Measurement of Ohmic Resistances, Bridge Circuits and Internal Resistances of Voltage Sources

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1 3 Car von Ossietzky niversity Odenburg Facuty V - Institute of Physics Modue Introductory Laboratory Course Physics Part I Measurement of Ohmic esistances, Bridge Circuits and Interna esistances of Votage Sources Keywords: OHM's aw, KICHHOFF's aws (KICHHOFF's current and votage aws), interna resistances of measuring instruments, WHETSTONE bridge, bridge circuit, votage source, interna resistances of votage sources, termina votage, strain gauge. Measuring program: Measurement of resistances with different ohmmeters, determination of a resistance by means of a current / votage measurement, WHETSTONE bridge, interna resistance of a function generator, specific resistance of tap water, bridge circuit for measuring aterations to resistances. eferences: // SCHENK, W., KEME, F. (HSG.): Physikaisches Praktikum, Vieweg + Teubner Verag, Wiesbaden /2/ WLCHE, W.: Praktikum der Physik Teubner Studienbücher, Teubner-Verag, Stuttgart /3/ EICHLE, H.J., KONFELDT, H.-D., SHM, J.: Das Neue Physikaische Grundpraktikum, Springer-Verag, Berin among others Introduction In the first part of this experiment, an insight into the different methods of measuring ohmic resistances wi be given. In particuar, it wi be shown to which degree rea properties of measuring instruments affect the resuts of a measurement, and which methods yied the best resuts in individua cases. In the second part of the experiment, the characteristics of rea votage sources are investigated. The main question here is how an important characteristic of such votage sources, namey their interna resistance, can be measured. dditionay, the specific resistance of tap water is measured, and the inear reationship between changes in resistance and votage in a bridge circuit is anaysed. 2 Theory 2. Kirchhoff's Laws Knowedge of KICHHOFF's aws 2 is a prerequisite for anaysing eectric networks (circuits) (see Fig. ). KICHHOFF s first aw (current aw) reads: The sum of a currents at a branch point equas zero. For this aw the sign convention is vaid that currents towards and off a network node are marked by contrary signs. It is of no importance, whether the affuent currents are marked positivey and the effuent currents are marked negativey. ppied to the circuit in Fig., KICHHOFF s first aw at the nodes and B reads as foows: : I I I = 0 B: I + I I = 0 () KICHHOFF s second aw (votage aw) reads: In a cosed oop of a network the sum of a components of the votage equas zero. Named after GEOG SIMON OHM ( ) 2 GSTV OBET KICHHOFF ( )

2 4 I, + _ I, I, 3 3 a b Fig. : B Circuit with a direct votage source with termina votage, the resistors,..., 3 as we as two nodes and B and two oops a and b. For the appication of this aw a sign convention is essentia as we. It reads: a) direction ( counting arrow ) is assigned to each votage running from the positive to the negative termina (e.g. votage source). b) direction ( counting arrow ) is assigned to each current, which marks the direction of movement of the positive charge carriers, i.e., the current fows from the positive to the negative poe per definition. ccording to OHM s aw, the direction of the votage over a resistance corresponds to the direction of the current I, fowing through and causing the votage drop. c) For the appication of the votage aw a direction of rotation has to be fixed (i.e., cockwise or counter-cockwise). Votages with counting arrows corresponding with the direction of rotation are counted as positive, the others as negative. ppied to the circuit in Fig., the KICHHOFF s second aw in the oops a and b for a counter cockwise direction of rotation reads: a: = 0 b: = 0 (2) With KICHHOFF s two aws and the reated sign conventions a eectric networks which are empoyed in the course of the introductory aboratory course can be described. Question : - How can the formuas for the parae and series connection of ohmic resistances be derived from KICHHOFF's aws? Which are the corresponding reationships? It is not aways easy to recognize whether resistances and other components in a network are in parae or in series connections. Two rues derived from KICHHOFF's aws can be usefu in making that decision: esistances are in parae if they show the same votage drop. esistances are in series if the same current fows through them. 2.2 Methods for Measuring Ohmic esistances 2.2. Determining the esistance from the Markings Fig. 2 depicts some common retai versions of resistors which are abeed with various kinds of markings (etterings or coor codes). In the simpest case, the vaue of the resistor is printed directy on the casing. Some inscriptions commony used for this purpose are 20 for 20 Ω, 47 for 4.7 Ω, 3k3 for 3.3 kω, or 5M6 for 5.6 MΩ.

3 5 Fig. 2: Common retai versions of resistors with various types of markings. Top oad resistors (power rating of severa W), bottom resistors for ow power appications (< W). Equay simpe is reading the coour key printed on most types of resistors. This coour key on the resistor consists of chromatic rings, which are aways arranged in such a way that the first chromatic ring is coser to the one end of the resistor than the ast chromatic ring is to the other end. Tabe shows how the vaue of a resistance can be determined by means of the coour coding. Tabe : Coour key for ohmic resistances. 3-4 rings st ring 2 nd ring 3 rd ring 4 th ring 5-6 rings st ring 2 nd ring 3 rd ring 4 th ring 5 th ring 6 th ring Coour st number 2 nd number 3 rd number Mutipier / Ω Toerance / % Temp. coeff. / 0-6 ΩK - back 0 0 ± 250 brown 0 ± ± 00 red ± 2 ± 50 orange ± 5 yeow ± 25 green ± 5 *) ± 20 bue ± 0 vioet ± 5 grey *) ± white *) ± 0 *) none ± 20 siver 0-2 ± 0 god 0 - ± 5 *) where the conductivity of god and siver ename interferes Question 2: - What is the vaue of a resistor for the coour coding ed (first ring) - Vioet - Brown - God? - What is the coour coding for a resistor of (3.9 kω ± 0 %)? Determining the esistance by Means of a Current / Votage Measurement When the two ends of an idea votage source (cf. Chapter 2.3), which deivers an adjustabe termina votage, are connected with the connecting wires of a resistor, the current I fows through the resistor and OHM s aw reads: (3) = I By measuring the votage using a votmeter and the current I using an ammeter, can thus be determined. Such a measurement can be carried out using the two circuits and B according to Fig. 3.

4 6 + _ I, I, V + V _ I, I V, V I V, V I, Fig. 3: Schatung Schatung B Two possibe circuits for measuring the resistance by means of a current/votage measurement. is connected with a direct votage source with termina votage. The current is measured with the ammeter, the votage is measured with the votmeter V. If idea measuring instruments were avaiabe, i.e., ammeter with a negigibe interna resistance and a votmeter with an infinitey high interna resistance, both circuits woud yied the same resut. However, an ammeter has an interna resistance of > 0 and a votmeter has an interna resistance of V <. Consequenty, a vaue for the resistance with an error is determined with each circuit. We wi now determine the reative error / for both circuits. Let I be the current through the ammeter, I the current through the resistor, and I V the current through the votmeter. is the votage drop across the ammeter, is the votage drop across and V is the votage drop across the votmeter. ccording to KICHHOFF's current aw we then obtain for the circuit : (4) I I IV = 0 and thus (5) I = I + IV sing this circuit, the vaue determined for the resistor M is given by (6) M V V = = I I + I V The deviation from the actua vaue (7) = I is hence given by: (8) = M Inserting Eq. (6) into Eq. (8) and using circuit, we obtain after some rearrangement and taking into consideration V = (votage aw) for the reative error: (9) = + V ccording to KICHHOFF's votage aw we obtain for circuit B: (0) V = 0 and thus

5 7 () V = + Considering I = I, the measured resistance M is: V V + (2) M = = = = + I I I For the actua resistance we refer again to Eq. (7). Inserting Eq. (2) into Eq. (8), we obtain for the reative error, if using circuit B: (3) = Question 3: - Sketch the graph of the reative error as a function of the resistance for circuits and B in a diagram. Is one of the circuits better than the other one, in principe? If not: in which case woud the individua circuits be preferred? - The two circuits are caed current-correct and votage-correct circuits, respectivey. Which name beongs to which circuit? Why? Measurement of esistance sing an Ohmmeter Instead of determining the resistance from a current/votage measurement, it can aso be directy measured using an anaogous ohmmeter (pointer instrument). In the simpest case such an ohmmeter consists of a votage source (battery), to which the resistance is connected, a variabe interna resistance i and an ammeter, by means of which the current through the resistance is determined. This current causes a neede defection, which is then read off an appropriate OHM scae. This scae is inversey proportiona to the current scae due to the reationship = /I. Since the votage source does not aways provide the same votage (ageing of battery), the ohmmeter has to be caibrated by adjusting i before beginning the measurement. For this purpose the contacts are shorted out and the neede defection is adjusted to 0 Ω using a reguating screw. Modern digita ohmmeters are structured differenty. Generay, they are integrated into mutimeters. Such instruments contain compex eectronic circuits with integrated microprocessors for measuring the required parameters (current, votage, resistance, frequency among others) and LCD eements dispaying the measured vaues Measurement of esistance sing the WHETSTONE Bridge By means of a WHETSTONE bridge 3 the vaue of a resistance can be determined without errors being caused by inadequate (rea) measuring instruments for the current, votage or resistance; however, a gauged comparative resistance is required for this purpose. We consider a WHETSTONE bridge-circuit ike the one in Fig. 4. homogenous resistance wire, usuay of constantan 4 with the specific resistance ρ ([ρ] = Ωm), the tota ength = + 2, and the cross-sectiona area is connected to the resistor to be measured and a gauged comparative resistor 3 as shown in the figure. The resistances of the two wires are: (4) 2 = ρ and 2 = ρ The votage, taken from a direct-votage source, is connected to this resistance network. The current fowing between the point P and the shiftabe tapping point Q aong the resistance wire is measured using an ammeter. 3 CHLES WHETSTONE ( ) 4 Constantan is an aoy consisting of approx. 60 % copper and approx. 40 % nicke, the specific resistance of which is neary constant across a wide temperature range (ρ Ωm at 20 C).

6 8 3 P + _ Q 2 2 Fig. 4: Wheatstone bridge with constantan resistance wire (yeow). (green) is the resistor to be measured, 3 is the resistor used for comparison. There is a position of the tapping point Q, where no votage is found between P and Q and therefore no current fows. In that case the votages over 3 and as we as at and 2 are equa. Such a WHETSTONE bridge is caed adjusted and we obtain: 3 (5) = = and thus 2 (6) = Thus, in the case of an adjusted WHETSTONE bridge, the resistance can be determined by measuring the engths and 2 and knowing the gauged resistance 3 from Eq. (6); insufficiencies of eectric instruments are then of no importance. This is the advantage of this measuring method, a so-caed compensation method. Question 4: - In Fig. 4, enter a the points where the circuit branches in the unbaanced WHETSTONE bridge as we as the currents that fow, incuding their signs. - In Fig. 4, draw a of the oops of the unbaanced WHETSTONE bridge as we as the votages in these oops, incuding their signs Bridge Circuit for Measuring terations to esistance Bridge circuits are, among other appications, used to convert sma aterations to resistance into proportiona votages. This is a standard method in many areas of sensor-measuring-technoogy. We consider, for exampe a bridge circuit with strain gauges (SG). SGs may be used for constructing force sensors. We wi get to know such a force sensor in the experiment Sensors.... Its theoretica foundation, however, sha aready be described here. The principe of a strain gauge is the eongation of a thin eectrica conductor of ength by means of an externa force F 5, which simutaneousy causes a decrease of its cross-sectiona area (Fig. 5). This aters its resistance, which is, according to Eq. (4), given by: (7) = ρ For a conductor with a circuar cross-section of diameter d we get: 4 (8) = ρ 2 π d 5 positive eongation is a stretching, a negative one a compression.

7 9 F Fig. 5: Diagram of a strain gauge (SG) on meta foi basis. thin meta foi (yeow) is pated on the carrier foi (gray) in meander form in order to increase the effective ength of the conductor whie keeping the size of the SG sma. The carrier foi is pasted on the work piece to be investigated and foows its deformations upon appication of a force F. The eongation changes the ength by Δ, the diameter d by Δd, and, depending on the materia, possiby the specific resistance ρ by Δρ. The resuting change in the ohmic resistance is given by the tota differentia Δ: (9) = ρ + + d = ρ + ρ 2 ρ d ρ d π d d d The reative change in resistance is thus: ρ d (20) = + 2 ρ d The reative change in ength / is defined as the eongation ε. (2) ε = The POISSON-number 6 µ is defined as the negative quotient of the reative change of the cross section d/d and the reative change of ength /, thus: d (22) µ = d : = d d ε Factoring out the vaue of ε = Δ/ in Eq. (20) and substituting Eq. (2) and Eq. (22) into Eq. (20) gives: ρ ρ (23) = µ ε : = k ε ε The expression within the brackets is the so caed k-factor of a strain gauge and depends on the materia of the strain gauge, exampes are k 2 for constantan and k 4 for patinum 7. The reative change in resistance by eongation increases with arger vaues of k. 6 SIMÉON DENIS POISSON (78 840) 7 For monocrystaine siicone (Si), k 00. Si-based SGs are, for exampe, used in pressure sensors which we wi get to know in the forthcoming experiment Sensors.

8 0 With the aid of a bridge circuit, this ateration to resistance is converted into a votage. For a quantitative description of the bridge circuit, we ook at the circuit shown in Fig. 6, which is set up anaogous to the WHETSTONE bridge represented in Fig V = - Fig. 6: Bridge circuit for measuring sma aterations to resistance of (here SG). The votage in the bridge diagona is measured with a votmeter V. In case the interna resistance of the votmeter V extends to infinity, the foowing reationships are vaid: (24) = = (25) + 2 = = 0 By combining Eqs. (24) and (25) we obtain: (26) 3 = 0 3 = The votage in the bridge diagona is: (27) 3 = 3 = We now consider the specia case of starting out with equa resistances,..., 4, one of which ( ) is subsequenty atered by the sma amount. In case of a bridge circuit with a strain gauge, woud be the resistance of the strain gauge and the change in resistance caused by mechanica deformation, hence: (28) = + = = : = With this, it foows that the votage from Eq. (27) is given by: (29) = 0 = 0 = Eq. (29) shows that the reationship between and is non-inear. If, however, <<, it hods: (30) 2 + 2

9 and thus: (3) 0 4 Near the point of baance ( << ), the ateration to resistance is thus reated to a votage in an approximatey inear manner, the ampitude of which can be infuenced by the operating votage (suppy votage) of the bridge, 0. In the set-up described above, one of the four resistors is repaced by a SG. ccordingy, this setup is commony caed a quarter bridge. In practice, one often uses a bridge circuit where two resistors are repaced by SGs which move in opposite directions of each other for a given deformation (Fig. 7). This arrangement is caed a haf bridge. One exampe is using two SGs to measure forces with a bending rod (Fig. 8), which we wi investigate in more detai in the experiment Sensors.... The two SGs are paced in the bridge circuit, so that the upper one that is being eongated repaces and the ower one being compressed repaces 2. It foows that: (32) = + 2 = 3 = 4 = By inserting Eq. (32) in Eq. (27), it foows for the haf bridge: 0 (33) = 2 This equation makes the advantage of a haf bridge compared to a quarter bridge cear: First, the reation between and is inear. Second, for the same change in resistance, the haf bridge produces an output votage of twice the magnitude. Thus, the sensitivity of the haf bridge is twice as high. In a fu bridge, a four resistors are repaced by SGs, which change pairwise ( / 4 and 2/ 3) in opposite directions. In this case we find for the votage : (34) 0 It is cear, that the sensitivity increases again by a factor of two = - V Fig. 7: Bridge circuit with two SGs (haf bridge). F Fig. 8: DMS Bending rod (green) with two strain gauges (yeow, German abbreviation is DMS). The rod (green) is fixed by a bock (gray) on the eft. The force F deforms the rod, so that the upper SG is eongated and the ower one is compressed. mechanica barrier (red) serves to protect the apparatus from overstraining.

10 2 2.3 Properties of ea Votage Sources 2.3. Interna esistance of ea Votage Sources n idea votage source provides a constant termina votage, which is equa to the constant source votage 0, independenty of the eectrica oad (the current it provides) at its connecting termina. Such idea votage sources cannot be reaized technicay. On the contrary, we are deaing with rea votage sources such as batteries, power units or function generators, the termina votage of which decreases with increasing oad. In order to describe this property of rea votage sources we use a mode in which the rea votage source is exchanged for an idea votage source G and an interna resistance i in series; Fig. 9 shows the corresponding equivaent circuit. When such a votage source is connected with a oad with an externa oad resistance according to Fig. 0, the oad current I fows through as we as through i. This current causes a votage drop I i at i by which the termina votage is reduced compared to the source votage 0. Thus we obtain: (35) = 0 I i If the source votage 0 is to be measured using an idea votmeter V in a circuit according to Fig. 0, then the oad current I has to be towards zero. This is achieved by a arge oad resistance. Exchanging the current I in Eq.(35) for / (according to OHM's aw) we obtain for the reationship between and : (36) = 0 + i i = 0 G = 0 i G I V Fig. 9: Equivaent circuit of area votage source without a oad. Fig. 0: Equivaent circuit of a rea votage source with oad resistance. From this equation we derive particuary for the case = i that the termina votage decreases to haf of the source votage. This aows us to determine the interna resistance of a rea votage source. Question 5: - Sketch the graph of the termina votage as a function of the oad resistance Matching a Device to a ea Votage Source Power Matching Whie connecting an eectrica device to a votage source it is often desirabe that the interna resistance of the device be dimensioned such that the maxima power can be taken from the votage source (power matching; appied e.g. in the transmission of high-frequency signas 8 ). The interna resistance of the device is the oad resistance, which is the votage source oad. The power P suppied to that resistance is given by: 2 (37) P = I = 8 Power matching in communication engineering at the same time prevents spurious signa refections, which wi be examined more cosey in the experiment Signa transfer (summer term).

11 3 Inserting Eq. (36) into Eq. (37) yieds: (38) P = 2 0 ( + ) i 2 Maxima power consumption is achieved when the interna resistance of the device equas the interna resistance of the votage source, hence if we have: (39) = i Question 6: - Sketch the graph of P as a function of. How do we get from Eq. (38) to Eq. (39)? What is the maximum power that can be drawn from the votage source? Votage Matching The object of votage matching, appied in high-current technoogy and in sound engineering, is to draw the highest possibe votage from the votage source. ccording to Eq. (36) the necessary condition for this is: (40) >> i Current Matching The object of current matching is to draw the strongest possibe current I from the votage source. This is, for exampe, used for charging accumuators. ccording to the OHM s aw we obtain: (4) I i + = 0 so that the condition for the strongest possibe current reads: (42) << i In this case the current is neary independent of the oad resistance. 3 Experimenta Procedure Equipment: Votage suppy (PHYWE (0-5 / 0-30) V), function generator (TOELLNE 740), severa digita mutimeters, digita oscioscope TEKTONIX TDS 02 / 02B / 202C / TBS 02B, resistance decade, side rheostat ( ges,5 Ω), unknown resistance in hoder, box for bridge circuit, pair of copper pates in hoder, water basin on verticay adjustabe hoder, meta measuring tape, caiper gauge, paper wipes. ttention: When connecting resistances with votage sources it has to be considered that the maximay aowabe dissipation power P max of the resistor must not be exceeded (P = I = 2 / < P max). Detais about the oad capacity P max of resistors are either found on the avaiabe components (e.g. resistor decade) or can be obtained from the technica assistant. Take care when operating the power suppy unit that no unintentiona current imitation is adjusted. Mutimeters of the type FLKE 2 provide ony a imited resoution for current measurements. Therefore, they are used ony as ohmmeters or votmeters in this experiment, not as ammeters. In mutimeters of the type MONCO DMT-300, fuses are bown easiy in case of an operating error. ppy specia caution to operating them!

12 4 3. Hints to the Measuring Instruments The used measuring instruments offer the possibiity to switch the measuring range manuay and party aso automaticay, which serves to indicate the measured vaue on the scae or numerica dispay of the measuring instrument with the highest possibe accuracy. For exampe, using a digita votmeter a votage of.78 V wi be indicated as.78 V in the measuring range 2 V, but wi be indicated as 2 V in the measuring range 200 V. When switching the measuring range of an ammeter a precision resistor ( shunt ) is added in parae to the interna resistance of the instrument. This resistor is rated such that the current fowing through the ammeter remains just about the same for a measuring ranges. naogousy, when switching the measuring range of a votmeter, a precision resistor is added in series to the interna resistance of the instrument, which is rated such that the votage drop is just about the same for a measuring ranges of the votmeter. Data on the interna resistances for measuring currents ( ) and for measuring votages ( V), which are dependent on the measuring range, are avaiabe for some of the measuring instruments used in the introductory aboratory course. Instead of an interna resistance, a votage drop is often stated (e.g. 20 mv, 200 mv etc.). In this case = / I max, I max representing the maximum current in the adjusted measuring range. For other measuring instruments, there are no data avaiabe on the interna resistances and/or V. In those cases it can be assumed that V is so arge (e.g. 0 MΩ) and is so sma (e.g. 0,5 Ω) that their infuence on the measurement resut is negigibe. Specifications of the tota measurement error of a measuring instrument and the precision of a measured vaue, respectivey, are found on the instruments or in the instrument manuas. These vaues generay consist of two parts. Their first - most significant - part is stated in percent of the measured vaue. Their second part can be stated in percent of the measuring range or in units of the ast decima of the measured vaue. The foowing exampes serve for expanation:.) direct votage of V is measured with the mutimeter FLKE 2 in the measuring range V. ccording to the manua the precision for this votage range is: ± (0.7 % of the measured vaue ) V. For the mentioned exampe, the precision thus is ± ( ) V = ± 0.09 V (rounded to two significant digits). This vaue is aso the maximum error for the measured vaue. 2.) direct votage of mv is measured with the mutimeter GILENT 272 in the measuring range mv. ccording to the manua the precision for this votage range is: ± (0.05 % + 5). The percent vaue refers to the measured vaue, the 5 refers to the ast shown digit of the measured vaue (here the 4 in 0.04 mv). The maximum error therefore is: ± ( ) mv = ± 0.20 mv (rounded to two significant digits). 3.2 Measurement of esistances The vaue of an unknown resistor (in the order of magnitude of kω) incuding the maximum error is to be determined by appying some of the methods described in Chapter 2.2. The foowing steps are to be taken consecutivey: a) Measurement using different ohmmeters: The vaue of the resistance is to be measured with at east five ohmmeters. Party aso ohmmeters of the same type may be used. number j is assigned to every ohmmeter. Prior to the measurements, the measuring instruments have to be adjusted to the measuring range that enabes the most precise measurement to be made. The maximum error j is given for each measured vaue j. The j are presented in a graph over j incuding error bars. b) Measurement of current/votage: s an exampe, circuit is set up according to Fig. 3. votage suppy serves as the votage source. Its interna resistance can be negected for this measurement. For at east ten different votages on the votage suppy the corresponding current is measured using an ammeter and the votage using a votmeter. Prior to these measurements, the range covering the vaues

13 5 to be measured has to be considered and the measuring ranges have to be adjusted accordingy. For each pair of vaues (, I) a resistance = /I is determined. Subsequenty, the mean and its standard deviation σ are cacuated from these data. Subsequenty, the measured votage vaues are potted against the measured current vaues I in a diagram and the maximum errors of and I are entered in the form of error bars. The parameters of the regression ine through the measured points are cacuated and the regression ine is drawn into the diagram 9. The sope of the regression ine (± σ ) is a good estimate for the unknown resistance vaue. This estimate is compared with the previousy found mean of (± σ ) and it is checked, whether both methods yied comparabe resuts. c) WHETSTONE bridge: WHETSTONE bridge is set up according to Fig. 4. gain a votage suppy serves as the votage source, and a resistance from a resistor decade as caibration resistance 3. This resistance is chosen to be about the same as the resistance to be measured:. In that case 2 is vaid for an adjusted WHETSTONE bridge and the error becomes minima for determining. The error of the vaue of cacuated using Eq. (6) is given by the maximum error. fter having determined the resistance by appying the different methods, a measured resuts from Chap. 3.2 are to be presented in a graph anaogous to Chap. 3.2 a) and compared. 3.3 Measurement of Interna esistance of a Function Generator With a circuit according to Fig. 0, the interna resistance (aso caed output resistance) of a function generator (FG) is to be determined. The equivaent circuit of the function generator consists of an idea votage source G and the interna resistance i (order of magnitude 50 Ω) in series. sinusoida votage is adjusted at the function generator (ampitude FG 4 V, frequency approx. khz) and initiay a oad resistance = 00 kω and thus >> i (resistance decade) is connected. The votage ampitude over is measured using an oscioscope. Its interna resistance of about MΩ can be negected. 0 FG is vaid for = 00 kω with sufficient precision. Subsequenty, the oad resistance is reduced by correspondingy switching the resistor decade to vaues between kω and 20 Ω. For each vaue of, the votage ampitude is measured and subsequenty is potted over. By means of graphic interpoation 0 of the curve the vaue for is extrapoated at which has been decreased to haf of 0. This resistance corresponds to the required interna resistance i of the function generator (see Chapter 2.3.). Hint: The maximum current at the owest possibe resistance (20 Ω) is I max = 4 V / 20 Ω = 200 m. The maximum momentary power at the resistance thus is P = I = 0.8 W and hence is beow the oad imit of the resistance decade of W. 3.4 Specific esistance of Tap Water Let us examine the set-up shown in Fig.. Two rectanguar copper pates of width b are mounted in parae at a distance. They are dipped into a water basin containing norma tap water. By ifting the water basin the pates can be punged into the water to a variabe depth d. ρ w being the specific resistance of the water, the ohmic resistance w of the water between the pates is given by (cf. Eq. (4)): (43) w = ρw b d 9 The cacuation of the parameters of the regression ine and its grafic representation are done using Origin. Hints are given in the accompanying seminar. 0 Graphic interpoation means: regression curve is drawn by hand through the measurement. Then the ine = 0/2 is drawn and its intersection point with the regression curve is determined. The vaue of the intersection point is read on the abscissa. Its maximum error foows from the reading accuracy of.

14 6 ~ d Fig. : Set-up for measuring the specific resistance of tap water (amperemeter and votmeter not drawn). By measuring the current I (ammeter) at an appied votage (votmeter) between the pates w (circuit, cf. Fig. 3) is determined for a number of vaues of the immersion depth d, within the range between 50 mm and 20 mm. For the singe vaues of, I and w errors must not be specified. Then, w is potted against /d. regression ine is drawn in the diagram and the specific resistance ρ w of the tap water incuding the maximum error is cacuated from its sope (measure and b!). emarks: In order to avoid poarization effects in the water, no direct votage but a sinusoida aternating votage without DC offset (turn off DC-Offset on the FG) is used, which can be obtained from a function generator ( eff 2 V at d 50 mm, frequency approx. 50 Hz). In spite of this, the water as an ionic conductor does not behave in the manner that we know from metaic conductors. For exampe, its resistance decreases with temperature, whie for metaic conductors, it increases. Therefore, we must assume that the measurement yieds ony a reference vaue for ρ W. In addition, ρ w deviates consideraby depending on the quaity of the tap water, so we need not compare the measured vaue with a iterature vaue. Measuring with an aternating votage requires the mutimeters to be switched into C mode! 3.5 Bridge Circuit for Measuring terations to esistance bridge circuit is set up according to Fig. 6 (,...,4 00 Ω, power suppy votage 0 5 V). 2 4 are sodered in a box. is adjusted with a resistance decade. The votage is measured in the bridge diagona for approximatey ten aterations to resistance of the resistor within the range between ± Ω and ± 0 Ω, hence for in the interva (90 0) Ω. is potted over and the inearity of the reationship is verified according to Eq. (3). The specific resistance of tap water at 20 C is in the order of magnitude of approx. (0 20) Ωm. For comparison: the specific resistance of copper at 20 C is about Ωm (Pease search for the corresponding reference from the iterature).