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FYSP104 / K2 RESISTANCE MEASUREMENTS This exercise familiarises you with methods that can be used to measure small, medium range and large resistances. Some limitations of these methods will also be encountered. 1 1.1 Introduction Measuring resistances Small resistances Small resistances of the order of ten ohm or less have resistances similar or even smaller than the ones of the cables and connectors needed in the measurement. The s.c. four-wire method is well suited for measurements of small resistances. For small resistances you may also use the volt-ampere method, the Wheatstone bridge or a digital general purpose meter (GPM). The effect of wires and connector is the larger the smaller the resistance under study is. Middle-range resistances Middle-range resistances (1 Ω - 10 kΩ) can be measured directly with a GPM or more accurately using the Wheatstone bridge. Large resistances For large resistances (over 10 kΩ) the effect of cables and connectors is negligible so GPM’s or special resistance meters are well suited. 1.2 The structure and functioning of the potentiometer Potentiometer is usually three-pole electronic device which resistance can be adjusted manually. The potentiometer includes a resistor which is connected to circuit from three points (poles). Two of them are solid, at the ends of the resistor and the third one is movable (fig.1). Thus the third pole divides the resistor in two pieces. In some potentiometers have fourth pole, but as in fig.1, it is actually the same as the pole 2. – 2 – The magnitude of the resistance can be adjusted with the third pole. If e.g. the third pole would be at the top of the resistor the resistance would be the entire resistance of the resistor. If the third pole would be exactly in the middle of the resistor, the resistance would be 50% of the entire resistance. Figure 1: The potentiometer 2 Methods 2.1 Volt-Ampere method When you do not need high accuracy a direct measurement is fast and convenient. Measuring the current through and voltage over the resistor gives rather good results in a wide resistance regime. Fig. 2 shows two couplings for volt-ampere method: s.c. short and long couplings. Figure 2: Short (left) and long couplings (right). – 3 – Use short coupling when Rx is small i.e. Rx << RV. Long coupling is suitable for large Rx i.e. Rx >> RA. In general these conditions are not valid so the internal resistances of the current and voltages have to be taken into account. Short coupling obeys the equation Rx = U = I − IV U . U I− RV When Rx is small, this simplifies to R x = (1) U . Correspondingly in long coupling one has to I subtract from primary result the internal resistance of the current meter: Rx = U −U A U − RA I U = − RA. = I I I (2) The last term in (2) can be neglected, if the resistance of the current meter is small compared to the resistance under study. In voltage measurements internal resistance of digital meters is typically 10 MΩ, whereas in the current measurements it depends on the scale chosen (0.01 - 100 Ω). 2.2 Two- and four wire-methods The straight measurement of the resistance is quick and easy when the accuracy doesn’t need to be very high. Then the s.c two-point method is used. The meter produces a small (constant) current, measures the voltage over the resistor and converts the result to resistance. Resistances of the cables affects to final resistance. In more sophisticated devices the four-wire method is available. There two pairs of cables are connected between the meter and the resistor. The meter produces a current (independent of the resistances of the cables and transitional resistance) and measures the voltage over the resistor. Because the voltage is measured straight from the poles of the resistor, the resistances of cables can be ignored (why?). – 4 – Figure 3: The circuit of the four-wire method 2.3 DC-bridges So called bridges are based on comparison to precision resistors; they are versatile and yield very good accuracy. For DC-bridges you need naturally a DC-power source. Bridges are frequently used in all kinds of precision measurement and surveillance instances. 2.3.1 Wheatstone bridge Wheatstone bridge shown in Fig. 4 is the most commonly used DC-bridge. When the bridge is brought to balance by adjusting one of the resistors, the reading of the galvanometer – a highly sensitive current meter - is zero. Thus I1 = I 4 and I2 = I3 R1I1 = R2 I 2 and R x I1 = R3 I 2 (3) The unknown resistance is given by equation RX = R1 R3 . R2 (4) Figure 4: Wheatstone bridge. R1 and R2 have constant resistors, R3 an adjustable resistor and Rx the resistor under study. – 5 – Equation (4) shows that Rx can be obtained by interchanging the positions of the galvanometer and the power source. However, this interchange may affect the sensitivity (why?). In most cases it is best to connect the galvanometer between the two largest resistors (why?). When you choose the coupling notice that every branch has to stand the power through it. 2 Measurements Equipment: Unknown resistor, ”small” resistor (~10 Ω), 22 Ω and 82 Ω resistors 100 Ω potentiometer (helipot) High-precision decade resistor (0 - 1000 Ω) LCR-meter Megger Digital GPM Finest or Uni-T 58 Digital GPM Good Will GDM-8055 Picoampere meter Keithley Autoranging Pico-ammeter DC-power source (e.g 4,5 V battery) Exercise 1. Familiarize yourself with the tuneable resistors used in this measurement (helipot and decade resistors, fig.5). Use GPM to find out, how to connect resistors. Especially the usage of helipot seems to be difficult (can you say why the resistance is what it is if you connect it differently?). Figure 5: Helipot and decade tuneable resistors – 6 – Exercise 2. Measure the resistance of a given, ”unknown”-resistor using the voltage-current method. The resistance is a few hundreds of ohm. Which one is more suitable, short or long coupling? Tune the voltage of the power source first to few volts (naturally, you don’t have to adjust the voltage of the battery). Exercise 3. Measure the resistance directly with the digital GPM. What could be the easiest way to estimate the systematic error caused by the cables? Exercise 4. Repeat the measurement with LCR meter (MEGGER) (fig. 6). Figure 6: LCR-meter Exercise 5. The four-wire method. Measure the resistance of “unknown” resistor with the digital GPM Good Will. Use both the two- and four-wire methods. Is there any difference between the results? Also measure the resistance of two connected cables by using the two point method. What do you see? In figure 7, the contacts of the GW meter are numbered. In two-wire method, connect cables to contacts 1 and 2. In four-wire method, you need two more cables and connect those from the resistor into contacts 3 and 4 (see also fig. 3). Figure 7: Good Will 8055 –meter Exercise 6. Measure the resistance of an “unknown” resistor Rx with the Wheatstone bridge-method. Choose resistors R1 and R2 to be 22 Ω and 82 Ω and resistor R3 tuneable precision resistor (decade resistor). A suitable voltage is about two volts (or a battery). When the bridge is in balance, check the resistance of each resistor with the GW -meter. – 7 – Then swap the positions of R1 and R2 and repeat the measurement. From both measurements, write down few (R3, IG)-values to determine the exact zero-point of the current. From the results calculate the resistance of the unknown resistor (including error limits). Justify whether you can ignore the resistances of the cables or not. You can use a GPM as a galvanometer. How do the resistors R1 and R2 affect to the measurements? Exercise 7. Finally, we investigate how much voltage and current meters cause error in short and long couplings. At this point at the latest we find out if we have chosen the “right” coupling in the exercise 2. Measure the resistance of the “small” and “large” resistors by using the short and long couplings. In short coupling, assume the resistance of the voltage meter to be infinite and in long coupling the resistance of the current meter to be zero. With “large” resistor you have to use the Keithley Pico-ammeter because the current is not very big. For same reason, you are advised to connect another 4,5 V battery in series with the previous one. With Keithley meter you need BNC adapter (fig.8). Figure 8: Keithley Pico-ammeter and BNC adapter. Note! Measuring the resistance of “small” resistor is very precise job to do. So before connecting, check the resistance of every cable. Do not use cables whose resistance is more than 0,05 Ω. Measure the resistances of the voltage and the current meters at the scales you chose. Use either two- or four-wire method (justify your choice). Take these results into account when determining the resistances of the resistors using equations (1) and (2). What do you see? Note 2! Use a GPM in mesuring the resistance of Keithley Pico-ammeter. GW and Keithley don’t want to work together. – 8 – 4 Results Write down the calculated results with estimated errors in the questionnaire form and answer to the questions presented in this manual, as well. Attach measurement notes in your report.