An Overview of Practical Capacitance Bridge Functioning. by Paul Moses

An Overview of Practical Capacitance Bridge Functioning by Paul Moses

INTRODUCTION The laboratory has a variety of bridges, both automatic and manual which can be used to measure the capacitance and dielectric loss of materials. The following discussion is applicable to the General Radio 1615 transformer ratio bridge, the Hewlett-Packard 4270 3 terminal bridge and all the Hewlett- Packard 4 terminal bridges: the 4192, 4194, 4274, 4275, and 4284. For primarily historical reasons bridges with 2 coaxial leads, like the General Radio 1615 and the HP 4270 are referred to as 3 terminal and bridges with 4 coaxial connections, like the newer HP s are referred to as 4 terminal bridges, even though this is not logically consistent. At a sufficient level of abstraction all the bridges can all be viewed as working in the same way. They apply a known voltage to the high side of the sample and a known current to the low side of the sample (fig. 1). Then the current is adjusted, either manually with the General Radio bridges or automatically by the HP LCR meters until a balance is reached at point A, which means that the voltage there is 0 with respect to the system common. At this point both the voltage across the sample and the current through it are know and the impedance can be calculated. This balanced condition is very important not only in finding the impedance but also because it creates a virtual ground which makes shielding possible. For more information please read the section on shielding. In the manual bridges the balance current is controlled by adjustable components which simulate the sample s capacitance and loss so the sample s capacitance and loss can be read off the front panel of the bridge. The HP bridges calculate a capacitance and loss based on the sample s impedance and a model. It is important to realize that the bridges don t measure dielectric constant or even capacitance, they measure the complex impedance and, one way or another, model to calculate capacitance. It is surprising but true that unless you have a sample of 0 dielectric loss it does not possess a unique capacitance and consequently no unique dielectric constant. If you don t understand this please read the section on impedance and capacitance. If you are not familiar with the use of complex numbers to describe the magnitude and phase of AC currents, voltages, and impedances please read the section on complex numbers. MANUAL BRIDGES The operation of the manual bridges is somewhat easier to understand than that of the automatic bridges. Our manual bridges use a center-tapped transformer to create a negative voltage which is applied to standard capacitors and resistors to make the balance current. Figure 2 is conceptually accurate although the actual construction is somewhat different. The transformer is broadband and extremely precise. An external generator is used to supply the measurement signal, which is usually between.1 to 10V and 60 Hz to 10 khz. The transformer has a total turns ratio of 2, but because of the center tap, the output is 2 voltages, each equal in magnitude to the generator signal with the signal in the reference arm exactly inverted with respect to the

generator. The person using the bridge looks at the signal present at the detector (the detector could be a lock-in amplifier, a tuned voltage detector, a current detector, or an ordinary AC voltmeter, depending on the sensitivity required) and adjusts the variable components until there is no signal at the balance point. The measurement circuit is then a voltage source connected to the sample high and a current source connected to the sample low. The current is determined by the inverted generator voltage divided by the impedance of the variable standard capacitor and resistor. This current source is only correct when the bridge is in balance, because it is only then that the voltage across the sample is equal to the (inverted) voltage across the standard components, see figure 3. Although, for technical reasons, this is not the actual scheme used in the bridges, it is possible to think of the standard capacitor and resistor as being constructed from discrete precision components as in figure 4. In this scheme the standard capacitance would be set by selecting one capacitor for each decade and the total capacitance would be the sum of all the decades, since the capacitors are connected in parallel. The standard resistors would be set similarly except that the resistances are connected in series. While this approach is perfectly reasonable it is difficult to achieve in practice for several reasons. First, in order to be able to read capacitance and loss to an adequate precision an excessive number of precision components are required and they have to be connected by an excessive number of switches. For example, to be able to read 999.999 pf one would need 9 standard capacitors plus precise open circuits for each decade for a total of 54 standard capacitors and 6 zero stray open circuits. There is a similar problem with the standard resistors. Second, the values required for the standard components are difficult. A precision capacitor as small as the resolution of the measurement is required and one might want to measure less than.001 pf. Such small capacitors are very difficult to manufacture precisely. Also, the resistance values required for loss measurements are impractical. For a D step of.1 at 100Hz with a sample of 1 pf a resistor of about 10^9 ohms is required. It is difficult to obtain such large and accurate resistors and when they can be found they have parallel capacitance which causes their impedance to deviate from purely resistive. To circumvent the first problem the properties of the transformer are exploited. Remember that the purpose of the standard components is to supply a current to the sample. Since the measurement is only taken when the bridge is in balance (0V at the detector) there is more than one way to supply current to the sample. For instance, suppose the sample were a lossless capacitor of 1 pf and the generator signal were 1 V. This sample current could be balanced by a signal of -1V applied to a standard capacitor of 1pF or a signal of -.1V applied to a capacitor of 10pF or any voltage applied to a capacitor as long as C*V = 1pFV. This property is used to reduce the number of standard components. The reference arm of the transformer is tapped for 0., -.1, -.2,..., -1. and one standard capacitor is used for each decade. Each capacitor is switched onto one of the taps and it allows its own CV current to flow, the total balance current is then the sum of the various decades. This

not only reduces the number of standard components (by a factor of 10) but also increases the size of the problematic smallest standard capacitor required by an order of magnitude. The problem of the resistors is solved by replacing the simple reference arm by the circuit of figure 6. I haven t noticed anything instructive about this trick so I am only going to state the result. This T network has no effect on the capacitance balance but the loss value at balance is given by D=ω*R(C t +C d ) This is extremely convenient because it reduces the size of the reference resistor required and because it makes the D balance independent of the value of the capacitance balance. This means that, at least at one frequency, the front panel can be labeled so that the reference resistor switches read out in terms of D. The balance condition of the bridge results several beneficial effects for the experimenter. First, because the balance point is held at exactly 0V, the high and low sides of the measurement circuit can be shielded from each other by ordinary metal shieding as long as the shielding is connected electrically to the system common. Also an electrode held at 0V (grounded) can be used as a guard for high impedance samples. For more information please see the section on virtual grounds and shielding. Second, loading of the transformer is not a first-order error because when the sample is loading the transformer this reduction in voltage is reflected into the reference arm of the transformer secondary and so it doesn t affect the bridge balance. In other words a 27.35 pf sample will still be balanced by the standard capacitor in the bridge when it is set to 27.35 pf. The only first order effect is that the measurement signal used will be reduced, for example to.95v instead of the 1V the generator might have been set to before the sample was plugged in. There is still a second order error resulting from resistance in the transformer winding and in the cable connecting the sample to the bridge which can have some effect on the answer. This can be minimized by keeping the cables as short as possible. In fact minimizing cable length is important for most measurements most of the time and should always be considered carefully (check the 1615 manual for more information) when attempting to measure larger samples at higher frequency. You should never use longer wires than necessary and you should always at least estimate the effect of cabling on your measurement either by calculating the impedances involved or by some direct measurement of the cable effect. Regardless of the bridge in use there are always effects from the cabling alone. As long as there is current flowing in a wire there is inductive impedance present. To some extent this impedance always appears in a measurement but at frequencies below 1 MHz with ordinary sample sizes (1 nf and less) it is not typically a problem. At higher frequency and with larger samples it can be serious. Another effect that is always present results from the distributed capacitance, resistance, and inductance in a coaxial cable. Please see figure 7. In any coaxial cable the distributed impedances act as a low pass filter. Once again the magnitudes of these impedances are such that they are not normally a large problem at low frequency but for the most accurate

measurements some sort of compensation is necessary. The newer automatic bridges have built in compensation software which allows you to measure the total cabling in your system in both and open circuit and short circuit states and the bridge will digitally subtract most, but not all, of the effect of the cables. The manual bridge is capable of a number of other functions and ranges but with a grasp of the above the additional complexities will be easy to understand when you have to use the bridge. AUTOMATIC BRIDGES The automatic bridges use the same principles as the manual bridges but perform the functions using electronics rather than passive components. For the moment consider a hypothetical 3 terminal automatic bridge as shown in figure 8. At this level of abstraction the bridge supplies a known voltage to the sample and measures the current through it with a phase sensitive ammeter. If the ammeter input impedance is 0 (an ideal ammeter) then the low side if the circuit is a virtual ground and so shielding has its normal function. This is in fact a perfectly reasonable configuration but is not completely practical because it is difficult to construct an ammeter which has low input impedance and high accuracy over the range of current and frequency required. For example a 1 pf sample at.1 V signal level at 100 Hz will result in only 62 pa of current while a 100 nf sample at 1 MHz and 1V will cause about 1 A to flow, a difference of 11 orders of magnitude. For electronic reasons it is much easier to construct a precision current source than it is to construct a precision ammeter. So the first improvement is shown by figure 9. This is just the familiar voltage and current source with a detector as explained above but now the voltage and current are generated by electronics. Since this is not a paper in op amp circuitry I am not going to make any attempt to explain the goings-on inside the automatic bridges. You just have to trust Hewlett-Packard. Because the companies that make automatic bridges want to sell as many as possible they want their bridges to work with the widest range of samples as possible. Since the difference between a capacitor, resistor and inductor (at any one frequency) is just the phase of the current passing through it, by building a current source which is adjustable over the entire +/- 90 deg. phase range they can build one bridge which will measure any type of component, so the only remaining problem (aside from the bridge s programming) is to allow for the greatest range of impedance magnitude possible. This is a problem for the electronics but once again that is not the subject of this paper. The problem of interest to us is lead impedance. With high impedance samples (relatively low value capacitors at relatively low frequency) the impedance of the leads is not usually a problem because it is very small compared to the sample impedance. But with samples of relatively low impedance (small value resistors and inductors and large value capacitors) lead impedance can

contribute very significantly to measurement error. The solution to this problem is to use a 4 terminal technique. This is the exact AC analog of the 4 terminal DC resistance measurement which you may be familiar with. Please refer to figure 10. Terminals 1 and 4 supply voltage and current as before. Terminal 2 is new, it measures voltage and it is brought out to the front of the bridge. If the lead impedance is significant then terminal 2 should be connected to the sample as close as possible to the sample. Terminal 2 is constructed to approximate an ideal AC voltmeter. As such it draws negligibly little current, so even if there is lead resistance it is no longer significant. Since essentially no current is flowing into terminal 2 the lead resistance doesn t create a voltage drop. Resistance in the leads from terminals 1 and 4 don t contribute error because the resulting voltage drop isn t measured as long as terminal 2 is connected next to the sample. These two terminals themselves function only as generator and current source so voltage drops in them are inconsequential. Current loss from terminal 4 would be significant but basically current in this lead is not interested in going anywhere except to the sample because of the virtual ground which is explained in its own section. Terminal 3 is not really new, it is just the detector but it has now been brought out to the front panel of the bridge so that it can also be connected next to the sample, once again eliminating the effect of lead resistance. Interestingly since the voltage at this point is 0 (when the bridge is balanced) this terminal doesn t have to be an ideal voltmeter or current detector. Since the voltage at both ends of this cable should be 0 there will be no current flowing in this cable regardless of the nature of the detector so lead resistance won t create a voltage drop. In addition to the 4 terminal configuration newer automatic bridges can also reduce the effects of lead impedance by compensating for it digitally. The idea is that before you perform a measurement of your sample you measure the impedance of your test fixture and cables in both the open circuit and short circuit states and the bridge corrects for these impedances in the final measurement. However these corrections are never perfect since they can be perturbed by the presence of the sample, movement of the cables, thermal expansion in the cables or test fixture, change of resistance in the cables and text fixture due to temperature changes, and simple imperfection in the measurement of the cable impedance. Furthermore if the cable impedance becomes significant compared to the sample impedance then the bridge is forced to subtract two similar numbers to find the answer, which always leads to error. Since each of the impedances to be subtracted have some percentage error their difference will unavoidably have a larger or much larger percentage error. As a rule in any careful measurement you should obtain some estimate of the lead impedance and perform the automatic compensation if the effect is at all significant. But if the effect is too large you cannot expect it to vanish enough by digital compensation and you have to address the problem directly. There is a further consideration in using the multiterminal bridges. The shields of the 4 terminals are not connected together internally. This is for 2 reasons, to reduce lead inductance and to reduce noise pick-up. Consider

figure 11, for the moment ignoring the voltage terminals. Since they don t carry significant current they don t contribute to this effect. Because the shields of the cables are not connected together inside the bridge the signal current is transmitted and returned in the same coaxial cable. In this way there is essentially no net current flowing through a loop around the cable and so approximately no magnetic field external to the cable and not much inductive impedance. If there were an internal connection in the bridge then some or all of the measurement current would flow through the bridge connection so the current in the cables would not be balanced and there would be a non-zero magnetic field outside the cable shown in fig 11.1. The 2 voltage terminals are also floated so that they can be connected to the measurement circuit at a point convenient to you. In general in any measurement set-up you should avoid joining conducting wires together at multiple points. This would be the case if the shields of the terminals were connected together in the instrument as illustrated in figure 12. This multiple connection makes a loop which can pick up environmental electromagnetic fields and add noise to your measurement. VIRTUAL GROUNDS AND SHIELDING One of the most important ideas to keep in mind when making low capacitance measurements is shielding. When any two conductors are held apart a geometrical capacitor is formed between them. More to the point, the 2 leads to the sample and the sample test fixture in a capacitance measurement also have a geometrical capacitance which will appear as additive error in your measurement unless steps are taken to eliminate the effect, figure 13. This stray capacitance will also affect the loss measurement in proportion to the ratio of stray capacitance to sample capacitance. In a philosophical sense this capacitance itself can never be eliminated but with a properly designed bridge and moderate attention to the sample set-up the stray capacitance can be harmlessly diverted. Contrary to popular belief simply interposing a piece of metal between the offending leads will not, by itself, reduce stray capacitance nor will grounding that piece of metal guarantee its shielding properties. The shielding property of conductors is not a fact of nature, it does not result directly from the laws of electromagnetics but is a consequence of the way that precision capacitance bridges are constructed and in sufficiently crude and inexpensive capacitance meters there may be no way to shield from strays without building external electronics. The ability to shield strays capacitance is a result of the virtual ground used in all sophisticated capacitance bridges (that I am aware of). Consider figure 2. I will illustrate this discussion with the transformer ratio bridge but exactly the same considerations apply to the automatic bridges. When the bridge is in balace point A is at 0V but is held there by the fact that the current flowing through the sample is precisely balanced by the current through the reference arm. Such a point, actively held at 0V, is called a virtual ground. Remember that the idea in a capacitance bridge is to know (at least in an abstract sense) the voltage applied to the sample and the resulting current flowing through the sample. Since point A (and the entire lead to point A) is at 0V, any capacitance existing between point A and a conductor held at 0V, like the bridge s shielding which is connected to the system common, will not cause

any current to flow from the low terminal, simply because there is no voltage across the capacitor, figure 14. Therefore this stray capacitance has no tendency to affect the measurement. In principle it is sufficient to shield only one side of the measurement circuit but practically speaking the other side is also always shielded both to help cover holes in the low side shielding and to reduce noise pick-up from environmental fields. Fortunately capacitance on the high side of the bridge can also be shielded without affecting the bridge balance. As shown in figure 15 capacitance in the high lead simply loads the system generator. As explained earlier this has no first-order effect on the manual bridges because the loading of the transformer is almost entirely reflected into the reference arm, and it has no serious effect on the automatic bridges because the applied voltage is measured seperately. Of course capacitance between the shields has no effect on the measurement because it affects neither the voltage nor the current being measured. IMPEDANCE It is important to understand that the natural property your sample possesses is not capacitance, dielectric constant, permitivity, or a probably anything else you are primarily interested in, but rather impedance. Dielectric loss is intrinsic and independent of the measurement but capacitance isn t. One way or another all measurement techniques (below the microwave frequency) measure bulk impedance and you must infer the capacitance in the presence of some loss. The usual technique involves asking the following question. Given a perfect capacitor and a perfect resistor, what values when connected in, say parallel, are required to give the value of impedance measured? This question can also be posed with regard to a series connection. This question can always be answered uniquely (for a single frequency) but unless the dielectric loss is 0 the series and parallel connection models will give different values for both the resistor and capacitor. For D<.1 the difference between the capacitors in the two models is negligible although the resistor values differ greatly. For a lossless capacitor the difference in the values for the series and parallel model resistor is infinite. The series resistance is 0 and the parallel resistance is infinite. The calculation is very simple. If Z(complex) is the impedance of the sample and w = 2πf then: SERIES: Z = R s + 1/iωC s R s = Re(Z) C s = (Im(Z)*ω) 1 PARALLEL: Y = 1/Z = 1/R p + iωc p R p = 1/Re(Y) C p = Im(Y)/ω where Y = conductance

From this calculated value of capacitance one infers the dielectric constant by the usual C=kε 0 A/t for parallel plate samples. Even this formula is only an approximation. It is based on the assumption that all of the electromagnetic flux is directly from one electrode to the other. Part of the assumption is that there is no stray capacitance through the surrounding air from one electrode to the other. This type of stray is called fringing capacitance and can be a serious problem with low permittivity samples and samples with fat geometries. The automatic bridges are normally used in the parallel model mode. By virtue of its construction, the manual bridge is usually used in the series mode although it has a feature that allows it to model a parallel conduction mechanism. This model inconsistency is not usually a problem because the manual bridges are normally used with low loss samples where the choice of model is not critical even though the parallel model is usually believed to be a better representation of the loss processes in a ferroelectric. COMPLEX NUMBERS Alternating currents and voltages are commonly described using complex numbers sometimes call phasors. Unlike DC signals, AC signals have a magnitude, frequency and a phase. In a typical application, all of the AC signals of interest have the same frequency so the only information that is unique to each signal is the magnitude and phase. It is convenient to combine these two numbers into one complex number. In the common problem there is one signal source that is used as the phase reference, for example the generator for a transformer ratio capacitance bridge. Other signals are described with reference to that one. For example, if the generator signal were a 1 Vpk-pk sine wave at 1kHz then it would be fully described by 1sin(2π*1000t)V. Its phasor would be 1 0 or 1. +0i = 1. The first representation is the polar form of a complex number, the magnitude (1.) and the phase angle (0.). The second form is the cartesian form 1+0i. The output of the transformer arm driving the reference arm would be -1 0 =1 180 or -1+0i=-1. In this representation impedances are immediately available. Just divide the (complex) voltage across a component by the (complex) current through it. A pure capacitor would then have an impedance given by 1./(iωC ). Since this number is purely imaginary the current through a perfect capacitor is exactly 90 degrees out of phase with the voltage through it. For example if a voltage of 1 0 at 1 khz is applied to a capacitor of 1 nf then the current through it is 0+6.28i µa= =6.28 90 µa. The important thing as far as this discussion is concerned is that impedances have both magnitude and

Im polar form: r θ y 0 θ x r Re cartesian form: x + yi 2 2 1/2 r = (x + y ) θ = Arctan y/x phase.

fig 1 D fig 2 Vo -Vo Co D Ro A

fig 3 Vo Vo -Vo Co D Ro Va If V = 0 then Vo i = ----------- Ro + 1 ωco ---- If the sample is assumed to be modeled by: Co = Ro fig 4 Then Co = Cs, Ro = Rs and since D = ωrc, D can be calculated. 0 pf.001 pf -3 10 pf digit D.009 pf 0 pf.01 pf -2 10 pf digit 10 9 Ω 2 * 10 9 Ω 10 9 Ω 2 * 10 9 Ω 10-3 Ω 2 * 10-3 Ω.09 pf 9 * 10 9 Ω 9 * 10 9 Ω 9 * 10-3 Ω 0 pf 100 pf 100 pf digit 900 pf

fig 5 0 1.01 pf i1 2.1 pf i2 i = i1 + i2 +...in 9 100 pf in

fig 6 fig. 7 fig. 8 I fig. 9 D fig. 10 V 1 2 V 3 I 4 fig. 11 (various clectronics) < < > > > < fig. 11.1 > < fig. 12

fig. 13 fig. 14 fig. 15 C stray Istray = ωcstray(va - Vshield) = ωcstray * 0 = 0