Computerized investigations of battery characteristics*

Transcription

1 FEATURES Computerized investigations of battery characteristics* P F Hinrichsen 1 Physics Department, John Abbott College, Ste Anne de Bellevue, Quebec H9X 3L9, Canada Hinrich@generation.net Abstract A computer interface has been used to measure the terminal voltage versus current characteristic of a variety of batteries, their series and parallel combinations, and the variation with discharge. The concept of an internal resistance demonstrates that current flows through the battery, determines the efficiency and serves to introduce Thevenin s theorem. Introduction Introductory electricity courses generally cover the characteristics of real resistors, capacitors, inductors and some semiconductors but use an idealized model of a battery, which assumes that the terminal voltage is independent of the current supplied. Once electromotive force, Ohm s and Kirchhoff s laws have been covered, a more realistic model [1, 2] which includes the internal resistance is appropriate and explains various phenomena, such as the dimming of the headlights when starting a car, why one cannot start it with eight D cells and the temperature rise of a shortcircuited cell. Many considerations govern the use of a specific battery technology for a given application [3 5]. However, the voltage regulation, and hence the internal resistance, is an important factor that can easily be measured in the student laboratory. The presence of an internal resistance in both the power source and the storage device clearly demonstrates that electrical energy is converted to heat both during charging and during discharging, * A more detailed (unpublished) version of this article was presented at the AAPT 2000 Summer Meeting in Guelph, Ontario, and is available by . 1 Present address: 11 Willowbrook Ave, Beaconsfield, Quebec H9W 3P8, Canada. and this should be included in any realistic discussion of energy conservation strategies. Two well known student misconceptions [6 8] are that batteries are constant-current sources [7, 8] and that current is generated at the positive and sinked at the negative terminal. The introduction of an internal resistance [9] emphasizes the idea that the current flows through the battery and thus may help in counteracting these misconceptions. Thevenin s theorem [10] and the ideas of input, output and impedance matching can be introduced by experiments on dry cells from which excessive power can be taken without the risk of expensive damage. Nine-volt zinc carbon batteries are ideal, and data from open to short circuit can be taken very rapidly, with only a small effect on the battery. Theory Single battery A battery of ideal EMF E and internal resistance r, connected to a load resistor R L as in figure 1, can be considered as a voltage divider so the terminal voltage V T is V T = E Ir. (1) /01/ $ IOP Publishing Ltd P HYSICS E DUCATION 327

2 P F Hinrichsen Figure 1. The model of a real battery of EMF E and internal resistance r connected to a variable load resistance R L. The EMF E and the internal resistance r can be obtained from the intercept and slope of a plot of V T versus I, see figure 2(a). The current is I = E/(r +R L ) and hence the power P L dissipated by the load is E2 R L P L = V T I = I 2 R L = EI I 2 r = (r + R L ). 2 (2) The load power P L, as represented by the shaded rectangle in figure 2, is a quadratic function of the current, and has a maximum of P max = E 2 /4r for a load R L = r. Series combination Two batteries of EMF E 1 and E 2 and internal resistance r 1 and r 2 respectively, when connected in series are equivalent to a battery of EMF E s = E 1 + E 2 and internal resistance r s = r 1 + r 2. Then for a load resistance R L, the current is I = E 1 + E 2 (3) r 1 + r 2 + R L and the short-circuit current I ss is between the individual battery short-circuit currents I s1 and I s2. The terminal voltage V T1 for one battery is V T1 = E 1(r 2 + R L ) E 2 r 1 (4) r 1 + r 2 + R L and the power supplied by it is P 1 = V T1 I = (E 1 + E 2 )[E 1 (r 2 + R L ) E 2 r 1 ]. (r 1 + r 2 + R L ) 2 (5) From equations (4) and (5) it can be seen that, for load resistances R L less than E 2 r 1 E 1 r 2 = ( Is2 I s1 1 ) r 2 (6) Figure 2. (a) The theoretically predicted linear relation between the terminal voltage and the current. The intersection with the load line of slope R L gives V T and I. (b) The load power P L = V T I as a function of the load resistance R L = V T /I. the terminal voltage V T1 and power P 1 become negative. This is not the same as the charging of a battery, for which the current is negative, V T > E, and at least some of the electrical energy is converted into chemical energy. In this case the second battery in series has a short-circuit current I s2 that is larger than I s1, allowing the current I to be greater than I s1 and thus the Ir 1 potential drop to be greater than E 1, making V T1 negative. The first battery still supplies power E 1 I, but this is less than the I 2 r 1 power dissipated by its internal resistance r 1, making the first battery a net user of energy. This is presumably one reason why manufacturers recommend that new batteries are not combined with old ones, and that battery technologies should not be mixed. 328 P HYSICS E DUCATION

3 Computerized investigations of battery characteristics Parallel combination The parallel connection is the classical threeresistor two-emf T circuit often used to illustrate Kirchhoff s laws [1], and is equivalent to a single battery of EMF E P = E 1r 2 + E 2 r 1 (7) r 1 + r 2 and internal resistance r P = r 1r 2 r 1 + r 2. (8) Thus E P is between E 1 and E 2, and the short-circuit current I sp = E 1r 2 + E 2 r 1 r 1 r 2 (9) is larger than for either battery. For identical batteries E P = E, r P = r/2 and I sp = 2I s,as shown graphically in figure 3. The current I 1 delivered by battery 1 is I 1 = (E 1 E 2 )R L + E 1 r 2. (10) R L r 1 + R L r 2 + r 1 r 2 Thus for high load resistances R L for which E 1 < E 2R L (11) R L + r 2 i.e. when the terminal voltage for the second battery alone would be higher than E 1, the current I 1 is negative and the second battery not only provides the load power but also charges the first battery. Given two identical batteries and a load R L how does one connect them for maximum power transfer to the load? The optimum connection depends on the ratio of the load resistance R L to the internal resistance r of the batteries, and this is a simple example of impedance matching. If one is free to choose the load resistance R L then the maximum power for either connection is twice that for a single battery (see figure 3) but occurs for R L = 2r and r/2 for the series and parallel connections respectively. For a predetermined load R L one should choose the series connection if R L >rand the parallel connection if R L <r. For a given load R L the increase in power for two identical batteries is P S P 1 = ( ) 2(r + RL ) 2 (12) 2r + R L Figure 3. The characteristics of a battery of EMF E and internal resistance r = E/I s, and of two such identical batteries in series (E s = 2E and r s = 2r) and in parallel (E P = E and r P = r/2). For a load R L the grey, white plus grey and pink plus grey rectangles represent the power delivered by the single battery, the parallel and the series combinations respectively. for series connection and ( ) P P 2(r + RL ) 2 = (13) P 1 r +2R L for parallel connection; thus as a function of load resistance the ratio varies from 1 to 4. If the load R L = r the power supplied is the same for the series and parallel connections and equals 4E 2 /9r. This can be generalized for two different batteries connected in either series or parallel. The load resistance at which the power is the same for either connection is then R L = E 1r E 2r 2 1 E 1 r 1 + E 2 r 2. (14) Apparatus and analysis A Pasco 500 interface [11] (±10 V, 12 bit, threeinput ADC) plus Macintosh 7600/132 running ScienceWorkshop [11] were used for the present measurements, but any similar system is adequate at the low precision and sampling rate required. Nine-volt zinc carbon batteries are cheap and ideal for the 10 V ADC, and the internal resistance of 8 15 is significantly larger than that of the P HYSICS E DUCATION 329

4 P F Hinrichsen Figure 4. The linear terminal voltage (full symbols) and parabolic load power (open symbols) as a function of the current for (i) a9vcarbon zinc battery (A, circles: E = 9.68 V, r = 11.8, P max = 2.0 W),(ii)a9Valkaline battery (B, squares: E = 9.05 V, r = 1.34, P max = 15.3 W), and (iii) six 1.5 V alkaline batteries in series (C, diamonds: E = 9.48 V, r = 0.55, P max = 38.1 W). 1 current probe plus connections. For alkaline C and D cells (r 0.1 ) a significant fraction of the observed internal resistance can be due to the battery holder, thus very short connections and a 0.10 probe were used. The load resistance R L should be varied from about 50 times to one tenth of the internal resistance, and in order to record points roughly evenly the rate of change of resistance should decrease as the load resistance is reduced. Typically two rheostats in series, or a ten-turn 500, 2 W Helipot, were used as the variable load. Graphs of terminal voltage versus current and calculated load power (V T I) versus current were displayed during data collection and provided essential visual feedback so that sufficient points are recorded at high currents. Online linear fits to V T versus I allow the student to extract values for E and r and compare the calculated maximum power, and the load resistance at which it occurs, with the power data. Subsequently the data were transferred to Excel [12] for offline comparison with equation (2). Results Battery characteristics Representative data for a new 9 V carbon zinc battery, a 9 V alkaline battery and six 1.5 V alkaline D cells in series are shown in figure 4. These data clearly demonstrate that, even though the EMF of the carbon zinc battery is larger than that of the alkaline battery, due to the higher internal resistance its ability to provide power is much less. This point is further emphasized by the six D cells in series, which can provide even more power. Comparative measurements were made on a selection of 9 V batteries, see figure 5(a). Note that it would take about eight carbon zinc batteries in parallel to achieve the same internal resistance as an alkaline battery. These data, which were taken in a continuous manner in about 20 seconds, should not be directly compared with manufacturers data [13], for which the load current is pulsed. The interesting characteristic of a nickel cadmium rechargeable battery after being charged for the first time is shown by the full symbols in figure 5(b). The open symbols are from a second run and show only a small deviation from linearity, and essentially the same short-circuit current. Data for a series of alkaline batteries are presented in figure 5(c) and show that the power capability depends on size; however, the N and AAA cells, despite their different shapes, have identical characteristics. 330 P HYSICS E DUCATION

5 Computerized investigations of battery characteristics Figure 5. (a) The characteristics of a selection of new 9 V batteries. (b) The characteristics of a rechargeable 9 V nickel cadmium battery. (c) The terminal voltage versus current for 1.5 V alkaline cells of different sizes. P HYSICS E DUCATION 331

6 P F Hinrichsen Figure 5. (continued). Figure 6. The terminal voltage of a9vnickel cadmium battery as a function of the current while charging (full symbols) and when discharging (open symbols). Battery charging To demonstrate that the internal resistance of a battery is the same when charging and when discharging, a partly discharged 9 V nickel cadmium rechargeable battery was connected in series with seven alkaline D cells and a 500 Helipot. The charging (full symbols in figure 6) and discharge characteristics (open symbols) were then measured separately. The deduced EMFs were identical and the internal resistances agreed 332 P HYSICS E DUCATION

7 Computerized investigations of battery characteristics Figure 7. The terminal voltage versus current (a) and the load power versus load resistance (b) for two 9 V carbon zinc batteries (full symbols) and for these batteries in series (open circles) and in parallel (open squares). The full lines are fits of equation (2) to the data. P HYSICS E DUCATION 333

8 P F Hinrichsen Figure 8. (a) The terminal voltage as a function of the current for a9vcarbon zinc battery (full squares), a9v alkaline battery (full diamonds) and the series combination (open circles). (b) The power delivered as a function of the current. to within 4%, thus confirming that the model describes the behaviour of real batteries for both charging and discharging. Combination of batteries The results for two 9 V carbon zinc batteries separately, in series and in parallel are shown in figure 7(a). The power as a function of the load resistance (figure 7(b)) is in good agreement with equation (2) and demonstrates the maxima at the different values of the load resistance. The combination of a9vcarbon zinc battery in series with a9valkaline battery of significantly different internal resistance is illustrated in 334 P HYSICS E DUCATION

9 Computerized investigations of battery characteristics Figure 9. (a) The terminal voltage versus current for (i) a9vcarbon zinc battery (A, full diamonds: E = 9.50 V, r = 7.89 ),(ii)a9vnickel cadmium battery (B, full circles: E = 8.27 V, r = 2.96 ), and (iii) the parallel combination (C, open squares: E = 8.78 V, r = 2.85 ). The open diamonds and open circles are for the individual currents from the carbon zinc and nickel cadmium batteries in parallel combination. (b) The power versus load current for the parallel combination. figure 8(a). The terminal voltage of the carbon zinc battery becomes negative when the current exceeds its short-circuit current. At this point the energy absorbed by the internal resistance of the carbon zinc battery exceeds that supplied by the EMF, so the net power supplied by it is negative, P HYSICS E DUCATION 335

10 P F Hinrichsen Figure 10. (a) The terminal voltage of a9valkaline battery as a function of the current and the total charge taken from the battery. (b) The variation of the EMF and internal resistance (for 0 <I<0.5 A) as a function of the total charge. see figure 8(b). The nominally 9 V carbon zinc and nickel cadmium batteries were chosen to illustrate the effect of connecting batteries of different technologies in parallel. The individual characteristics (the full symbols in figure 9(a)) were essentially the same as those derived from the terminal voltage V TP and the individual currents I 1 and I 2 when the batteries were connected in parallel (the open symbols in figure 9(a)). For high load resistances the carbon zinc battery, due to its higher EMF, charges the nickel cadmium battery and would therefore deliver more power to the load than the parallel combination. As predicted by equation (10), when the load is reduced the nickel cadmium battery starts to supply current and power, and for load currents more than 0.5 A it supplies more than half the power, see figure 9(b). 336 P HYSICS E DUCATION

11 Battery discharge The variation of E and r with battery use depends on many factors [5]. To investigate this variation the V T versus I characteristic was repeatedly measured for a9valkaline battery and plotted against the total amount of charge that had been delivered by the battery, see figure 10(a). For a new battery the internal resistance at low current is somewhat higher that at larger currents, i.e. the terminal voltage is not quite a linear function of the current. Subsequent measurements are much more linear but become curved in the opposite sense as the battery is discharged, and for severely discharged alkaline batteries the terminal voltage versus current even develops a positive slope at low load, see figure 10(a). It can be seen that as the battery is discharged the EMF initially decreases but then remains almost constant. Surprisingly the internal resistance initially decreases (this was observed for other batteries) but then increases, see figure 10(b). Thus the capacity to deliver power decreases with use due to changes in both the EMF and the internal resistance. Conclusion These and many other investigations of battery characteristics can be made rapidly, without significant discharge of the battery, and require only elementary computer interfaces. The model of a battery as an ideal source of EMF in series with an internal resistance represents the behaviour of many real batteries surprisingly well [13] and emphasizes the fact that the current flows through the electrolyte of the battery as well as the external circuit. The linear V T versus I characteristic is not limited to just the first quadrant, but extends from the second quadrant, negative current while charging (see figure 6), to the fourth quadrant, negative terminal voltage (see figure 8). Received 14 February 2001, in final form 26 March 2001 PII: S (01) Computerized investigations of battery characteristics References [1] Tipler P L 1999 EMF and batteries, Kirchhoff s laws Physics for Scientists and Engineers 3rd edn (New York: Freeman) vol. 2, pp [2] Saslow W M 1999 Voltaic cells for physicists: Two surface pumps and an internal resistance Am. J. Phys [3] Lindon D 1995 Handbook of Batteries (New York: McGraw-Hill) [4] TuckCDS1991 Modern Battery Technology (Chichester: Ellis Harwood) [5] Energizer Batteries, [6] Cohen R, Eylon B and Ganiel U 1983 Potential difference and current in simple electric circuits: A study of students concepts Am. J. Phys [7] McDermott L C and Shaffer P S 1992 Research as a guide for curriculum development: An example from Introductory electricity. Part I: Investigation of student understanding Am. J. Phys [8] Shaffer P S and McDermott L C 1992 Research as a guide for curriculum development: An example from Introductory electricity. Part II: Design of instructional strategies Am. J. Phys [9] Greenslade T B 1996 Output resistance Phys. Teacher [10] Horowitz P and Hill W 1980 Thévenin s theorem The Art of Electronics (Cambridge: Cambridge University Press) pp 8 10 [11] Pasco Scientific, Foothills Blvd, Roseville, CA , USA product catalog [12] Microsoft Excel 98 spreadsheet (Microsoft Corporation, 1998) [13] Duracell Batteries, Peter Hinrichsen received his doctorate in nuclear physics from Manchester University in Since 1973 he has conducted research at the University of Montreal while teaching at John Abbott College. He is an avid sailor and has been an official at Olympic regattas, measuring the moment of inertia of sailing dinghies. P HYSICS E DUCATION 337