Measurement of Mutual Inductance By Heaviside Mutual Inductance Bridge

Heaviside Mutual Inductance Bridge:

         Heaviside Bridge (shown in the below figure) measures mutual inductance in terms of a known self- inductance.The same bridge, slightly modified, was used by Campbell to measure a self-inductance in terms of a known mutual inductance

Let       M = unknown mutual inductance, 
            L1 = self-inductance of secondary of mutual inductance
            L2 =known self-inductance, and
          R1,R2,R3,R4   = non-inductive resistors. 

At balance voltage drop between b and c must equal the voltage drop between d and c. Also the voltage drop across a-b-c must equal the voltage drop across a-d-c, Thus we have the following equations at balance 

It is clear from the above equation, that L1, the self inductance of the secondary of the mutual inductor must be known in order that M be measured by this method. 

In case         R3 = R4

we get           M = (L2-L1)/2        and      R1= R2 

This method can be used for measurement of self-inductance. Supposing L2 is the self-inductance to be determined. From above equations, we get 


Campbell's Modification of Heaviside Bridge:

         The below figure shows a modified Heaviside bridge.This modification is due to Campbell.This is used to measure a self-inductance in terms of a mutual inductance. In this case an additional balancing coil R is included in arm ad in series with inductor under test.An additional resistance r is put in arm ab. Balance is obtained by varying M and r. A short circuiting switch is placed across the coil R2, L2 under measurement.Two sets of readings are taken one with switch being open and the other with switch being closed. Let values of M and r, be M1 and r1 with switch open, and M2 and r2 with switch closed. 
With switch open from above equation we have,


Hence                        L2 = (M1-M2)(1 +R4/R3)

With switch open, we have,                   R2+ R=(R1+r1)R4/R3 

and with switch closed 

                                  R2 = (r1-r2) R4/R3 

This method is a good example of the methods adopted to eliminate the effects of leads etc.

When we have equal ratio arms R3 = R4

and therefore from above equations we get, 

                               L2 = (M1-M2)  and R2 = (r1-r2)

Heaviside Campbell Equal Ratio Bridge:

              The use of balancing coil in the above method reduces the sensitivity of the bridge.The below figure shows Heaviside Campbell equal ratio bridge. This is a better arrangement which improves sensitivity and also dispenses with the use of a balancing coil. In this method the secondary of the mutual inductor is made up of two equal coils each having a self-inductance. 
           One of the coils is connected in arm ab and the other in arm ad. The primary of mutual inductance reacts with both of them. L2, R2 is the coil whose self-inductance and resistance is to be determined. The resistances R3 and R4 are made equal.Balance is obtained by varying the mutual inductance and resistance r. 

At balance      I1R3 = I2R4      but  R3 = R4

and therefore      I1 = I2 = I/2     as      I = I1 + I2

Writing the other equation for balance


Equating the real and imaginary terms 

               R2 = R1 + r     and           L2 =2(Mx+My)=2M 

Thus the magnitude of inductance which can be measured with this method is twice the range of the mutual inductor.The values calculated above include the effects of leads etc. In order to eliminate these effects, we take two readings with switch open circuited and another with switch closed.Let M1, r1 be the readings of M,r with open circuit and M2, r2 with short circuit. 

                         R1 = (r1-r2and  L1 = 2(M1-M2) 
        In this post we have learnt Measurement of Mutual Inductance By Heaviside Mutual Inductance Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ Comment below if you have any queries!
Measurement of Capacitance By Schering Bridge

Measurement of Capacitance By Schering Bridge:

The connection and phasor diagram of Schering Bridge under balance conditions are shown in the below figure. 

                C1 = capacitor whose capacitance is to be determined

                r1  = a series resistance representing the loss in the capacitor 

           C2 = a standard capacitor.This capacitor is either an air or a gas capacitor and hence is loss-free.However, if necessary, a correction may be made for the loss angle of this capacitor, 

                 R3 = a non-inductive resistance, 

                 C4 = a variable capacitor, and 

           R4 = a variable non-inductive resistance in parallel with variable capacitor C4. 

Must Read:



Equating the real and imaginary terms, we obtain 

                                    r1 = R3C4/C2

and                              C1 = C2 ( R4/R3)

Two independent balance equations are obtained if C4 and R4 are chosen as the variable elements. 

Must Read:

Dissipation factor, 

                  D1 = tan δ = ωC1r1 =ω(C2R4/R3) x (R3C4/C2) = ωC4R4  

         Therefore values of capacitance C1, and its dissipation factor are obtained from the values of bridge elements at balance. 

        Permanently set up Schering bridges are some times arranged so that balancing is done by adjustment of R2 and C4 with C2 and R4 remaining fixed. Since R3 appears in both the balance equations and therefore there is some difficulty in obtaining balance but Schering Bridge has certain advantages as explained below :

          The equation for capacitance is C1 = (R4/R3) C2 and since R4 and C2 are fixed, the dial of resistor R3 may be calibrated to read the capacitance directly. 

        Dissipation factor D1 ωC4R4  and in case the frequency is fixed the dial of capacitor C4 can be calibrated to read the dissipation factor directly. 

       Let us say that the working frequency is 50 Hz and the value of R4 is kept fixed at 3,180 Ω.

Dissipation factor,

                 D1 = 2π x 50 x 3180 x C4 = C4 X 10

       Since C4 is a variable decade capacitance box, its setting in μF directly gives the value of the dissipation factor. . 

        It should, however, be understood that the calibration for dissipation factor holds good for one-particular frequency, but may be used at another frequency if correction is made by multiplying by the ratio of frequencies.

Must Read:


        In this post we have learnt Measurement of Capacitance of Schering Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ Comment below if you have any queries!
Measurement of Capacitance By De Sauty's Bridge

Measurement of Capacitance By DeSauty's Bridge:

          DeSauty's Bridge is the simplest method of comparing two capacitances. The connections and the phasor diagram of DeSauty's Bridge are shown in the below figure.

Let                   C1 = capacitor whose capacitance is to be measured

                       C2  = a standard capacitor, and 

                 R3, R4  = non-inductive resistors. 


At balance, Done

               The balance can be obtained by varying either R3 or R4.The advantage of DeSauty's Bridge is its simplicity.But this advantage is nullified by the fact that it is impossible to obtain balance if both the capacitors are not free from dielectric loss.Thus with DeSauty's Bridge only loss-less capacitors like air capacitors can be compared. In order to make measurements on imperfect capacitors (i.e., capacitors having dielectric loss), DeSauty's Bridge is modified as shown in the below figure.This modification is due to Grover. 

             Resistors R1 and R2 are connected in series with Cand C2 respectively. r1 and r2 are resistances representing the loss component of the two capacitors.
At balance, Done Done

          The balance may be obtained by variation of resistances R1, R2, R3, R4.The above figure (b) shows the phasor diagram of the DeSauty's Bridge under balance conditions.The angles δ1 and δ2 are the phase angles of capacitors C1 and C2 respectively. 

Dissipation factors for the capacitors are : 

                   D1 = tan δ1 = ωC1r1             and            
                   D2 = tan δ2ωC2r2

From above equation, we have Done

              Therefore, if the dissipation factor of one of the capacitors is known, the dissipation factor for the other can be determined.De Sauty's Bridge does not give accurate results for dissipation factor since its value depends on difference of quantities R1R4/R3 and R2.These quantities are moderately large and their difference is very small and since this difference cannot be known with a high degree accuracy, the dissipation factor cannot be determined accurately.


        In this post we have learnt Measurement of Capacitance By De Sauty's Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ Comment below if you have any queries!
Measurement of Self Inductance by Owen's Bridge

Measurement of Self Inductance by Owen's Bridge :

Owen's bridge may be used for measurement of an inductance in terms of capacitance.The below figure shows the connections and phasor diagrams, for Owen's bridge, under balance conditions. 


          L1 = unknown self-inductance of resistance 

          R2 = variable non-inductive resistance, 

          R3 = fixed non-inductive resistance, 

          C2 = variable standard capacitor, 

          C4 = fixed standard capacitor

At balance, 

Advantages of Owen's Bridge: 

1)Examining the equations for balance, We find that we obtain two independent equations in case C2 and R2 are made variable.Since R2 and C2, the variable elements,are in the same arm,convergence to balance conditions is much easier. 

2)The balance equations are quite simple and do not contain any frequency component. 

3)Owen's Bridge can be used over a wide range of measurement of inductances

Disadvantages of Owen's Bridge

1)Owen's bridge requires a variable capacitor which is an expensive item and also its accuracy is about 1 percent. 

2)The value of capacitance C2 tends to become rather large when measuring high Q coils. 

        In this post we have learnt Measurement of Self Inductance by Owen's Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @ Comment below if you have any queries!
Measurement of Self Inductance By Anderson's Bridge

Measurement of Self Inductance By Anderson's Bridge:

       Anderson's Bridge, in fact, is a modification of the Maxwell's inductance capacitance bridge.In Anderson's Bridge, the self-inductance is measured in terms of a standard capacitor.Anderson's Bridge is useful applicable for precise measurement of self-inductance over a very wide range of values.The below figure shows the connections and the phasor diagram of the bridge for balanced conditions. 


Let  L1 = self-inductance to be measured, 
      R1 = resistance of self-inductor, 
        r = resistance connected in series with self-inductor, 
r, R2, R3, R4 = known non-inductive resistances, and 
       C = fixed standard capacitor

At balance, 

       An examination of balance equations reveals that to obtain easy convergence of balance, alternate adjustments of r1 and r should be done as they appear in only one of the two balance equations. 

Advantages of  Anderson's Bridge

1. In case adjustments are carried out by manipulating control over r1 and r, they become independent of each other. This is a marked superiority over sliding balance conditions met with low Q coils when measuring with Maxwell's bridge. A study of convergence conditions would reveal that it is much easier to obtain balance in the case of Anderson's bridge than in Maxwell's bridge for low Q-coils. 

2. A fixed capacitor can be used instead of a variable capacitor as in the case of Maxwell's bridge. 

3. Anderson's Bridge may be used for accurate determination of capacitance in terms of inductance. 

Disadvantages of Anderson's Bridge : 

1. The Anderson's bridge is more complex than its prototype Maxwell's bridge. The Anderson's bridge has more parts and is more complicated to set up and manipulate. The balance equations are not simple and in fact are much more tedious. 

2. An additional junction point increases the difficulty of shielding the bridge.

        In this post we have learnt Measurement of Self Inductance By Anderson's Bridge.You can download this article as pdf,ppt.If you have queries you can mail us @  Comment below if you have any queries!
What are ac bridges & General form of a.c bridges

What are A.C Bridges ?         

       Alternating current bridge methods are of outstanding importance for measurement of electrical quantities.Measurement of inductance, capacitance, storage factor, loss factor may be made conveniently and accurately by employing a.c. bridge networks.The a.c. bridge is a natural outgrowth of the Wheatstone bridge.An a.c. bridge, in its basic form, consists of four arms, a source of excitation, and a balance detector. 
           In an a.c. bridge each of the four arms is an impedance, and the battery and the galvanometer of the Wheatstone bridge are replaced respectively by an a.c. source and a detector sensitive to small alternating potential differences.The usefulness of a.c. bridge circuits is not restricted to the measurement of unknown impedances and associated parameters like inductance, capacitance, storage factor, dissipation factor etc.
        These circuits find other applications in communication systems and complex electronic circuits.Alternating current bridge circuits are commonly used for phase shifting, providing feedback paths for oscillators and amplifiers, filtering out undesirable signals and measuring frequency of audio signals. 
         For measurements at low frequencies, the power line may act as the source of supply to the bridge circuits. For higher frequencies electronic oscillators are universally used as bridge source supplies. These oscillators have the advantage that the frequency is constant, easily adjustable, and determinable with accuracy.The waveform is very close to a sine wave, and their power output is sufficient for most bridge measurements.A typical oscillator has a frequency range of 40 Hz to 125 kHz with a power output of 7 W. 

           The detectors commonly used for a.c. bridges are : 

(i) Head phones, 

(ii) Vibration galvanometers, and 

(iii) Tuneable amplifier detectors. 

              Head phones are widely used as detectors at frequencies of 250 Hz and over upto 3 or 4 kHz. They are most sensitive detectors for this frequency range. When working at a single frequency a tuned detector normally gives the greatest sensitivity and discrimination against harmonics in the supply. 
           Vibration galvanometers are extremely useful for power and low audio frequency ranges. Vibration galvanometers are manufactured to work at various frequencies ranging from 5 Hz to 1000 Hz but are most commonly used below 200 Hz as below this frequency they are more sensitive than the head phones. 
            Tuneable amplifier detectors are the most versatile of the detectors. The transistor amplifier can be tuned electrically and thus can be made to respond to a narrow bandwidth at the bridge frequency. The output of the amplifier is fed to a pointer type of instrument. This detector can be used, over a frequency range of 10 Hz to 100 kHz. 
            For ordinary a.c. bridge measurements of inductance and capacitance, a fixed frequency oscillator of 1000 Hz and output of about 1 W is adequate. For more specialised work continuously variable oscillators are preferable with outputs upto 5 W. The high power may be necessary on some occasions, but in practice it is better to limit the power supplied to the bridge. 
         Another practice which is usually followed is to use an untuned amplifier detector. The balance detection is sensed both orally by head phones, and visually by a pointer galvanometer having a logarithmic deflection (In avoid damage to the galvanometer which may be caused by unbalance). 

General Equation for A.C Bridge Balance:

            The below figure shows a basic a.c bridge.The four arms of the bridge are impedances Z1,Z2,Z3 & Z4.


          The conditions for balance of bridge require that there should be no current through the detector.This requires that the potential difference between points b and d should be zero.This will be the case when the voltage drop from a to b equals to voltage drop from a to d, both in magnitude and phase. In complex notation we can, thus, write : 


On Substitutions we get, 

                                   ZZ4 = ZZ3  

or when using admittances instead of impedances

                                  Y1 Y4 = YY3  

           Above two equations represent the basic equations for balance of an a.c. bridge.Equation ZZ4 = ZZ3  is convenient to use when dealing with series elements of a bridge while Equation Y1 Y4 = YY3 is useful when dealing with parallel elements.Equation ZZ4 = ZZ3 states that the product of impedances of one pair opposite arms must equal the product of impedances of the other pair of opposite arms expressed in complex notation.This means that both magnitudes and the phase angles of the impedances must be taken into account. 
           Considering the polar form, the impedance can be written as Z = Z∠θ, where Z represents the magnitude and θ represents the phase angle of the complex impedance.Now that equatin can be re-written in the form 

                              (Z1∠θ1)(Z4∠θ4) = (Z2∠θ2)(Z3∠θ3)  

Thus for balance, we must have ,

                         Z1 Z4 ∠θ1 + θ4 = Z2 Z3 ∠θ2 + θ3 

       The above equation shows that two conditions must be satisfied simultaneously when balancing an a.c. bridge.The first condition is that the magnitude of impedances satisfy the relationship :
                                          ZZ4 = ZZ3

The second condition is that the phase angles of impedances satisfy the relationship : 

                                 ∠θ1 + θ4 = ∠θ2 + θ3 

           The phase angles are positive for an inductive impedance and negative for capacitive impedance.

If we work in terms of rectangular co-ordinates, we have 

                     Z1= R1 + jX1 ;                       Z2= R2 + jX2 
                    Z3= R3 + jX3            and         Z4= R4 + jX4 

For balance,             ZZ4 = ZZ3 

or     (R1 + jX1)(R4 + jX4) = (R2 + jX2)(R3 + jX3

    or    R1 R4 - XX+ j(X1R4 + X4R1= RR3 - XX4 + j(X2R3 + X3R2
         The above equation is a complex equation and a complex equation is satisfied only if real and imaginary parts of each side of the equation are separately equal. Thus, for balance, 

                             R1 R4 - XX4 = RR3 - XX4 
                               X1R4 + X4R1 = X2R3 + X3R2 

        Thus there are two independent conditions for balance and both of them must be satisfied for the bridge to be balanced. 

General Form of A.C Bridges:

           As an example let us consider, the bridge circuit in the below figure. R3 and R4 are non-inductive resistances, L1 and L2 are inductances of the negligible resistance and R1 and R2 are non-inductive resistors.Therefore at balance, 

                                     ZZ4 = ZZ3 

or                  (R1 + jωL1) R4 = (R2 + jωL2) R3 

Equating the real and imaginary parts separately, we have, 


           Thus if L1 and R1 are unknown, the above bridge may be used to measure these quantities in terms of R2, R3, R4 and L2.We may deduce several important conclusions from the above simple example.They are : 

1.Two balance equations are always obtained for an a.c. bridge circuit.This follows from the fact that for balance in an a.c. bridge, both magnitude and phase relationships must be satisfied. This requires that real and imaginary terms must be separated, which give two equations to be satisfied for balance.

2.The two balance equations enable us to know two unknown quantities.The two quantities are usually a resistance and an inductance or a capacitance. 

3.In order to satisfy both conditions for balance and for convenience of manipulation, the bridge must contain two variable elements in its configuration. For greatest convenience,each of the balance equations must contain one variable element, and one only. The equations are then said to be independent. 
       In the bridge of below figure, R2 and L2 are obvious choice as variable elements since L2 does not appear in the expression for R1, and R2 does not figure in the expression for L1 and hence the two balance equations are independent. The technique of balancing is to adjust L2 till a minimum indication is obtained on the detector, then to adjust R2 until a new smaller minimum indication is obtained. Then L2 and R2 are alternately adjusted until the detector shows no indication. 
             The process of alternate manipulation of two variable elements is rather typical of the general balancing procedures adopted in most a.c. bridges. When two variables are chosen such that the two balance equations are no longer independent, the bridge has a very poor convergence of balance and gives the effect of sliding balance. The term Sliding Balance describes a condition of interaction between the two controls. Thus when we balance with R2, then go to R3 and back to R2 for adjustment, we find a new apparent balance. 
           Thus the balance point appears to move, or slide and settles only gradually to its final point after many adjustments.It may be emphasised here that in case the two balance conditions are independent, not more thane two or three adjustments of the variable elements would be necessary to obtain balance. In case we choose the two variable components such that the two equations are not independent the balance procedure becomes laborious and time-consuming. 
           For example, if we choose R2 and R3 as variable elements, the two equations are no longer independent since R3 appears in both the equations. There are two adjustments, one resistive and the other reactive that must be made to secure balance.For the usual magnitude responsive detector, these adjustments must be made alternately until they converge on the balance point. The convergence to balance point is best when both the variable elements are in the same arms. 

4.In this bridge circuit balance equations are independent of frequency. This is often a considerable advantage in an a.c. bridge, for the exact value of the source frequency need not then be known. Also, if a bridge is balanced for a fundamental frequency it should also be balanced for any harmonic and the wave-form of the source need not be perfectly sinusoidal. On the other hand, it must be realized that the effective inductance and resistance for example, of a coil, vary with frequency, so that a bridge balanced at a fundamental frequency is never, in practice, truly balanced for the harmonics.
             To minimize difficulties due to this the source wave form should be good, and it is often an advantage to use a detector tuned to the fundamental frequency.Further while the disappearance of the frequency factor is of advantage in many bridges, some bridges derive their usefulness from the presence of a frequency factor ; such bridges must then be supplied from a source with very good wave-form and high frequency stability.Alternatively, they may be used to determine frequency.

      In this post we have learnt What are ac bridges & General form of a.c bridges.You can download this article as pdf,ppt.If you have queries you can mail us @  Comment below if you have any queries!