Using a Schering A.C. Bridge at low frequency to determine

condenser properties.

David.R.Gilson16th November 1998

Abstract

This experiment was an exercise in balancing a standard Schering A.C. bridge circuit to determine unknown impedances. These circuits are very sensitive to noise and care had to be taken to earth out any noise. This was done by using Co-axial leads for the transmission lines and utilising the lead screens as a path for the noise current to go to earth. The exercise consisted of measuring four capacitances (and loss factors) and then similarly analysing a butterfly capacitor with two different dielectrics, Air and Medicinal Paraffin. The data from the Medicinal Paraffin was used to calculate the dielectric constant of the Paraffin. The calculated value was found to be 1.93±0.01. Which is close to the published value of 2.2.

1) Introduction

[1] Usually, the most precise means of measuring a complex impedance with an alternating current (A.C.) is to use some type of A.C. bridge. A generalised A.C. (Wheatstone) bridge is shown is figure one.

Figure 1. A generalised A.C. (Wheatstone) bridge.

For an unknown impedance to be determined, the bridge has to be balanced, the voltage across the detector has to be zero. This means that not only the voltages at both sides should have the same amplitude, but the same phase too. Once this is achieved, two separate conditions are satisfied, which respectively involve the real and imaginary parts of a complex impedance. It is now clear that there are two balancing conditions which must be satisfied simultaneously. Such a property means that the two balancing conditions must be independent of each other. This is an important influence in the design of bridge circuits for such use. For this, one would choose a bridge which had two variable impedances which were independent and exclusive of each other in both of the balancing conditions. With such a bridge one of the impedances can be varied until a minimum reading on the detector is reached and the second variable impedance can be varied with until a new minimum is observed, then finer control can be achieved by returning to and varying the first impedance again. This process would be repeated as required.

It is also highly advantageous for the balancing conditions to be independent of frequency. This is because it is almost certain that no power supply could give a perfectly sinusoidal output, and it is therefore equally certain that there would be additional harmonics. This becomes a problem when one is observing and attempting to minimise very small signals. The harmonic voltages would make such a small signal (which is obviously of comparable amplitude) indiscernible.

Resistances and capacitances are the variable impedances used to reach the balanced state. This is because variable inductances use a movable contact which introduces a large intrinsic resistance. Since a general rule in these matters are that the impedances of all the "arms" should be of the same order of magnitude for the best performance, variable inductances are quite undesirable and are hence not used.

The balance condition for the A.C. bridge shown in Figure One is similar to that of it's D.C. counter part,

Z 1

Z 2=

Z 3

Z 4 (1)

Both of the balance conditions are contained in this complex equation. This makes sense when you consider that the real and imaginary conditions must be satisfied simultaneously. The A.C. Schering bridge is particularly used to measure capacitance and the same general principals outlined above also apply to this circuit.

Figure Two. A circuit diagram of an A.C. Schering bridge.

A diagram of a general Schering bridge is shown in Fig. Two. The unknown capacitor (or condenser) is represented by the series grouping of C and R on the same arm of the bridge. C1 is a good standard condenser, whose magnitude should be of the same order as that of the condenser under test (Note, this requirement may not be a possible if the condenser under test is unknown!). R1 is a fixed (non-reactive) resistance, and R2 is a variable resistor shunted by a variable condenser C2. The balance condition can now be expanded to the following,

jC1R 1j C =R1 1

R2 jC 2 (2)

Separating the real and imaginary parts gives both of the balance conditions,

C=C1

R2

R1 (3)

R=R1

C2

C1 (4)

These conditions fulfil the A.C. bridge prescription that has been discussed so far. Only one of the variable components appears in each equation and there is no frequency dependency either. Also, the unknown capacitance is obtained in terms of a standard capacitor and the ratio of two resistances. C2 is only used in the determination of R, the unknown resistance. With a good condenser R will be small and a high degree of accuracy in the calculation of R is uncommon.

To find the values of C and R in the circuit, equations 3 and 4 can be rearranged to linear functions, where the unknowns (C & R) are the gradients of each function respectively. By changing the value of R1 in the circuit shown in Fig. 2 different balance values of C2 and R2 will be found. So eq. 3 and 4 respectively become,

C R1=R2 C1 (5)

RC1

C 2=R1 (6)

Now the unknown values can be determined by taking the gradient from experimental values of the known impedances.

There is a particular piece of apparatus that can be used in this type of investigation. The apparatus referred to is the Butterfly Capacitor (BC). The technical aspects of this are discussed in more detail in the second section. The BC however can be considered as a series of parallel plate capacitors in parallel which is held inside of a brass container . The container can be filled with different substances. This means that dielectric properties of different materials (liquids and/or gases) can be investigated.

If the capacitance of the BC in a vacuum is known and the capacitance for a particular dielectric medium can be measured, the dielectric constant of the medium can be determined simply by the equation,

m=Cm

C0 (7)

2) Experimental ProcedureThe bridge circuit that was to be used throughout the whole experiment was constructed (schematically at least) as shown in Fig. 2. C1, the fixed capacitor was a standard laboratory capacitance, for this exercise the value of C1 was (0.304±0.1%)µF. Both resistances were variable (non reactive) resistors. Even though R1 in the circuit is supposed to be a fixed resistor, it was required to be changed to change the conditions of the bridge so that different values of R2 and C2 could be found at the balanced state of the bridge. A good range to use for the resistors is between 0 Ohms and 10 Kilo-Ohms. The variable capacitor C2 was a standard laboratory variable capacitor. A good range for this (if the reader wishes to repeat the experiment) is between 10-1 and 10-4 µ-Farads.

The driving voltage was supplied from a standard frequency generator which had a variable amplitude and frequency. Since this experiment depends on low (audio in fact) frequencies, so it should be made sure that any frequency generator used is capable of producing output at such frequencies. In this experiment the signal generator was constantly held at 8 KHz and 2 volts output amplitude. A standard Cathode Ray Oscilloscope (CRO) was used as a detector in the bridge, this is preferable to other methods used in the past for this type of experiment as a much higher resolution is possible than audio detectors for example.

All the components were connected by screened co-axial leads. All of the component terminals were threaded axles (apart from the variable capacitor) with a hole through for wires to be placed in. The axles then had a screw-down cap to hold any connected wires secure, this is standard equipment that can be found in virtually any laboratory. The variable capacitor had a spring loaded terminal, this type of terminal is just as suitable as the former terminal type.

The bridge is very susceptible to noise. This is a problem when attempting to observe near balance signals on the CRO. If there is too much noise the low-near-balance signals would be indiscernible from the noise and would obviously detrimentally affect the accuracy of any final readings. Fortunately this difficulty can be circumvented. If short screened leads are used and all of the screens are connect to earth points, the noise current can be minimised. However any point should be connected to an earth point once only, otherwise earth loops occur which would also spoil any readings. In this experiment each component had it's own earth (all the bridge elements had metallic casing connected to an earth terminal which provided a good earth connection).

The first part of the exercise was to examine four condensers of unknown capacitance (and loss factor). In Fig. 2 the series combination of C and R is modelled as the unknown condenser. The resistor R1 was held constant while R2 and C2 were adjusted (alternately) until a minimum signal was observed on the CRO. The values of all the bridge elements were then recorded. The process would then be repeated with a different value of R1 until a satisfactory number of readings were taken (in this case five reading were taken for each condenser, which provided ample precision for each value of capacitance).

The next part of the exercise was to examine the condenser properties of a Butterfly Capacitor (BC) with two different dielectrics. The BC consisted of (for those who are unfamiliar with the device) a brass cylindrical container whose lid accommodated the actual capacitor and terminals. The capacitive part was a series of interlaced parallel metal plates, which could be considered (for the sake of simplicity and the observed internal connections) as an arrangement of single (parallel plate) capacitors in parallel. This was on the underside of the lid, so that when in place, the capacitor was encased in the container. The terminals were on the top of the lid, there were two circuital terminals and one for earth as with the other bridge elements.

First of all, the BC was examined dry, the dimensions of the capacitor were measured so that the vacuum capacitance could be determined (for future reference), this was done by using standard vernier callipers. Then the BC was connected to the bridge circuit with Air as it's dielectric, the same process was used as before with the BC in the place of C and R (See Fig. 2). Next, the BC container was filled with Medicinal Paraffin and the BC was replaced in the container. So now, the Paraffin was the BC's dielectric. When doing this, it should be made sure that (and was in this experiment) that the BC is clean and free from contaminants which would effect the BC's dielectric properties. Again the same process of setting R1 and then varying R2 and C2 until balance is achieved and then setting R1 again and so on, was done on the BC with a Paraffin dielectric.

3) Results

Examination of unknown standard condensers

As suggested by equations 5 and 6, the capacitance and resistance of each condenser could be calculated from taking the gradient of plots of combinations of respective data. Recalling equation 5, C R1=R2 C1 .

Below are plots of R1 versus R2.C1 for the data collected in this experiment.

Figure Three. A plot of R1 versus R2.C1 for the first unknown condenser.

Figure Four. A plot of R1 versus R2.C1 for the second unknown condenser.

Figure Five. A plot of R1 versus R2.C1 for the third unknown condenser.

Figure Six. A plot of R1 versus R2.C1 for the fourth unknown condenser.

Below is a table showing the numerical values of the capacitances and the respective errors along with the values and errors of the resistances of the condensers. All these values were calculated using least square fit analysis, including the errors. See reference [2] for the source of these formulae.

Capacitance

Resistance

Condenser

C(µ

F)

±∆C(µF)

R(Ω)

±∆R(Ω)

One 0.673

0.028

0.21

0.67

Two 0.487

0.025

1.07

0.48

Three

0.231

0.029

2.59

4.95

Four 0.457

0.030

4.05

6.00

Table One. Capacitances and resistances of unknown laboratory condensers

As can be seen, the uncertainty in the capacitive values are reasonable (no more than 12.6% for the largest). The resistances have extremely large uncertainties. Since the accuracy of the bridge components are very good, there is little intrinsic error for the resistive readings. There were actually difficulties in taking these readings which shall be discussed in the next section.

Examination of a Butterfly Capacitor with various dielectric mediums.

The next stage of the experiment was analysing the condenser properties of a BC (butterfly capacitor). The internal geometry of the BC was quite complicated and could have meant higher uncertainty in calculated quantities. Simplifying assumptions were hence made about the capacitor geometry (see last section for a longer discussion of this point). Below is a list of spatial values for the internal structure of the BC.

Plate spacing (m) 0.00065Plate width (m) 0.03008Plate depth (m) 0.04090Error in linear measurement (m) 0.00001Plate area (m²) 0.00123Error in area (m²) 0.00058Number of plates 68.00000

Table Two. Spatial dimensions of simplified BC structure.

From these values the capacitance of the BC in vacuum can be calculated. As stated earlier, the BC was examined with Air and medicinal paraffin as dielectrics. The capacitance and loss factors were calculated in the same way as the other unknown condensers. Below are the graphs used to determine the capacitance for Air and Paraffin.

Figure Six. A plot of R1 versus R2.C1 for a BC with an Air dielectric.

Figure Seven. A plot of R1 versus R2.C1 for a BC with a medicinal paraffin dielectric.

Below is table showing the capacitance and loss factor (with errors) for air and paraffin dielectric, the vacuum capacitance is also included.

Capacitance ResistanceDielectric C(nF) ±∆C(nF) R(Ω) ±∆R(Ω)Vacuum 1.140 0.008 Not/Applicable.Air 1.200 0.090 85.1 24.5Med. Paraffin 2.190 0.060 7.39.10-11 4.27.10-11

Table Three. Capacitances of BC with various dielectrics.

It can once again be seen that the capacitances (not including the vacuum capacitance) were calculated with satisfactory accuracy but again the loss factor accuracy is very low.

One final calculation in this exercise was to calculate the dielectric constant of the medicinal paraffin. This can be calculated directly from the values shown above, using equation 7. Once calculated the value 1.93±0.01 was obtained. The error here is dependent on the error in the measurements of the capacitor dimensions used to determine the vacuum capacitance and the statistical error from the determination the paraffin dielectric capacitance.

4) Discussion

Fortunately, the results were as much as they were expected to be. The one part of this experiment that was unsatisfactory was the determination of all the loss factors. The errors were too large to make any sensible judgements on the magnitudes calculated. As already mentioned, these large errors are not due to inaccuracies in the bridge circuit elements. However (as already pointed out), the Schering bridge circuit is very susceptible to noise. The screened leads went some way to eliminating this problem by connecting the screens to earth. In construction of the circuit, some of these leads were cut with excess length to allow manoeuvrability without disrupting any of the terminal connections. However, the longer the lead, the larger the cross section that can pick up noise. In that case, it would reduce the noise levels further if the leads were made as short as possible.

In the determination of the dielectric constant of medicinal paraffin there were several limitations. Firstly the concentration of the paraffin was not known, any dilution would change it's electromagnetic properties, which would obviously change the dielectric constant. Also, the internal geometry of the BC was quite complicated, so to calculate the vacuum capacitance, the geometry had to be modelled as a much simpler structure and also the error in the vacuum capacitance was dependent on the error of the spatial measurements of the capacitor, which in turn effected the error of the dielectric constant.

In this exercise, the circuit that was constructed was reliable for measuring capacitances of condensers, but not for measuring loss factors. A good improvement would be to decrease the length of the leads as much a possible without straining the terminal connections.

5) Conclusion

The purpose of this exercise was to use an A.C. Schering bridge to determine the capacitance and loss factor of any particular condenser. The first four standard condensers were successfully examined, albeit with inaccurate loss factors. The butterfly capacitor was equally successfully examined with two different dielectrics (Air and Medicinal Paraffin) again though, with inaccurate loss factors.

The dielectric constant of medicinal paraffin (at room temperature and pressure) was measured to be 1.93±0.01. This is close to the published value of 2.2 (see reference [3]). However, the concentration of the published value is unknown as is the sample used in the experiment, which could in itself account for the difference (apart from other factors discussed earlier).

6) References

[1] Bleany B.I. and Bleany B., 1957, "Electricity & Magnetism" 2nd Ed., Oxford Science Publications, Chapter 16, pp 434,..,440.

[2] Kirkup Les, 1994, "Experimental Methods, an introduction to the analysis and presentation of data". John Wiley and sons, Chapter 6, pp 109,114,115.

[3] GWC Kaye & TH Laby, 1973, "Tables of Physical and Chemical Constants", 14th ed. Longman group limited.