An Inexpensive Impedance Bridge
July 1944 QST

Here is an amazingly detailed article on how to construct and operate a near-lab-quality impedance bridge out of relatively inexpensive components. It appeared in a 1944 issue of QST magazine. A bridge is used to determine the precise value of a resistor, capacitor, or inductor. Prior to modern, easily affordable digital impedance meters, both amateurs and professionals relied on such devices for lab and field work. Why might you need to measure the value of a component when most are marked with a value? One common application is when a variable version of a component (or components) is soldered into the circuit while tweaking for optimal performance, and then the variable is replaced either with a single fixed component or a fixed component with a smaller-range variable component (the latter provides adjustment, but over a smaller range of values). It is not uncommon when doing the initial tuning on a complete home-built transceiver to have many variable components in place initially, and then solder in fixed versions later. This design centering process provides good reference values for future designs and makes the final product more affordable and compact, since variable are almost always more expensive and larger in physical size.

An Inexpensive Impedance Bridge - The Principles and Construction of a Laboratory-Type Instrument for C, L and Measurements

By Athan Cosmas*

Into the life of every ham there comes a time when the exact measurement of some value of C, L or R is required. It may be of a resistor which is to be used with a delicate relay, a coil for some type of filter, or perhaps a condenser which is needed in a special circuit. What to do? If there is no school laboratory handy, or if there are no friends who happen to own an expensive instrument such as the General Radio Type 650-A Impedance Bridge, the best "out" is to build an impedance bridge which will do the work.

The bridge shown in the accompanying photographs will enable the making of all the measurements which usually are required in ham work. It has many of the fine features of the G-R bridge which it emulates. It will, of course, lack several of the fine points which contribute to the nicety and high accuracy of the expensive laboratory instrument i but it may be made from inexpensive parts, most of which the average ham has on hand, and it will have high enough accuracy for the average type of amateur measurements. The only hard-to-get item is the galvanometer.

Range

The complete circuit diagram of the instrument is given in Fig. 1. It includes a switching arrangement whereby any of the basic bridge circuits shown in Fig. 2 may be obtained.

In Fig. 1, when selector switch S₂ is in the position marked R, the circuit is that of the Wheatstone bridge, shown in Fig. 2-A. With this arrangement any resistance value from 0.01 ohm to 1 megohm can be measured when it is connected across the terminals at the right marked R.

When the switch is turned to either of the positions marked CD or CDQ, the circuit is that of the capacity bridge shown in Fig. 2-B. Any capacity between 100 μfd. and 10 μμfd. connected across the C-L terminals can be measured with either of these arrangements. This circuit also provides for two ranges of power factor, 0 to 0.1 with S₂ in the CD position and 0 to 1 with S₂ in the CDQ position.

With the switch thrown to the LDQ position, the circuit is that of the Maxwell bridge shown in Fig. 2-C. This circuit is used to measure the inductance of coils having values of Q up to 10. In the LQ position, the circuit is changed to that of the Hay inductance bridge shown in Fig. 2-D. With it, coils having values of Q up to 1000 can be measured. The inductance range is from 10 microhenries to 100 henries with either circuit.

For the benefit of those who have not had occasion to work with bridge circuits of this sort R, in the past, a brief explanation of the operating principles will be given.

Resistance Measurement

Referring to Fig. 2-A, the fundamental bridge circuit consists of four resistance arms. Two of these arms, R_a and R_b, are made up of fixed resistance values which are selected by a dual tap switch, S₁. The third arm, R_v consists of a calibrated variable resistor (in this case the resultant of R₉ and R₁₀ in parallel, because a variable unit of proper taper could not be obtained), while the fourth arm is composed of the unknown resistance, R_u.G is a d.c. galvanometer which, in effect, indicates the voltage differential between the midpoints of the upper and lower branches.

The object in adjusting the bridge is to arrive at a balanced condition where no current flows through G. In order that no current shall flow through G, it is obvious that its terminals must be at the same voltage. For this to be true, the galvanometer must have each of its terminals connected at the same percentage of the total resistance in each arm. For instance, if R_b has three times the resistance of R_a, then the unknown resistance, R_u, must have three times the resistance of the variable resistor, R_v, when the latter is set for zero galvanometer current. Since R_v is calibrated, it is a simple matter to determine the value of the unknown resistance.

From this reasoning we can set down the following proportion for the condition of zero current through the galvanometer:

From this we obtain

It is apparent that the unknown resistance, R_u, must always be equal to the value of resistance at which the variable resistor, R_v, is set, times a multiplying factor represented by the ratio R_b/R_a. lf some fixed value is selected for R_a, then a change in R_b alone will change the multiplying factor. Thus, the several resistances (R₃, etc.) represented by R_b may be considered as multipliers for the range of R_v.

As an illustration, in the instrument shown in the photographs R_v is 10,000 ohms, R_a is also 10,000 ohms (except for the highest resistance range, G in Fig. 1), while the tap switch, S₁, changes R_b in steps of 10 to 1; i.e., 1 ohm, 10 ohms, 100 ohms, etc., up to 100,000 ohms. The multiplying factors which can be applied to the resistance setting of R_v are, therefore,

  , etc.

or, in decimal equivalents, 0.0001, 0.001, 0.01, etc. Since the useful range of R_v is assumed to be from 100 to 10,000 ohms, the successive ranges of resistance measurements which can be made by the bridge are from 100 X 0.0001 = 0.01 ohm to 10,000 X 0.0001 = 1 ohm when R_b = 1 ohm; from 100 X 0.001 = 0.1 ohm to 10,000 X 0.001 = 10 ohms when R_b = 10 ohms; from 100 X 0.01 = 1 ohm to 10,000 X 0.01 = 100 ohms when R_b = 100 ohms etc.

Therefore, with the particular values selected for this bridge, the maximum resistance measurable in each range is equal to the value of R_b selected by the tap switch, S₁.

In the wiring diagram of Fig. 1, R₁ and R₂ are the resistors represented by R_a, while R₃ to R₈ are the resistors represented by R_b. R_v represents the resultant of R₉ and R₁₀ in parallel. When S₁ is turned to the last tap (G), R_a is changed from R₁ (10,000 ohms) to R₂ (1000 ohms). In this position, R_b (which represents R₈) has a value of 100,000 ohms. The .multiplying factor for this range is, therefore,

As R_v is varied from 100 to 10,000 ohms, the resistance-measuring range runs from 100 X 100 = 10,000 ohms to 10,000 X 100 = 1,000,000 ohms = 1 megohm.

In use, the unknown resistance is connected to the terminals marked R, the CRL multiplier switch, S₁, is turned to the approximate value and the CRL dial, controlling R₁₀, is adjusted for zero galvanometer current, The CRL dial reading is taken and the multiplying factor indicated by the position of S₁, as shown in Table I, is applied to the CRL dial reading to obtain the value of the unknown resistance.

If the bridge is very far off balance when the battery voltage is applied, excessive current may flow through the galvanometer. While R₁₄ serves to limit this current, a push-button switch may also be incorporated in series with the galvanometer so that the battery voltage may be applied only momentarily until the arms are adjusted for an approximate balance.

Labels on components refer to similar components and labels in Fig. 1 and to designations in the text. The respective dials controlling each variable unit also are indicated. In A, a DeJur student galvanometer or its equivalent is suitable for G. The galvanometer is strictly necessary only for low-resistance measurements. The 1,000-cycle a.c. source and headphones may be used for measuring the higher resistance values as well as for inductance and capacity.

Capacity Measurement

When selector switch S₂ is in the CD position, the circuit becomes that of the capacity bridge shown in Fig. 2-B. This arrangement is similar to the Wheatstone bridge circuit used for resistance measurements except that two of the arms contain capacities - one the unknown capacity, C_u, and opposite it a known capacity, C_s. The principle of obtaining a balance is much the same as that described for the Wheatstone bridge. In order to obtain voltage drops across the condensers, it is obvious that an a.c. source must be used instead of a battery. This is provided by a 1000-cycle generator. In place of the galvanometer a pair of headphones is used as an indicator, and the bridge is balanced when the arms are adjusted to give minimum response in the headphones.

Since the impedance of a condenser is in inverse proportion to its capacity, the expression for a balance becomes

From this we obtain

From the above, we see that the ratio R_v/R_b is the multiplying factor to be applied to the standard capacity to obtain the unknown value, C_u. The highest capacity range is made available when R_b is set at one ohm. The multiplying factor then becomes when R_v = 100 ohms, and = 10,000 when R_v = 10,000. At the other end of the range, when R_b is set at 100,000 ohms the multiplying factor is reduced to = 0.001 when R_v = 100 ohms and = 0.1 when R_v = 10,000 ohms. The standard represented by C_s is C₂ in Fig. 1. It has a value of 0.01 μfd., to which the above multiplying factors are applied when determining the value of the unknown capacity. The total capacity range varies from 100 μfd. when  = 10,000 to 0.00001 μfd. = 10 μμfd. when  = 0.001. Table I shows the factor by which the CRL dial reading should be multiplied to obtain the capacity in μfd. for each of the ranges set by S₁.

Power Factor

When making capacity measurements with the bridge it will be found impossible to obtain a complete balance unless the power factor of the condenser under measurement happens to be the same as that of the standard condenser, because of the difference in phase shifts. A condenser with a power factor greater than zero may be represented by a pure capacity (a condenser without losses) in series with a resistance. Therefore, if the losses of the condenser used as a standard are negligible, the power factor of the arm containing the standard may be made the same as the power factor of the arm containing the unknown capacity by adding resistance (R₁₁ or R₁₃ in Fig. 2-B) until the circuit is in balance. The setting of the series resistance for balance thus serves as a means for measuring the power factor.

A close approximation of the power factor of a condenser is given by the ratio R/X, which is known as the dissipation factor. Here R is the equivalent series resistance and X the reactance of the condenser. The latter is equal to where f is the frequency of the applied voltage in cycles and C the capacity of the condenser in farads. Therefore, in Fig. 2-B,

p.f. =  (R_s) (2πfC)

As an example, we know that the frequency is 1000 cycles and the capacity 0.01 μd. = (0.01) (10^-6) farads. Substituting these values, we obtain

p.f. = (R_s) (6.28) (1000) (0.01) (10^-6)

      = (R_s) (0.0000628)

R_s represents either of the variable resistances, R₁₁ or R₁₃, in the standard arm in Fig. 2-B. R₁₁ is in the circuit when S₂ is in the CDQ position. It has a maximum resistance of 16,000 ohms and is controlled by the dial marked DQ. At full scale the power factor of the standard arm is (16,000) (0.0000628) = 1. When S₂ is in the CD position the circuit is the same except that R₁₃, with a maximum resistance of 1600 ohms, is substituted for R₁₁. R₁₃ is controlled by the dial marked D. When R₁₃ is set at maximum the power factor indicated is 0.1. If the DQ dial is marked 0 to 10, its readings should be multiplied by 0.1 to obtain the correct power factor. (See Table III.) Similarly, the D dial reading should be multiplied by 0.01.

In practice, S₁ is first set to the appropriate range for the capacity to be measured. The CRL dial controlling R₁₀ is then varied for minimum response in the headphones. Finally, the D or DQ dials and the CRL dial must be carefully juggled back and forth for minimum response. When the positions giving the lowest possible response are found, dial readings of capacity and power factor can be made.

Inductance Measurement

When selector switch S₂ is turned to the LDQ position, the circuit becomes that of the Maxwell inductance bridge shown in Fig. 2-C. The Hay inductance bridge of Fig. 2-D is obtained with S₂ in the position marked LQ. The circuits are the same insofar as the measurement of inductance is concerned; they differ only in the ranges of Q which may be measured.

Since the impedance of a coil is proportional to its inductance while that of a condenser is in inverse proportion to its capacity, the condition for balance in the circuits of Fig. 2-C and 2-D is given by

From this we see that the product of R_bR_v is the factor by which the numerical value of C_s must be multiplied to obtain the value of the unknown inductance. Both inductance and capacity are expressed in units of similar order; i.e., in henries and farads. In the circuits of Figs. 2-C and 2-D, C_s represents C₁, which has a capacity of 0.1 μfd., while R_v may be varied from 100 ohms to 10,000 ohms and R_b from 1 ohm to 100,000 ohms, as before.

The smallest multiplying factor is obtained when R_v and R_b are at their minimums of 100 ohms and 1 ohm respectively. Then the factor becomes 100 and L_u = (100) (0.1) = 10 μh. (μh. because C_s is expressed in μfd.). The largest multiplying factor is obtained with the maximum values of resistance for both R_v and R_b, which are 10,000 ohms and 100,000 ohms, respectively. The factor at this end of the range is (10,000) (100,000) = 10⁹ and L_u = (10⁹) (0.1) = 10⁸ μh. = 10² h. = 100 h. Therefore, the range of the instrument on inductance measurements is from 10 μh. to 100 h.

As in the case of capacity measurements, it will be found necessary to balance resistive components as well as reactive components in the nonresistive arms. The amount of resistance which must be added in the capacitive arm to obtain minimum response in the headphones may be used as a measure of the Q (or X/R) of the coil. Since the reactance of the standard condenser, C_s is given by

When selector switch S₂ in Fig. 1 is in the LDQ position for the Maxwell bridge circuit of Fig. 2-C, C_s = C₁ = 0.1 μfd. and R = R₁₁, which is the variable resistor controlled by the DQ dial and which has a useful range of 160 to 16,000 ohms. The frequency is, of course, 1000 cycles, as before. Substituting these values in the above equation,

(the factor 10^-6 in the above denominator being necessary in converting to farads). At the other end of the range of R₁₁,

Thus the range of this circuit in measuring Q is from 0.1 to 10.

When S₂ is in the LQ position to give the Hay bridge circuit of Fig. 2-D the procedure is the same, except that R₁₂, which has a useful range of 16.5 to 165 ohms, is substituted for R₁₁. This gives a range of Q from 10 to 1000.

Constructing Resistance Standards

Most of the constructional details may be observed from the photographs. If the case is made of sufficient size, the galvanometer, battery and 1000-cycle source can be included in the unit for greater convenience.

The absolute accuracy of measurements made with the bridge naturally will depend upon the accuracy of the fixed resistors and condensers used as standards, as well as the calibration of the variable resistors. Ordinary copper magnet wire may be used in constructing homemade fixed resistance standards of values up to 10,000 ohms. Reference to the wire table in the Handbook (see pages 401 and 427 in the 1944 edition) will show the approximate resistance of copper wire of various sizes. For instance, the table shows that No. 28 wire has a resistance of 66.2 ohms per 1000 feet, or 0.0662 ohms per foot. Therefore, a length of about 16 feet will have a resistance of approximately 1 ohm.

The standard resistors (R₁ through R₈) must be of the noninductive type. Fig. 3 shows the method used in winding the lower-value resistors on a thin strip of Bakelite. The two ends of the wire first are soldered to the terminals at one end of the strip. The two half-lengths of wire are then wound in opposite directions around the Bakelite strip and the loop end fastened to the other end of the strip.

This method was used in making the 1-, 10- and 100-ohm standards. For the 1000- and 10,000- ohm units half-inch Bakelite rod was used, grooves being cut in the rod so that the windings could be made in pies. Each pair of adjacent pies was wound in opposite directions. Resistance wire rated at 80 ohms per foot was used, wound 250 ohms per pie for the 1,000-ohm units and 2500 ohms per pie for the 10,000-ohm unit. Two 50,000-ohm meter multipliers, rated at 1 per cent accuracy, were connected in series to provide the 100,000-ohm standard.

Calibration

The most accurate means available should be used in checking the resistance of the standards. A local serviceman or a school laboratory may have a resistance bridge which can be borrowed to make the calibrations. The wire-wound units can be adjusted to exact values by removing the insulation from the loop end and twisting the loop until the correct value is obtained.

An accurate calibration must also be obtained for the R₉R₁₀ combination. The curve should be checked at as many points over the range of R₁₀ as possible. If a 10,000-ohm resistor with a logarithmic taper is available it may be used to replace the parallel combination. When building this unit a potentiometer of this type could not be obtained locally, and so the combination of R₉ and R₁₀ was used to obtain an approach to the desired logarithmic characteristic.

Once the fixed resistance standards and R₁₀ are calibrated, it is a relatively simple matter to calibrate R₁₁, R₁₂ and R₁₃ by simply connecting them to the R terminals of the bridge. These three units preferably should also have a logarithmic taper.

Condensers having capacities as close as possible to..the required values of 0.1 μfd. and 0.01 μfd. should be used for the capacity standards. Both should be of the mica type, to minimize loss errors. C₁ may be made up of a combination of smaller-capacity units in parallel, if necessary.

The accompanying tables (II and III) show how the dials should be marked to be direct reading.

Fig. 4 shows the circuit of an inexpensive generator suitable for the 1000-cycle signal source required for measuring capacity and inductance. The frequency can be checked with sufficient accuracy by matching it up with the second B above middle C on a correctly tuned piano. The buzzer should be enclosed in a sound-proof box.

Posted July 20, 2021
(updated from original post on 12/21/2012)