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Carl von Ossietzky Univers ity Oldenburg Faculty V In Carl von Ossietzky Univers ity Oldenburg Faculty V In

Carl von Ossietzky Univers ity Oldenburg Faculty V In - PDF document

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Carl von Ossietzky Univers ity Oldenburg Faculty V In - PPT Presentation

Experimentalphysik 2 Elektrizitt und Optik Springer Verlag Berlin among others 2 TCKER H Taschenbuch der Physik Harri Deutsch Frankfurt 3 ORIES R CHMIDT ALTER H Taschenbuch der Elektrotechnik Harri Deutsch Frankfurt Introduction In this experim ID: 77859

Experimentalphysik Elektrizitt

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17 Carl von Ossietzky University OldenburgFaculty V Institute of PhysicsModule Introductory Laboratory Course Physics Part IMeasurement of CapacitiesCharging and Discharging of CapacitorsKeywords:Capacitor, parallelplate capacitor, dielectric, element, charge and discharge curves of capacitors, phase shift, 0bQCU AUb- Q0+ Q0dE + 18 is termed thecapacitance of the capacitor. Its unit is ARAD []CFAsVCV (1 C = 1 OULOMBFor a parallelplate capacitor in a vacuum the capacitance is exclusively determined by thegeometry of its arrangement. It is directly proportional to the areaof the plate and inversely proportional to the distancebetweenthe plates: ~ACd Question How can the proportionality be illustrated? Hint: Consider the electric field and the voltage in a charged parallelplate capacitor that is separated from the voltage source following chargingand whose plates are pulled apart afterwardsSee to it that the chargeremains constant.Applying the proportionality coefficient we obtain: CAd0 (in a vacuum)is called the electric field constantpermittivityof vacuum. It is calculated from two internationally determined constants, namelythe speed of light c(in vacuum) and the magnetic field constantpermeabilityof vacuum, and can therefore be stated with an optional precision (cf.back page of the cover of this script). We confine ourselves to ourdigits here: 0021218854110:,cAsVm By putting an electric insulator (dielectric) between the plates of the capacitor the capacitance is increased by the factor CAdr0 (in matteris termed relative permittivity relative dielectric constant), the product is called permittivitydielectric constantis a numerical value dependent on the insulating material used. It is, e.g. for air at C and normal pressure 1.0006, for water at 20°C: 81, for different kinds of glass: , and for ceramics (depending on kind): . In a vacuum Question How can we explain the increase in capacitance due to the dielectric? (Hint: Attenuationof the electric field.)Many different types of capacitors are commonly available in retail. They come in a variety of casings, and theircapacitances spanseveral orders of magnitudFig. shows some examples. Named after ICHAEL ARADAY(1791 1867)HARLES UGUSTIN DE OULOMB(1736 1806)In an alternating current circuit, is dependent on the frequency of the fed voltageThementioned values are approximate values for the case of low frequencies within a range below 1 kHz. 19 Fig. Common retail versions of capacitors of differenttypeand casings. The capacitances of the depicted types vary between several picofarad (pF) and several microfarad (F).Charging and Discharging of a CapacitorDischargingLet us firsttake a look at the discharging of a capacitor. We are particularly interested in knowing how long the discharging takes and how it develops with time. For this purpose we examine a charged capacitor with capacitance according to Fig. which is discharged via a resistance uch an arrangement is called resistanccapacitance element. At an optional time after closing the switch S we obtain (cEq.(1) QtCUt()() 䘀椀朮 㨀⁄椀猀挀栀慲最椀渀最昀⁡⁣慰慣椀琀漀爀⁶椀愠愠爀敳椀猀琀) is the momentary charge of the capacitor and ) the momentary voltage across the capacitor. According to IRCHHOFF's law this voltage equals the voltage acrossthe resistance , so that we obtainwith the momentary current UtRIt()() 吀栀攠捵爀爀敮琀 ) is caused by the decreasing(hence the minus sign) charge of the capacitorwith timeHence ItdQtdt()() 䔀煳⸠⠀㜩⠸⤀Ⱐ愀湤 ⠹⤀挀潭扩湥⁴漠礀椀攀氀搠琀桥⁤椀昀昀攀爀攀渀琀椀愀氀⁥煵愀琀椀潮 昀潲 琀桥 摩猀挀桡爀最椀湧昀⁴栀攀⁣愀灡挀椀琀漀爀㨀 RC 吀桥⁳漀氀畴椀潮昀⁴桩猀⁤椀昀昀攀爀攀湴椀愀氀⁥煵愀琀椀潮⁵湤攀爀⁴桥 椀湩琀椀愀氀 挀潮搀椀琀椀潮) = reads: QtQetRC()0 The product has the unit [] = F = (V/A)(As/V) = shusrepresents a time period , the socalledtime constantwhich has the following meaning: at a time the charge has decreased to a value /e, which is about the 0.368fold of the initial value: SCR 20 (12 ()()0.368tRCQtQ==→==≈⋅ 䘀漀爀⁴栀攀⁴業攀 t = Thalflife time, within which the charge has decreased to halfof the initial value, we obtain: RCRC If a discharge process shall be observedit is easier to look at the decreasing voltageacross the capacitornstead of observing the decreasing chargeof the capacitor according to Eq. (11)Applying Eqs. (1)and (7), Eq. (11)yields: UtUtRC()0e The voltage drop, which can be very easily measured using, for example, an oscilloscope, has the same temporal variation as the decrease in charge.Hence, Eq. (14) yields an important relation for measuring capacitances in practice. Measuring the voltage at two different times and , we obtain (cf.Fig. ㄀㄀ ㈀㈰㨀攀UtUUUtUU==== 吀桥愀琀畲愀氀 氀潧愀爀椀琀桭昀⁅焀⸠⠀㄀㔀⤀⁹椀攀氀摳 ㄀ ㈀ 汮汮汮汮=−=− 䠀攀湣攀Ⱐ椀琀⁦潬氀潷猀㨀 ㄀㈱汮汮汮UttURC−== 愀湤⁦椀湡氀氀礀㨀 ≈∆ The equationabove is the basis for all capacitance measurements in this laboratory session. In order to be stringent, it would be necessary to replace ln() by ln({}) (likewise for , etc.) in equation (16)and the following, since the logarithm is only defined for a numerical argument (e.g. {}), but not for quantities having an associated unit (e.g. ). To simplify the presentation we omit the curly brackets, silently implying the numerical value of the given physical quantity.Many multimeters employ this principle for measuringcapacities. 21 Fig. Course of discharge of a capacity.Let us now observe the chargingof a capacitor with the capacitance with the help of a real voltage source according toFig. The real voltage source can be considered an ideal voltage source G in series with the source voltage and a resistance (the internal resistance of a real voltage source). According to IRCHHOFF's law we obtain an optional time after closing the switch S) is the charging current)Fig. : Charging of a capacitor via a real voltage source 0RC()d()()()()()QtQtQtUUtUtRItRCtC=+=⋅+=+ 䠀攀湣攀⁩琀⁦潬氀潷猀眀楴栀 QCU 搀⠩QtRCQ+−= 吀桥⁳漀氀畴椀潮昀⁴桩猀⁤椀昀昀攀爀攀湴椀愀氀⁥煵愀琀椀潮⁲攀愀摳㨀 ⠩㄀攀QtQ=− 吀栀攠琀椀洀攠捯渀猀琀慮琀 states the time period within which the capacitor is charged to the (11/e)fold of its maximumcharge Analogous to the dischargingof the capacitor, for the easily observablevoltage increase of the capacitor we can write: ()1eUtU=− Question Plotthe development of Eqs. (14)and (22)for the time interval [0; 5] for the values nF and using U1U2Ut1t2t S= U0GRC I 22 Interconnection of Several CapacitorsThe total capacitance of an arrangement consisting of several capacitors can be calculated by applying IRCHHOFF's laws. Forseries connectionof capacitors with the capacitances we obtain (cf. Fig. for 2): 11 䙯爀⁡ parallel connectionone obtains(cf. Fig. for = 2): 1niiCC 䘀椀朮  㨀⁓敲椀敳⁣漀渀渀散琀椀漀渀昀⁣愀瀀慣椀琀漀爀猀 䘀椀朮  倀慲慬氀敬⁣漀渀渀散琀椀漀渀昀 捡瀀慣椀琀漀爀猀 inusoidal Excitation of a lementSo far we have studied the behaviour of a capacitor which is charged or discharged oncevia a resistance. In order to understand the behaviour of capacitors in alternating circuits we will now observe the reaction of a element, which means a setup consisting of resistance and capacitor,upon sinusoidal excitation. We look at a setup according to Fig. An ideal voltage source provides the alternating voltage with the angular frequency ()cos()UtUt 䘀椀朮 㨠刀䌀⁥汥洀攀渀琠眀椀瑨 猀椀渀甀猀漀楤愀氠攀砀挀椀瑡瑩漀渀䄀湡氀潧潵猀⁴漠䔀焮 ⠀ㄹ⤀楴⁦漀氀汯眀猀 昀爀漀洀 䤀刀䌀䡈但䘀✀猠瘀漀氀琀慧攠氀慷㨀 䜀 剃搀⠩⠩挀漀猀⠀⤀⠩⠩QtQtUtUtUtUtR==+=+ 䠀敮捥楴⁦漀氀汯眀猀㨀 搀⠩⠀⤀挀漀猀⠀⤀ QtRCCUt+−= 䤀琠楳甀爀⁡業 琀漠摥琀攀爀洀椀湥⁴栀攀⁴攀洀灯爀愀氀⁤攀瘀攀氀潰洀攀湴昀 For this purpose, it is sufficient,according to (7), to find the temporal development of ). From the considerations presented in Chapter we know that the capacitor cannot be charged or discharged infinitely rapidly. This means that the course of arging ) cannot follow the voltage ) instantaneously, but rather with a certain temporal delay. Therefore, we expect a phase shiftof ) compared to ). Thus, wetry to solvethe differential uation(27)by setting Of course, the ansatz ) = sin() would also achieve its purpose; however, the form with the cosfunction has become established in physics. C12C 1CC2 ~ U (t)GRC 23 (28 cos( 䈀礀⁩湳攀爀琀椀湧⁅焮 ⠀㈸⤀椀湴漠䔀焀⸀ ⠀㈷⤀眀攀潷⁨愀瘀攀 琀漀⁤攀琀攀爀洀椀湥⁴栀攀⁵湫湯眀渠煵愀渀琀椀琀椀攀猀 . Following some calculations which are most easily done using complex quantities, appendix in Chap. obtain for the maximum charge Qof the capacitor: RC 愀湤⁦漀爀⁴桥 phase shift between ) or ) and arctan()RC=− 愀湤 琀愀渀RC=− 爀敳瀀散琀椀瘀敬礀⸀䘀爀潭⁅焮 ⠀㌰⤀眀攀 汥愀爀渀 瑨愀琠is always negative. Tcharge ) always lags behind the voltage ). For the limit 0 we obtain 0° and for the limit it follows: 90°.With therelationship: costan1()1RC++ 眀攀扴愀椀渠批⁩渀猀攀爀琀椀湧⁅焮 ⠀㌲⤀椀湴漠䔀焮 ⠀㈀㤩 挀漀猀QCU 䌀潭灡爀椀湧⁅煳⸠⠀ㄩ愀湤 ⠀㌀㌩眀攠氀敡爀渀⁴栀慴 琀栀攠洀慸椀洀甀洀⁣栀慲最攠漀昀⁴栀攠捡瀀愀捩琀漀爀 椀猀漀眀敲 戀礀⁡⁦慣琀漀爀 漀昀⁣漀猠under sinusoidal excitation than under a direct voltageof magnitude For the limit 0 we obtain and for the limit it follows that= 0.Question How can these extreme cases be illustrated?We will now calculate the temporal course of the current ) through the loop according toFig. . We have: d() 䤀湳攀爀琀椀湧⁅焮 ⠀㈀㠩椀湴漠⠀㌀㐩愀湤 瀀攀牦漀牭椀渀最⁴栀攀⁤椀晦攀牥渀琀椀愀琀椀漀渀礀椀敬搀猺 000()sincoscosItQtQtItwwϕwwϕwq=−+=++=+ 眀椀琀栠琀桥⁣畲爀攀渀琀⁡洀灬椀琀畤攀  == 愀湤⁴栀攀⁰桡猀攀⁳桩昀琀 between the current ) and the voltage 24 (37 2 Using the relationship tan(/2) = 1/tan, we obtain from Eqs. (37)and (31) tan 䔀焮 ⠀㌀㠩獨漀眀猀 琀栀慴 椀渀 琀栀攠捡獥 0 the current precedes the voltage by 90° (/2). In the , however, current and voltage are in phase (0°). With increasing frequency the phase shift between current and voltage decreases from 90° to 0°.ImpedanceThe impedance(or apparent resistanceis an important parameter for the description of electrical circuits. It will be treated in more detail in the experimental physics lecture of the second semester. For this reason, we will restrict ourselves here to a few remarks on impedance.The impedanceis defined as the total resistancean electrical circuit poses to an alternating voltage of angular frequency It follows that ). The unit of impedance is Ohm =Ω 䄀渠椀洀灥摡湣攀⁩渠愀渠䄀䌀 挀椀爀挀畩琀 眀椀氀氀Ⱐ椀渠最攀湥爀愀氀Ⱐ椀湦氀略湣攀⁴桥 amplitudeand the phaseof the current in a circuit. Thus it is practical to represent it as a complex quantity: ReImZZiZ=+ 䘀椀朮 猀桯眀猀 as a pointer in the plane of complex numbers. The real part of is the (ohmic) resistanceof a circui ReRZ 吀桥⁩洀愀最椀湡爀礀⁰愀爀琀昀 is called reactance X Im 吀栀甀猬⁷攠挀慮⁷爀椀琀攠昀漀爀 (according to equation(39) ZRiX=+ 吀桥愀最湩琀畤攀昀(i.e. the length of the arrowin Fig. ) is given by: 22ZRX=+ 湤⁴栀攀⁰桡猀攀Ⰰ洀攀愀湩湧琀栀攠慮最氀攠潦⁴桥⁡爀爀潷 眀楴栀琀桥⁒攀慸椀猠椀猠最椀瘀敮⁢礀㨀 愀爀挀琀愀渀≈∆ 坩琀栀⁴栀攠慢漀瘀攬from E(39)or (42)can be written in polar formas: eiZZ 䤀渀⁧敮敲愀氀Ⰰ⁴栀攠total resistanceis not a pure ohmic resistance!In an AC circuit with capacitor C and coil L, the reactance is composed of an inductivecomponent caused by L, and a capacitivecomponent caused by C. More about this in the second semester. 25 Fig. : Impedance as a pointer in the plane of complex numbersIn analogy to Ohm’s law is given by the ratio of the voltage amplitude to the current amplitudeFor the element in Chap. it follows (given by E(36) ==+ 䌀潭灡爀椀猀潮昀⁅⠀㐶⤀眀楴栀⁅⠀㐳⤀猀桯眀猀⁴栀愀琀 is composedof aohmicresistanceand a capacitivereactance). In the case 0 we have 1/, i.e., is mainly determined bythe capacitor which blocksthe circuit in this case. For , however, the situation is inverse: that case 1/0, i.e., the capacitor does not block and is mainly determined by the hmicresistancexperimental ProcedureEquipment:Digital oscilloscopeEKTRONIXTDS 1012 / 1012B / 2012C / TBS 1102B, function generator (OELLNER 7401, output resistance 50), Multimeter (GILENT34405A), voltage supply, stopwatch, resistor decade, single capacitors on mounting plate (approx. 10F, approx. 10nF), plate capacitor (aluminium; 0.200.17) with dielectric (PVC plates of variable thickness, (1, 2, 3)mm), 5 coaxial cables of different length, switch, metal measuring tape, tape measure, calliper gauge.HintIn the following circuit diagrams those components are drawn inredwhose quantities (capacitance or resistance) are to be measured (Fig. Fig. ) or above which signals are measured (Fig. ). The dashed frames surround the equivalent circuit diagrams of the instruments which are used to measure the required quantities, such as the function generator or the oscilloscope. Besides the input and output resistances and the capacitances of the instruments, oftenanother capacitor is drawn into the circuit diagram. represents the capacitance of all cables required for the measurementsetup (capacitance of connecting cablesIn order to simplify the text we will often use the terms input capacitance, the capacitance of connecting cables, the capacitor etc. when we mean capacitors of the capacitancesor etc.Determiningthe Input Resistance of an Oscilloscopefrom the Discharge Curve apacitorThe input resistance of an oscilloscope is to be determined from the discharge curve of a capacitor with the capacitance Fig. . For this purposeis charged via the internal resistance of a voltage source (voltage supply; initial voltage V), then is separated fromthe voltage sourceopen switch) and the discharge of via is observedhe input capacitance of the oscilloscope, the capacitance of connecting cables and the capacitance are in parallel. e choose so we can neglect and here, measurewith multimeter GILENT34405A).According to Eq. (18) the time difference ∆t = tis measured ten timesusing a stopwatch within which the voltage decreases from the value to the value (measure and ). The input resistance of the oscilloscope, including the maximum error, is determined from the mean value of according to Eq.(18)The values for and may be assumed to be error free (exact) for this purpose. ImReZ R X 26 Fig. Equivalent circuit for voltage supply, capacitor C, connecting cables (with capacity and oscilloscopewith the input resistance to be measured. MeasuringapacitancesDescription of the Measuring MethodThe procedure applied in experiment to measure the time difference ∆t = tis well suited if the time constant is largeFor small time constants it is ideal to periodically charge and discharge the capacitor and to measure the time difference by direct observation of the discharging curve with an oscilloscope. Periodic charging and discharging can be achieved by connecting the capacitor with a function generator (FG) and providing a periodic squarewave voltagewith an amplitude (e.. The FG then serves as a voltage source with an incorporated electronic switchFig. shows the related equivalent circuit diagram. Fig. Equivalent circuit for function generator FG, connecting cables (with capacity ) capacitance to be measured, and oscilloscope. Refer to the text for other labels.omparisonwithFig. shows two differences:Besides the capacitance of the connecting cables (the input capacitance of the oscilloscope and the capacitance to be measuredthe “output capacitanceof the has to be taken into accountThese three capacitiestogether form the total capacitance the measuring AOKFCCCC=++ 桥 䘀䜀⁡猠慮 敬散琀爀漀渀椀挠猀眀椀琀捨搀漀攀猠渀漀琀 獥瀀愀爀愀琀攠琀栀攠瘀漀氀琀慧攠獯甀爀捥 眀椀琀栀⁲敳椀獴慮捥from the circuit (likethe switch S inFig. ), but only causes a periodic charge reversal of the capacities and ue to the charge reversalis performed via Therefore determines the time constantof the RC element together with and In this case, Eq. (18)therefore reads ≈∆ 䄠爀攀愀氀 猀焀甀愀爀攀睡瘀攀signal from a FG never has edges with slope ∞. Rather, e.g. the falling edge resembles the discharging curve of a capacitor with capacitance . This quantity is described as output capacitance according to an equivalent circuit here.It is of no importance to the measurement, whether the capacitor is charged and then discharged or periodically commutated, as in this case. This does not influence the time response. SpannungsquelleOszilloskop Oszilloskop CF 27 Eq. (48)provides the possibility to determine an unknown capacitance by measuring and , provided that andare known.For the function generators used in the laboratory courseThis results in a small value of the time constant of the capacitor discharge, leading to a small (and hence difficult to measure) time difference For this reason,an external resistancefrom the resistance decade is placedin series with in a setup according toFig. and Fig. 13 in order to achieve a total resistance of GFDRRR=+ 琀桵猀 椀渀捲攀慳椀湧琀栀攠琀椀洀攠搀椀昀昀敲敮捥 (48)then becomes ≈∆ 䘀椀朮 䌀楲挀甀楴⁦爀漀洀 䘀椀朮 眀椀琀栀⁡摤攀搠爀攀猀椀猀琀潲 Fig. icture of the circuit from Fig.showing the function generator on the left, the oscilloscope on the right, and the resistance decade with resistor in the centre. is located between the two black terminals of the resistance decade. The yellow terminal is a support contact without an electrical connection to BNCT connector is inserted in the cable connecting the resistance decade and the oscilloscopein orderto connect the capacitor for which the capacianceis to be determined.From this follows, that the capacitance is given by: ≈∆ Preliminary MeasurementsIn order to determine an unknown capacitance from(51)the value of the total capacitance of the circuit needs to know in addition to the resistance is determined by setting up the circuit according to Fig. with 0 (i.e. withoutthecapacitancto bemeasured. A BNCT piece is included in the circuit (Fig. to connect the capacitance which is to be determined for each subsequent measurement. can now be determined using E(50)For this purpose, the discharge curve of is displayed on the oscilloscope and the time difference associated with the voltage drop from to is measured.For measuring thequantities, the digital oscilloscope can be operated in the mode → AcquisitionMean value. In this operation mode, the influence of signal noise is minimized. Oszilloskop CF DR 28 and may be taken as exact values for calculating the maximum error of For a maximum error of 0.01in accordance with the accuracy of the resistance decade, may be usedOnce these preparations have been made, unknown capacitances added to the circuit can bemeasuredHinEqs. (18)and (51)hold for the discharge of a charged capacitor from an initial voltage to 0 V. The voltage levels and are positive at all times . If, however, a rectangular voltage with amplitude is applied to the capacitor, it follows that the maximum voltage is +and the minimum voltage is Fig. , left ordinate). Hence, the resulting reloadingcurve may include negative voltage values. In this case, qs. (18)and (51)cannot be applied, since the logarithm function is only defined for arguments having a positive value.This problem can be solved by recognising that the temporal evolution of a reloading curve from the voltage +to has the same shape as the discharge curve of a capacitor having an initial voltage of 2and a minimum voltage of 0Fig. , right ordinate). Thus, adding the amplitude to all voltage values recorded from the oscilloscope ensures that and are always positive, and hence qs. (18)and (51)can be used.This method requires, that the rectangular voltage signal doesnot have any DC component (Offsetknob on the FG must be set to OFF) and that its amplitude is known.It follows that must be measured once. To facilitate reading the voltage levels off the oscilloscope, it is recommended to place the signal symmetrically about the centre (horizontal) line of the scale (“0” in Fig. , left ordinate). In this case, and can be determined simply by reading the scale marks on the oscilloscope’s screen and can be determined by using the time cursorsFig. Charge reversalcurve of the capacitor upon applying a rectangular voltage of amplitude without DCoffset (left ordinate)The same temporal course results for a rectangular voltage with amplitude andoffset (right ordinate, blue). Thehorizontal lines indicate the scale ticks of the oscilloscope.Determination of the Capacitance of Coaxial CableIn this part of the experimentthe capacitance of coaxial cableaddedto the existing (coaxialcableshaving a total capacitance ), is to be measured. The simplest method to achieve this is to connect the extraneous cables to the BNCT connectorFig. is thus connected parallel to Fivecoaxial cables different lengths (measure the lengthare connected in turn to the BNCpiece.For each cable,the quantities and are measured and the capacitance is calculated according to (51)Stating the errorfor the individual values of may be omitted. As a result the mean value of the capacitance of the coaxial cable per meter including the standard deviation of the mean is to be stated and to be compared with the valuefrom literaturefor coaxial cables of the type RG 58 C/U (101pF/m). Determining the Relative Permittivityof PVCFollowing the method described in Chapter the capacitance of a plate capacitor with the dielectric PVC between its plates is to be determined. The objective is to determine the relative permittivity of PVC from a series of capacitance measurements with varying thickness of the dielectric.Theplate capacitor consists of two equal aluminium plates of the area with a PVC plate of equal size and thickness between them. The capacitor is connectedbetween function generator and oscilloscope in addition and in parallel to the existing connecting cables. It is connected to the BNCpiece by a coaxial U1U2Ut1t2t -U0+U002U0U00 29 cable having laboratory plugs on the other endOne of the aluminium plates is put on the laboratory bench and connected to the negative pole” of the function generator(outer contact of the BNCconnector). The PVC plate is put on this plate and the second aluminium plate is put on top of it and connected to the other pole of the function generator.Measurements are done for PVC plate sizesof (3, 4, 5, 6) mm (measure with a calliper gauge and with a metal measuring tape). is determined for each size (Eq.(51)). For further analysis, is plotted over 1/can be determined (Eq. (6)from the slopeof the regressionline and can be compared with the literature value (Eq.(6)hase Shift Between Current and Voltage in an lementUsing a setup according toFig. the phase shift between the cosinusoidal output voltage of the function generator and the charge and discharge current of the capacitor with dependence on the angular frequency is to be measured. We neglect the internal resistances as well as input and output capacitances of the function generator and the oscilloscope for this experiment.The output voltage of the function generator can be measured directly using the oscilloscope (symbolized by the “voltmeter” in Fig. ). he current is measured via a small detour: causes a voltage drop, R I , that is in phase with and can also be measured with the oscilloscope (The measurement of is carried out for an element with and measure both values with multimeter GILENTat frequencies(1,100)kHzThe amplitude of shall amount to approx. 5V at kHzis plotted vs. with maximum error for . Into the same diagram the theoretical expected values fare plotted too and are compared with the measured dataFig. Setup for measuring the phase shift between and in RCelement.Practical hints:When carrying out the experiment it should be considered that the reactance 1/(of the capacity is a function of so that the voltage amplitudes also vary with The phase shift can best be determined by measuring the time difference of the passages through zero by both voltages and compare withthe experiment “OscilloscopeConsider at the connecting of the cablesfor the measurement of and that the outer contacts of the BNC sockets of the oscilloscope are on the same potential! Consequently this also applies to the outer contacts of the BNC plugs at the coaxial cables!Question How large is the phase shift between the voltage at the capacitor () and the current ? How can thephase shift be measured? This additional cable increases the total capacity of the connecting cables in the experimental setup. It is thus necessary to (re)measure the total capacity of the measuring apparatus prior to connecting the parallel plate capacitor. terature value according to /3/3.1 … 3.5 (without stating frequency). CRVV12FGFR~ UFG 30 AppendixCalculating with complex quantities, Eqs. (29)and (30)are easy to derive. In a complex form the formulasin Eqs. (25)and (28), respectively, can be written as ()eitUtU QtQ 䤀湳攀爀琀椀湧⁢潴栀⁥煵愀琀椀潮猀⁩渀琀漠䔀焮 ⠀㈶⤀愀湤⁰攀爀昀潲洀椀湧⁴桥⁤椀昀昀攀爀攀湴椀愀琀椀潮⁷攀戀琀愀椀渀⁡昀琀攀爀⁤椀瘀椀猀椀潮 批  it 〰 UiRQQ=+ 䠀敮捥楴⁦漀氀汯眀猀㨀 iUQiRC 吀桥攀昀琀⁳椀摥昀⁅焮 ⠀㔀㔩椀猠common way to represent a complex number polar notationof modu| and the phase angle (argument :ehere:e,zzzQzQϕϕ== 吀栀攠洀潤畬畳漀映is given by zzz * being the complex conjugated to which is obtained by changing the sign of the imaginary unit and ). For the moduluswe thus obtain:     ㄱ㄀UUUUCRCiRiRRwww+−+ 吀栀椀猠椀猠琀栀攠爀敳甀氀琀杩癥渀⁩渀 䔀焮 ⠀㈀㤩坥⁵猀攠愠secondcommon method to represent complex numbers to calculate the phase angle, namely ReIm:zzizi=+=+ 桥爀攀the real part(Re)and the imaginary part(Im)of . From these quantities the phase angle can be calculated as π00arctanπ00abab+⇔<∧≥−⇔<∧< 䤀渀爀摥爀⁴漠愀灰氀礀⁅焮 ⠀㘰⤀Ⱐ眀攀⁨愀瘀攀⁴漠挀潮瘀攀爀琀⁅焮 ⠀㔵⤀椀湴漠琀桥⁦潲洀潦⁅焮⠀㔹⤀Ⰰ 琀栀愀琀⁩猠眀攠洀甀獴⁳数愀爀愀琀攀 琀栀攠爀敡氀⁡渀搀⁴栀攠椀洀慧椀渀慲礀 瀀慲琀 昀爀漀洀 敡捨 漀琀栀攀爀⸀⁆潲 琀桩猀⁰畲瀀潳攀⁷攀⁨愀癥琀漀⁥汩洀楮愀瑥 from the denominator, for which the fraction is appropriately extended. Eq. (55)then becomes: 31 (61 UiRiiRRiRiRww==−=≈≈++∆∆ 爀漀洠䕱⸀⠀㘱⤀眀攠捡渀⁲敡搀昀昀 and 02221UCRC UCRC 䄀琀琀攀渀琀椀潮畳琀⁢攀⁰愀椀搠琀漠琀桥⁦愀挀琀⁴栀愀琀⁴桥爀攀⁩猀⁡⁰漀猀椀琀椀瘀攀⁳椀最渠椀渠琀桥⁤攀昀椀湩琀椀潮 攀煵愀琀椀潮⠀㔹⤀⸠吀桵猀Ⱐ琀桥 渀攀条琀椀癥⁳椀杮昀 in Eq.(61)belongs to the imaginary part By insertingEq.(62)into Eq.(60)we obtain: arctanarctan==− 吀栀椀猠椀猠琀栀攠爀敳甀氀琀 杩癥渀⁩渀 䕱⸀⠀㌀〩