( ) Laboratory 6 Impedance measurements rev 6e

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1 Laboratory 6 meance measurements rev 6e Goal: meance measurements using the irect metho, with the LCmeter, or the inirect metho, using the DC brige. Fig. b. The series (left) an the arallel (right) circuit moels. Theory summary n the sinusoial regime, the imeance an the amittance are efine as an Y, where an reresent the hasors of the voltage an the electric current intensity, from Fig. a. Generally, this units are comle values, their algebraic form being: + jx, Y G + jb the series resistance; X the series reactance (with X >0 for inuctive imeances, an X <0 for caacitive imeances); G arallel conuctance; B arallel suscetance (with B >0 for caacitive amittances, an B <0 for inuctive amittances). The relationshi between the amittance an imeance units are: G B X G B + G + B The moel of a lossy reactance Let us consier a lossy reactance, with the quality factor Q, at frequency f. There eist two moels for this circuit: series an arallel, shown in Fig. b. X can be the reactance of an inuctor, or a caacitor: X L ωl Fig. a X C ωc For the two moels the quality factors Q s an Q are efine: X X s ωls Qs ω C s s s s B Q ω C G X ωl Being efine for the same hysical comonent, the two quality factors must be equal: Q s Q Q The tangent of the loss angle, D is: D Q The relationshis between the elements of the two moels, at a given frequency f, X X s + X s ( + D ) are: Q s ( ) + Q The equivalence relationshi between reactances can be written, in eenence with the nature of the reactance, i.e., caacitive or inuctive, as follows: L Ls ( + / Q ) () C C + / Q s ( ) The rincile of quariolar measurement (4-terminal or 4-wires measurements) When small value imeances are measure, or the testing leas are long (measurement at a istance), the imeance of the leas an the contact resistances is negligible no more, being comarable with the imeance. The measuring rincile uses two terminals at each en of the imeance. air of terminals is use to inject the current through the unknown imeance, an the other is use to measure the voltage ro on. Due to the 4 terminals, the connetion is calle quariolar. The two airs of terminals are to be connecte in the roimity of the boy of the imeance.

2 3 4 z 3 z z Fig a: The quariolar moel. Fig. b: The biolar moel. The quariolar connection is given in Fig. a; two searate airs of wires are connecte: - from source to, the (arasitic) imeances of the connection wires are z, z 3. - from the voltmeter to, the imeances of the wires are z,. One can observe that z an are in series with the voltmeter which has a very high inut imeance such that they are negligible. z an z 3 are connecte in series with the current source, with high internal imeance, such that they become negligible too. Therefore, this scheme enables the minimization of the effect of the 4 small value unwante imeances, by utting them in series with high value imeances alreay in the circuit. Fig. b shows the biolar connection. n this case, the measurement instrument contains the source an the voltmeter. is connecte by two wires only, z an z 3. Thus, the aths of the current an the voltage cannot be searate anymore, the measure imeance being the sum of the measure one an those of the wire: m + z + z3 causing a systematic error: s z + z3 ε For eamle, when measuring a resistance, by using some cables with resistance r, one obtains a systematic error, when the biolar (two wires only) configuration is use: s r ε DC briges. The Wheatstone brige. n Fig. 3 the schematic of a Wheatstone brige can be seen, where r g is the internal resistance of the source, an V is the internal resistance of the null inicator (or the voltmeter). Consiering r g negligible an V very high, the unbalance voltage is given by: z z z z 3 3 Eg z3 r g E g [ ] Fig. 3 [ 3], V 3 [ 4] The brige is balance when 0. t the equilibrium, between the resistances of the brige the relationshi (4) is vali: 3 4 (4) The ratio of the brige,, is efine as the ratio of any two ajacent resistances which are connecte at the same terminal of the voltmeter, when the brige is balance. 4 3 or (5) 3 4 The ajustment factor σ (also calle the imbalance of the brige) is efine as: 4 40 σ 40 where 40 is the value of the resistance 4 when the brige is balance (resistances,, 3 are fie values). For σ 0 the voltage can be aroimate by using the eression: S E σ The sensitivity of the brige, S, is: Eg S ( ) [ ] 0,5 0,0 0,0 α - 4 From (6), one can euce that S()S(/). This is the reason of choosing ratio to be either / or /, if the resistances are ajacent an connecte to the same terminal of the voltmeter. The eenence of the sensitivity S of is eicte in Fig. 4. t can be observe that for α, S(α)S(α - ). g S 5 9 α Fig. 4 (6)

3 5 6 Measurements. Measuring resistances with the LC-meter The LC-meter is a evice that enables automatic measurement of two arameters of an imeance, at the choice of the user (soft key MODE). nitialize the LC-meter using MEN SETP ECLL CLBTON YES() EXT. Measure three ifferent value resistances using the LC-meter: SPEED MED, DSPLY VLE, MODE /Q, CCT SEES. These settings can be change by ushing the soft keys situate at the right of the islay. The efault working frequency is khz (verify on the islay; if it is not, ush FEQ (same as - ), write the esire value an ush ENTE). Determine the absolute errors, Δ, an the relative errors, ε, of the value measure by the LC-meter,, when comare to the nominal value, nom (the one you fin on the measure resistance, elicitly written, or color coe). Δ - nom, ε Δ/ nom. Measuring the resistance of a wire se an comare the biolar (T) an the quariolar (4T) configurations to measure a very small value resistance (the resistance of a wire, which is below Ohm). a) connect any available wire to the testing leas of the LC-meter. Observe that the aater connecte at the LC-meter uses 4 terminals, the measurement being quariolar. Each of the two crocoile clis connects terminals of the LC-meter at one terminal of the unknown imeance, as seen in Fig. a. Write own the value for the resistance of the wire. b) Measure the same wire using the ohmmeter (multimeter, key Ω), which only has terminals so the measurements are biolar. Write own the inicate value, biolar. Comute the relative error of the biolar measurement when comare to the quariolar one. Which is the reason for such a great ifference between the values measure at oints a an b? c) Connect the two alligators of the ohmeter, irectly to each other (without the wire between them). Write own the inicate value, connection wires. Note this is the systematic absolute error ( ) when measuring the wire. ) Determine the value of the resistance of the measure wire by alying a correction of the systematic error, wire, as follows: subtract the value of the resistance of the alligator cables from c from the value measure at oint b. Comute the relative error of this value when comare to the value etermine at oint a. This value is smaller, greater or equal to the relative error from b? Why? Which is the reason for which at oint a you in t have to etermine the resistance of the connection wires? 3. Measuring caacitors an inuctors a) measuring caacitors Measure two caacitors (of ifferent tyes an values): a stirofle (olistirene) caacitor between F a multilayer ceramic caacitor between nf... 00nF Fig. 5 ceramic caacitors stirofle se MODE C/D, series moel (CCT SEES) an write own the values C s an D. Select the arallel circuit (CCT PLL) an write own the value C. The value of the D factor remains the same. Comute Q D Which caacitor tye has the higher Q? Comare the values C s an C. Justify the results of the comarison. Determine the arasitic resistance of the caacitors: set MODE C/ an write own the arasitic resistance for the series moel ( s ). Base on the relationshi between s an, an comute Q, etermine the arallel resistance. Which of s an is greater? b) measuring inuctances Measure the inuctance eistent at the working station. Set MODE L/Q an measure the series moel (L s an Q), (CCT SEES), an the arallel moel (L an Q), (CCT PLL). Comute the quality factor Q calc from the relationshi () between L s an L. Measure the value of the resistance for the series moel s. What is the hysical cause of S? Comare the Q factor for inuctances an for caacitors. 4. Measuring a C grou a) Measure a resistor aroun 0 kω on moe C/, an the smallest caacitor available at the table (< 500F) on C/D moe (series: C S, D) se them to buil a series C grou on the solerless breaboar. Connect the crocoile clis of the LC-meter irectly to the en terminals of the series grou (on t use aitional wires). Follow the stes at oint 3, to measure the grou. The working frequencies are khz an 00KHz. Determine the elements of the series moel (C s, D) an of the

4 7 8 arallel one (C, D). Determine Q. Comute the quality factor Q calc of the D grou using the relationshi between the series an the arallel moel (). Determine the resistance of the grou, using C/ moe, series ( S) an arallel ( P ). lso comute X C /ωc S at the frequencies. b) reeat the measurements for a arallel grou built on the breaboar using the same,c values. c) elain the results: what haens to the Q of the grou comare of the Q of the caacitor? n which case (S/P) the change is greater, an why? we measure a real series or arallel grou, hysically built from comonents in series or arallel, using the LC-meter set to the equivalent (mathematical) S/P moel. n which case the values we measure for C are closer to the value of C measure ineenently (without )? at the frequencies, in which case the measure C values iffer to a greater etent from the ineenent C value? Note that greatest errors are not at the same frequencies (large or small) in the two cases (S/P). Elain why, taking into account the value of X C. 5. Measuring resistances with the DC brige a) balancing the brige On the breaboar, buil a DC brige. The nominal values for resistances,, 3 are [..5kΩ], 0kΩ an 0kΩ. 4 is a variable resistor (also calle trimmer or semi-ajustable otentiometer). Some eamles of trimmers, an their equivalent schematic, are given in Fig. 7. The resistances measure between the three terminals,,3 of a trimer follow the rule: ct. where the value 3 is the nominal value of the trimmer (a fie value, ineenent of the ajustment). The central terminal is calle a cursor an its osition eens on the ajustment screw. Thus, for a KΩ trimer, eening on the osition of the cursor, one can measure, for eamle: 00Ω, 3900Ω, 3 KΩ Fig. 7 trimmer observing the rule ct. Observe that, if either the terminals - or -3 are connecte in the circuit, eening on the ajustment, one can measure a resistance between 0 an ma value ( 3 ). Do not connect terminals an 3, because trying to ajust the otentiometer is useless, in this case. Observe that the remaining terminal can be connecte at the cursor, as in Fig. 3, or it can be let unconnecte, with the same result- either way, from an equivalent oint of view the trimer must have two terminals in the circuit of the brige. Fig. 8 gives one otion for the circuit from Fig. 3 on the solerless breaboar. The wires from the left art of the schematic are use to connect the DC suly, set at 3V. B E 3V [] [3] [4] V 3 4 [] 3 Fig. 8: Eamle Wheatstone brige on the rotoboar - Measure the 3 resistances (before connecting them in the circuit). Draw the schematic of the brige, utting on the scheme the chosen resistances an their measure values. Write own the ens of the two iagonals of the brige (- an 3-4). f you on t write the measure value on each resistance (observing their ositioning on the breaboar), you will not be able to ientify the two nominalvalue 0k resistances, once you ve isconnecte them from the LC-meter! - Set the DC suly at E3V (from the switch). Measure with the DC voltmeter, the eact value of the suly voltage, E. emark: for this alication, the olarity of the suly voltage is not relevant. - Connect the suly in a iagonal of the brige an the voltmeter in the other iagonal. Balance the brige, ajusting the otentiometer until the inication of the voltmeter is 0 (or the smallest value, in absolute value, that can be ractically obtaine). emark: By ajusting the trimer at the etreme values, you shoul obtain both ositive an negative voltages, an 0V when the brige is balance. f, no matter how much you ajust the

5 9 0 trimer, the sign of the voltage oesn t change, it means you in t wire correctly the brige, an it cannot be balance. -Measure the value of the trimmer for which the balance was obtaine; o not remove the trimmer, however make sure it is electrically isconnecte from the circuit (for instance, temorarily remove 3 an isconnect the ower suly). f you on t searate the trimmer from the rest of the circuit, the measurement will be wrong, since you can see that the series grou is in arallel to 4. se two wires going from the holes in the rotoboar corresoning to the trimmer, to the ohmmeter. Note the terminals of the trimmer which were use ( out of 3: for instance on fig. 8, terminals an were use). Measure the balance value 40 measure - Comare the value 40 measure for the otentiometer (when the brige is balance) with the value for balance 40 calc comute with the relation for balance briges (4), using the measure values for resistances,,3. Elain the reason for the ifference between the values, while the two, 40 measure an 40 calc are base on the values of the resistances measure with the same ohmmeter, so the ohmmeter error is not relevant. Which are the sources of this error? emark: the voltmeter has a floating inut none of its inut lugs is connecte to the case; the DC suly has the outut floating. f a non-floating suly an a non-floating voltmeter were use (for eamle, the c.a. millivoltmeter with BNC lug an the c.a. generator with BNC outut), the grouns of the two aaratus woul be common: the groun woul aear between oints an 4 (or,4) from Fig. 3, utting in short-circuit the resistor, or 3. b) etermining the configuration for the maimum sensitivity The relation (4) is the same, no matter the iagonals in which the voltmeter an the DC suly are connecte. Determine the eerimental iagonal that ensures maimum sensitivity (6). For the configuration alreay wire on the breaboar, reconnect the otentiometer an slightly vary it until the voltmeter inicates 0mV. fter that, switch the voltmeter an the DC suly (by connecting the voltmeter on iagonal 3-4 an the suly in the iagonal -) an, without any ajustment of the value of the trimer, rea the inication of the voltmeter, which will be ifferent from smaller or greater. Which iagonal is more sensitive? Justify! Comute the ratio of the brige (relation (5)) an its sensitivity S (relation (6)) for the two situations, -, S -, 3-4, S 3-4, base on the values measure for the 4 resistances. For which of the two cases the comute sensitivity is greater? For what value of, S is maimum? Comare with the eerimental euctions.. For an inuctor L 400mH an Q50 are measure, at the frequency fkhz. Determine the resistance an the value of the inuctor for the series moel, L s.. Comute the quality factor for a C series grou, with C s0nf an s50ω, at the frequency khz. 3. Comute the quality factor for a C arallel grou, with C s0nf an MΩ, at the frequency khz. 4. For an inuctor L s 0mH an Q0 are measure, at the frequency fkhz. Determine the resistance s an the value of the inuctor for the arallel moel, L. 5. For a caacitor C s 00nF an Q000 are measure, at the frequency f0khz. Determine s an the tangent of the loss angle, D tgδ. 6. resistor is measure using the biolar connection (two terminals only). The value of the resistance is 50Ω. The resistance of the cables is 0,5Ω. Determine the systematic error when measuring the resistor. 7. For the brige in Fig. 0 comute the resistance when the brige is balance an the sensitivity S. 8k 0 k Fig k k k Fig k 8. For an inuctive imeance, L 0mH an L s 00mH are measure. Determine the quality factor of the imeance. 9. Show that S has the same value, no matter the efinition for the ratio of the brige or. 0. For a Wheatstone brige, the imbalance voltage has the values mv for 4., 0kΩ an mv for kΩ. Determine the value for 40 to balance the brige.. Establish the iagonal in which the voltmeter shoul be connecte to maimize the sensitivity of the brige from Fig. 9. Comute the ratio an the sensitivity, after this choice. Prearatory questions