References: Circuits with Resistors and Diodes Edward M. Purcell, Electricity and Magnetism 2 nd ed, Ch. 4, (McGraw Hill, 1985) R.P. Feynman, Lectures on Physics, Vol. 2, Ch. 22, (Addison Wesley, 1963). Equipment: Pasco Digital Function Generator, Tektronix Digital Oscilloscope, Resistor Boxes, Capacitors, Ammeters, Voltmeters, DC Power Supply, and Diodes. Introduction: In the laboratory, we first review the simplest circuit, in which a DC power source is connected to a resistor. Electrical power is defined and measured in this system. A similar circuit is constructed using an AC source, and power relations in this case are examined. Combinations of resistors in series and parallel connection are discussed, and the standard resistance boxes of the laboratory are used as an example. Optionally on the part of the student, this material can be clarified by building resistive circuits. The diode is introduced, and the concept of the ideal diode as a circuit element is defined. A crude DC power supply is constructed from resistors and diode, to demonstrate the essentials of instrument power sources. At the option of the instructor, the effect of adding a capacitor as part of the supply can be examined. It has the effect of something the pulsating DC and so indicates how real DC supplies are constructed. No details of capacitance are given, as it is the subject of the next laboratory. The section An Introduction to the Oscilloscope in the Brown Physics Lab Manual may be useful. 140625 1
The Simple DC Circuit In Fig.1, a DC power source is shown connected to a resistor. The arrow shows the direction of (conventional) current. The ammeter (A) measures the current through the resistor R. Actually, it measures the combined current flowing both through R and the voltmeter (V) but the latter requires very little current to register a deflection of the meter movement, so the ammeter current can be taken as entirely that through R alone. The circuit is complete, because the two points, m and n, are both grounded. This places them both at the same potential at all times, so they can be considered to be the same point. Looked at from the stand point of current, the latter leaves the ammeter at its negative terminal, travels to and through the entire earth, and re-enters the grounded negative post of the power supply. m n Construct the circuit of Fig.1 using the HP power supply, and two Agilent multimeters, one connected and set for a voltage measurement, the other for a current measurement. Use the resistance box set at 10 kω. Be sure not to connect the negative terminal of the ammeter directly to the power supply, although that would be completely correct electrically. Instead, make the ground returns by connecting the ammeter terminal to a ground on the AC power strip, and the negative terminal of the power supply to the ground terminal on the face of the power supply itself. We want to demonstrate the uses that can be made of the ground Fig.1 terminal, and it will be very relevant in the AC measurement made below. Turn on the power supply, and raise its voltage until 10 V DC is dropped across the resister. The current should then read as one milliampere (ma) on the ammeter. Leave this circuit set up as you prepare the next observation. Neither the meters nor the power supply can be used in the next circuit, which uses AC power. 140625 2
The Simple AC Circuit In Fig. 2, an AC power source is shown connected to a resistor. A signal generator will be used as the power source, which is why we wanted to use a high resistance in the DC circuit to which it will be compared. A signal generator only supplies currents in the milliampere range. This particular signal generator, as the markings on its output terminals indicate, has one of its terminals internally grounded (that is, connected to the ground terminal of its AC power cord). Because of this internal ground, the circuit of Fig. 2 is actually an electrically closed loop, although it may not appear to be. The circuit should be wired in the following way. A lead is connected from the ungrounded terminal of the signal generator to one of the resistor terminals of the resistance box (R-box). The other resistor terminal is connected by a very short lead to the black ground terminal also on the R- box. In spite of its marking, this ground terminal of the generator does not go to a real ground, as the similarly marked terminal of the generator does, but is more a statement of what should be grounded if possible (it connects the outer metal case of the R-box). So complete the circuit by a lead from the ground terminal of the R-box to a true ground point on the AC power strip. Signal observation is made with the scope (A-channel). The arrangement of grounds on this instrument should also be well understood. Look at the signal input connector of the B-channel. It will not be used, but shows you the type of coaxial connector called BNC that is often used in equipment handling high frequency signals. Neither the outer ring nor the center pin of the BNC connector is internally grounded, but there is a preferred arrangement if a ground connection is to be made. It is the outer ring that should be grounded, because this gives the greatest protection against noise and signal loss the reasons are complicated. Now the A- channel of the scope has been equipped with a double banana plug, and one of the two prongs corresponds to the outer ring, the other to the center pin, of the BNC connector. The banana plug 140625 3
that corresponds to the outer ring, the one that is preferred for ground connection, is the one that has a tab in the molded plastic case. Notice that only one of the two plugs has such a tab. Making this ground connection is simple, since there are now several points where ground is located the terminal of the R-box, the marked terminal of the signal generator, or those on the power strip. The scope signal terminal (the un-tabbed banana plug) is fixed to look at the voltage dropped across the resistor by connecting it to the ungrounded side of the resistance. Set the R- box to 10 k Ω, the same as was used in the DC circuit. The signal generator should be set for sine output, and a frequency of 1000 Hz or so to start with. Set its amplitude so that a signal is observed on the scope that has a peak-to-peak value of 2 10 2 1/2 = 20 2 1/2 = 28.3 VAC If you look at that value, you have a sine wave whose amplitude is one half the value, or 14.2 VAC, and so has an RMS value (0.707 of the amplitude) of 10 VAC. So you have an RMS voltage equal to the magnitude of the DC voltage that was placed across an identical resistance value in the DC circuit. You are dissipating power in this resistor at the same rate as in the DC case. Further, the frequency does not enter, and so the statements hold true if you vary the frequency over a very large range. The limit of this statement is reached when the AC frequency reaches regions where other forms of reactance, such as capacitance or inductance, must be taken into account. These are the subjects of later laboratories. Resistors in Combination When two or more resistors are connected together so the current that passes through one resistor must also pass through the next, the resistors are said to be connected in a series. An example of such a connection is given in Fig. 3(a). It is useful to be able to find the value of a single resistor that would be electrically equivalent to a number of resistors in series. This is easily found from Ohm s law, and the fact that when the voltage drop across the equivalent resistance is equal to that across the set of resistors, the current through it must be the same as that through the set. Calling the equivalent resister R, and the series resistors it replaces R 1, R 2, etc., we have 140625 4
V = IR 1 + IR 2 + IR 3 +. Eq. (1) = I (R 1 + R 2 + R 3 +.) so that the equivalent to resistors in series is the sum: R=R 1 + R 2 + R 3 +. Eq. (2) Just as often, it is useful to be able to find the equivalent single resistor for a number of resistors connected in parallel. The term means that the separate resisters R 1, R 2, and so on, are connected so the same voltage is dropped across each of them. Fig. 3(b) shows an example of the parallel connection. Here the equivalent resistor R must have a value such that when the voltage dropped across it is equal to that dropped across each parallel resistor R 1, R 2,, the current through R must equal the sum of the currents through the resistors it replaces. We have I = I 1 + I 2 + I 3 +. Eq.(3) So that by repeated use of the Ohm s law V/R = V/R 1 + V/R 2 + V/R 3 + Eq. (4) 140625 5
giving the relation for the equivalent resistance 1/R=1/R 1 + 1/R 2 + 1/R 3 +. Eq. (5) In the special case of two resistors in parallel, Eq. (5) becomes R=R 1 R 2 / (R 1 + R 2 ) Eq. (6) Eq. (1) and Eq. (5) can easily be used to show that the net resistance of series resistors is always greater than the largest single resistor in a set, and the net resistance of resistors in parallel is always less than that of the smallest resistor in the set. As a special case, any resistor in parallel with zero resistance, like a short length of wire, makes no contribution -the net resistance is zero. If these statements are not clear, you should discuss them with the lab instructor. The resistance boxes themselves offer an example of combined series and parallel resistance, where the parallel resistances have zero value. Fig. 4 shows the schematic of a few of the repeating sections of the R-box. The numbers 1, 2, 3, 4 are resistance values in one column of the R-box, and we suppress some multiple of 10 that defines which column. If all four switches are up (open), current floes through 1+2+3+14 = 10 ohms. If S were down (closed) the current that went through the corresponding resistor would now go around it through the wire and switch. Now in going from A to B current would flow through 0+2+3+4 = 9 ohms. By closing the right switches, we can obtain any value of resistance from zero to ten ohms. The whole box simply repeats this technique in each column, and the different columns are themselves connected in series with one another. We will not build circuits to demonstrate these rules for resistors in combination, as there will be many opportunities to exercise your knowledge of them. This is a general remark, and for the particular student who has not seen them before, or is uncertain when and how they could be applied, by all means use the laboratory equipment, in 140625 6
particular groups of resistance boxes each treated as a single resistor whose value you can set, to make connections that will, clarity the rules. The diode exercises that follow are useful, but not as elementary and important to you as the understanding of this present material. The Diode For a resistor, a plot of current versus voltage would look like the straight R-line of Fig. 5. Here V represents the voltage difference between one particular end of the resistor and the other, and I represents the current flowing from that end to the other. Positive values of V mean that the chosen end is at a higher potential than the other, and positive I means that current flows from the reference end to the other end. Negative V means that the current flows back through the resistor from the other end to the reference point. Different values of resistance would be represented by straight lines of different slope- to be precise, the slope of the R-line is the reciprocal of the resistance since I/V=1/R by Ohm s law. Now suppose we were looking at a very low resistance. The slope of its R-line would be large, getting closer to the current axis, and meaning that that very small voltage would be dropped across the resistor even when large currents were going through it. In the limit of zero resistance, the line would coincide with the current axis. Conversely, for a very high resistance we would have a line almost coinciding with the voltage axis, as very large voltage would lead to very small current through the resistor. The diode is a circuit element that presents a very small resistance to the flow of current in one direction, and a very large resistance to flow in the opposite direction. It is represented in Fig. 5(a) by two straight line segments, one of which represents very low forward resistance, the other very high reverse resistance. 140625 7
The diode symbol contains an arrowhead pointing in the direction in which conventional, positive current encounters very low resistance, and a bar, as shown in Fig. 5(b). The arrowhead represents one of two sections of a real diode, called the anode. The bar represents the other section, called the cathode. The cathode end of the diodes you will use is distinguished by a black band marked around the tubular body. Physically, these sections are made of two different kinds of semiconductor crystals. Where they are joined, in a careful manufacturing process, a potential barrier develops against the flow of quasi-free electrons from the one crystal into the other. This is what gives the diode its directional property. The potential barrier is like a hill, and in one direction it has to be climbed with great effort, while in the other direction the electrons fall downhill with ease. In the next circuit you will see, to some approximation, how a low voltage power supply such as the might be made. Instead of using the 60 Hz wall power directly, one of our approximations will be to use a higher frequency since wave from the signal generator. The principle is exactly 140625 8
the same, and when we are done you will have an inkling as to why the frequency increase is convenient. In Fig. 6, the AC current flows easily clockwise through the diode, so the voltage dropped across it is small. Therefore essentially all the voltage from the generator is dropped across the resistor during the half cycle when the generator voltage is positive with respect to ground. But when this voltage goes negative, the resistance of the diode becomes very large (order of mega-ohms), so essentially all the voltage is dropped across the diode rather that the 10 k Ω resistor. Looked at on the scope, the voltage across the diode would be as in Fig. 6(c) when the voltage from the generator is as shown in Fig. 6(b). The display of Fig. 6(b) and 6c can be set up on the scope using its dual trace feature, described below. First, introduce the diode into the AC circuit that you constructed before. The most convenient place is on the R-box itself. As it was set up, there is a lead from one side of the resistor terminal to the ground post of the R-box. Replace this lead by the diode, with the cathode hand toward the ground post. Fig. 6(a) is then the corresponding circuit diagram, with R=10 k Ω. To make the desired display, put another banana plug on the B signal input of the scope. Place a lead from its tabbed banana plug to that of the A signal input, so they are both at ground. Connect its signal banana plug to the R-box post where the diode is connected to the resistor. Then the A-channel is looking at the sine wave output of the signal generator, as before, and the B-channel is looking at the voltage across the diode. By setting the channel selector button from A to alt, the beam of the scope will alternately sweep according to the A- channel signal and the B-channel signal. Be sure to set the B sensitivity (volts/cm) to the same value as that of the A channel. You may have to adjust the vertical positions of each signal, 140625 9
and their sensitivities, to see both signals clearly. What you have done is changed alternating current into unidirectional current, the first step toward obtaining DC power. The process is called rectifying the AC signal 1. What we need now is some way of storing the energy briefly, so that we can feed it back into the circuit during those time periods when we have no current now flowing in the resistor. One such device is the capacitor, the subject of the next laboratory. Just to see its effect, add the capacitance box (Cbox), which physically resembles the R-box, at the point indicated in Fig. 7. Note that the two terminals of the capacitor are the red and black ones- the green ground terminal is not used. Keep the A-channel where it was before. Start with zero capacitance, then push switches up on the capacitance box. Larger values of capacitance will store larger values of energy. Just note what you see. 1 More exactly, this is half-wave rectification, since only half the sine wave is retained. A more complicated circuit would be able to use both halves of the AC signal, which is called full-wave rectification. 140625 10