easily push the current through the series L-C circuit, because the circuit has no voltage drop across it at all! Of course, there is always a voltage across the inductor, and there is always a voltage across the capacitor, since they do have a current passing through them at all times; however, at resonance, these voltages are exactly out of phase, so that the net effect is a current passing through the capacitor and the inductor without any voltage drop at all. The L-C series circuit acts as a short circuit for AC currents exactly at the resonant frequency. For this reason, a series L-C circuit is used as a trap to conduct signals at the resonant frequency to ground. A Voltage-Driven Parallel L-C Circuit An AC source that provides a voltage drives an inductor having inductance and a capacitor having capacitance, all connected in parallel. Recall that the currents and through the inductor and capacitor are not in phase with their respective voltages and. In particular, for a sinusoidal voltage driver, which of the following statements is true? Answer not displayed Page 9 of 17

10 What is the amplitude of the current through the voltage source? In other words, if the time-dependent current is, what is the value of? Hint B Hint B.4 Voltage in a parallel circuit Current through the capacitor The current through the inductor Total current Express the amplitude of the current in terms of,,, and. = Answer not displayed With and representing the respective magnitudes of the currents through the inductor and capacitor, which of the following statements is true? Answer not displayed Part D The L-C circuit is one example of a system that can exhibit resonance behavior. The resonant frequency is. At this frequency, what is the amplitude of the current supplied by the voltage source? Express your answer in terms of and any other need terms from the problem introduction. = Answer not displayed Part E Reactance and Current Consider the two circuits shown in the figure. The current in circuit 1, containing an inductor of self-inductance, has an angular frequency, while the current in circuit 2, containing a capacitor of capacitance, has an angular frequency. If we increase and decrease, both bulbs grow dimmer. If we keep and constant, we can achieve the exact same effect of decreasing the brightness of each bulb by performing which of the following sets of actions? Hint A.1 Hint A.2 Hint A.3 How to approach the problem Inductive reactance Capacitive reactance.4 Determine how can be changed.5 Determine how can be changed increasing and decreasing increasing both and decreasing and increasing decreasing both and As you found out, the reactance of these circuits can be changed not only by varying the frequency of the current, but also by changing the characteristics of the elements in them, i.e., by changing Page 10 of 17

11 the inductance of the inductor and the capacitance of the capacitor. Now combine the capacitor, the inductor, and the bulbs in a single circuit, as shown in the figure. What happens to the brightness of each bulb if you increase the frequency of the current in the new circuit while keeping and constant? Hint B.1 How to approach the problem In the new circuit, bulb 1 is still in series with the inductor, and so is bulb 2 with respect to the capacitor. Therefore, to determine how their brightness may change, you need to analyze how the current in both the inductor and the capacitor may change at higher frequencies. Note that because the branch of the circuit containing the inductor and bulb 1 is in parallel with the branch containing the capacitor and bulb 2, if the current in one branch decreases, the current in the other branch increases..2 Find which element experiences a decrease in current at higher frequencies Which element of the circuit will experience a decrease in current at higher frequencies? Hint B.2.a Hint B.2.b Hint B.2.c Current in circuit elements Inductive reactance Capacitive reactance the inductor the capacitor both the inductor and the capacitor neither the inductor nor the capacitor Since inductive reactance is proportional to frequency, for a given voltage, high-frequency currents will have a much smaller amplitude through the inductor than through the capacitor. Both bulbs become brighter. The brightness of each bulb remains constant. Bulb 1 becomes brighter than bulb 2. Bulb 2 becomes brighter than bulb 1. Both bulbs grow dimmer. Since inductive reactance is proportional to frequency, for a given voltage, high-frequency currents will have a much smaller amplitude through the inductor than through the capacitor. That is, the inductor tends to block high-frequency currents. The opposite situation occurs if the frequency is decreased. The capacitor will block low-frequency currents and bulb 2 will grow dimmer. A High-Pass Filter A series L-R-C circuit consisting of a voltage source, a capacitor of capacitance, an inductor of inductance, and a resistor of resistance is driven with an AC voltage of amplitude and frequency. Define to be the amplitude of the voltage across the resistor and the inductor. Find the ratio..1 Find Assume that the amplitude of the current in the circuit is. Write down an equation for, the amplitude of the input voltage of the circuit. Hint A.1.a Hint A.1.b How to approach the problem Combined impedance Express your answer in terms of,, the reactance of the capacitor, and the reactance of the inductor. Page 11 of 17

12 =.2 Find Assuming that the amplitude of the current in the circuit is, write down an equation for the output voltage of the circuit. Express your answer in terms of,, the reactance of the capacitor, and the reactance of the inductor. = Express your answer in terms of either,, and, or,, and. = For the following questions it will be useful to write the voltage ratio in the following form:. Which of the following statements is true in the limit of large ( )? Hint B.1 Implications of large In the large limit,, so you can approximate. is proportional to. is proportional to. is proportional to. is close to 1. Which of the following statements is true in the limit of small ( )? Hint C.1 Implications of small In the small limit,, so you can approximate. In addition,, and. is proportional to. is proportional to. is proportional to. is close to 1. When is large,, and when is small,. Therefore, this circuit has the property that only the amplitude of the low-frequency inputs will be attenuated (reduced in value) at the output, while the amplitude of the high-frequency inputs will pass through relatively unchanged. This is why such a circuit is called a high-pass filter. Constructing a Low-Pass Filter A series L-R-C circuit is driven with AC voltage of amplitude and frequency. Define to be the amplitude of the voltage across the capacitor. The resistance of the resistor is, the capacitance of the capacitor is, and the inductance of the inductor is. Page 12 of 17

13 What is the ratio?.1 Find.2 Find Express your answer in terms of either,,, and or,, and. = Answer not displayed Power in ac Circuits Alternating Current, LC circuit A capacitor with capacitance is connected in parallel to two inductors: inductor 1 with inductance, and inductor 2 with inductance, as shown in the figure. The capacitor is charged up to a voltage, at which point it has a charge. There is no current in the inductors. Then the switch is closed. Since the two inductors are in parallel, the voltage across them is the same at any time. Hence,, where and are the reactances of inductors 1 and 2, and and are the currents through them. Use this equality to express in terms of. Hint A.1 The reactance of an inductor Express your answer in terms of. = Answer not displayed What is the effective inductance of the inductors 1 and 2 in the circuit? Hint B.1 Formulas for effective inductance Express your answer in terms of. = Answer not displayed Find the maximum current through inductor 1. Page 13 of 17

14 .1 Use conservation of energy to find the total maximum current Express your answer in terms of,, and. = Answer not displayed Alternating Voltages and Currents The voltage supplied by a wall socket varies with time, reversing its polarity with a constant frequency, as shown in the graph. What is the rms value of the voltage plotted in the graph? Hint A.1 RMS value of a quantity with sinusoidal time dependence A quantity that varies with time as (or ) has a maximum value equal to and an rms value given by..2 Find the maximum value of the voltage What is the maximum value of the voltage plotted in the graph? Express your answer in volts. = 170 Express your answer in volts. = 120 This is the standard rms voltage supplied to a typical household in North America. When a lamp is connected to a wall plug, the resulting circuit can be represented by a simplified AC circuit, as shown in the figure. Here the lamp has been replaced by a resistor with an equivalent resistance = 120. What is the rms value of the current flowing through the circuit? Hint B.1 Ohm's law in AC circuits Ohm's law can still be applied to an AC circuit, provided the values used to describe all physical quantities are consistent. For example, Ohm's law can be written using maximum values of voltage and current, or alternatively using rms quantities. Express your answer in amperes. = 1.00 What is the average power dissipated in the resistor? Hint C.1 Average power in an AC circuit Page 14 of 17

15 Express your answer in watts. = 120 The instantaneous power dissipated in the resistor can be substantially higher than the average power. However, since the voltage supplied to the resistor varies in time, so does the instantaneous power. Therefore, a better estimate of the energy dissipated in an AC circuit is given by the average power. For example, the power rating on light bulbs is in fact the average power dissipated in the bulb. The RLC Circuits A Series L-R-C Circuit: The Phasor Approach Learning Goal: To understand the use of phasor diagrams in calculating the impedance and resonance conditions in a series L-R-C circuit. In this problem, you will consider a series L-R-C circuit, containing a resistor of resistance, an inductor of inductance, and a capacitor of capacitance, all connected in series to an AC source providing an alternating voltage. You may have solved a number of problems in which you had to find the effective resistance of a circuit containing multiple resistors. Finding the overall resistance of a circuit is often of practical interest. In this problem, we will start our analysis of this L-R-C circuit by finding its effective overall resistance, or impedance. The impedance is defined by, where and are the amplitudes of the voltage across the entire circuit and the current, respectively. Find the impedance of the circuit using the phasor diagram shown. Notice that in this series circuit, the current is same for all elements of the circuit: You may find the following reminders helpful: In a series circuit, the overall voltage is the sum of the individual voltages. In an AC circuit, the voltage across a capacitor lags behind the current, whereas the voltage across an inductor leads the current. The reactance (effective resistance) of an inductor in an AC circuit is given by. The reactance (effective resistance) of the capacitor in an AC circuit is given by. Hint A.1 Hint A.2 Hint A.3 Finding individual voltages Finding the overall voltage Combining the vectors Express your answer in terms of,, and. = Answer not displayed Now find the tangent of the phase angle between the current and the overall voltage in this circuit. Express in terms of,,, and. = Answer not displayed We may be interested in finding the resonance conditions for the circuit, in other words, the conditions corresponding to the maximum current amplitude produced by a voltage source of a given amplitude. Finding such conditions has an immediate practical interest. For instance, tuning a radio means, essentially, changing the parameters of the circuitry so that the signal of the desired frequency has the maximum possible amplitude. Imagine that the parameters,,, and the amplitude of the voltage are fixed, but the frequency of the voltage source is changeable. If the frequency of the source is changed from a very low one to a very high one, the current amplitude will also change. The frequency at which is at a maximum is called resonance. Find the frequency at which the circuit reaches resonance. Hint C.1 Analyzing the impedance Page 15 of 17

17 , where is the voltage across the capacitor, is the amplitude of the current through the capacitor, and is the capacitive reactance. Hint B.2 Current at the resonance frequency Recall that at the resonance frequency the impedance is equal to the resistance. As a result, the current in the system is given by. Since we know the maximum voltage that the capacitor can handle, we should be able to combine this equation and the equation for capacitor voltage to determine a maximum source voltage. Express your answer in volts to three significant figures. = 36.8 Summary 8 of 16 problems complete (46.49% avg. score) of 35 points Page 17 of 17

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