The Maxwell’s bridge or Maxwell’s Wein bridge is an AC bridge used to find or measure the unknown self-inductance. There are different types of ac bridges used for the measurement of self-inductance in which Maxwell’s bridge is the most commonly used bridge. The bridge uses the principle of null-deflection i.e., by balancing the impedances on the bridge arms. The equation obtained when the bridge is balanced (i.e., ratio of impedances are balanced) can be used to determine the unknown inductance connected to it.

#### There are two types of Maxwell's bridges used to find unknown inductance,- Maxwell’s inductance bridge
- Maxwell’s inductance capacitance bridge.

## Maxwell’s Inductance Bridge :

In Maxwell's inductance bridge the unknown self-inductance to be measured is compared with the known inductance. Hence the unknown self-inductance and internal resistor of an inductor can be measured with Maxwell's inductance bridge. The circuit diagram is shown in the below figure.

#### Let,- L
_{1} = Unknown inductance to be measured - R
_{1} = Resistance of the Unknown inductance - R
_{3}, & R_{4} = Standard non-inductive resistances - R
_{2} = Standard Variable resistance - L
_{2} = Standard Variable inductance with fixed resistance r_{2}.

From the above figure, the impedances of the respective arms are given as,
Under balanced condition (i.e., when detector shows null deflection), we have,
On equating the real and imaginary parts on both sides, we get,
Hence, the unknown self-inductance and resistance of the inductor are obtained in terms of known standard values. Also, both the equations are independent of frequency term.

_{1}= Unknown inductance to be measured_{1}= Resistance of the Unknown inductance_{3}, & R_{4}= Standard non-inductive resistances_{2}= Standard Variable resistance_{2}= Standard Variable inductance with fixed resistance r_{2}.### Phasor Diagram of Maxwell’s Inductance Bridge :

In Maxwell's inductance bridge, the resistance R_{2} is the decade resistance box. The resistor R_{3} and R_{4} can have their values as from 10, 100, 1000, and 10,000 Ω by the successive adjustment of L_{2} and R_{2}. The balance of the bridge can be adjusted. The phasor diagram of Maxwell's inductance bridge under balance condition is shown below.

## Maxwell’s Inductance Capacitance Bridge :

Maxwell’s inductance capacitance bridge is also similar to Maxwell’s inductance bridge seen above. But here the unknown self-inductance to be determined is compared with the standard known capacitor. The circuit diagram for Maxwell’s inductance capacitance bridge is shown in the below figure.

#### The advantages of using a standard capacitor in Maxwell's bridge are as follows,- The standard capacitors used in Maxwell's bridge are of low cost compared to the stable and accurate standard inductors.
- These capacitors are small in size.
- The capacitors used in this bridge are lossless capacitors as there is less possibility of losing energy.
- The capacitor is independent of the external fields. Whereas, the stray magnetic fields can be eliminated by providing proper shielding on the standard inductor.
- In Maxwell's inductance bridge, there is a requirement of adjusting the current flow through the inductor to show the rated value of inductance.

Let,- L
_{1} = Unknown inductance to be measured - R
_{1} = Resistance of the unknown inductor - R
_{2}, R_{3} = Standard non-inductive resistances - R
_{4} = Standard non-inductive variable resistance - C
_{4} = Standard variable capacitor.

From the above figure, the impedances of the respective arms are given as,
Under balanced condition (i.e., when detector shows null deflection), we have,
Equating the real and imaginary terms on both sides, we get,
Thus, the bridge can be balanced by varying R_{4} and C_{4}. The Q-factor or storage factor of the inductor is given by,

_{1}= Unknown inductance to be measured_{1}= Resistance of the unknown inductor_{2}, R_{3}= Standard non-inductive resistances_{4}= Standard non-inductive variable resistance_{4}= Standard variable capacitor.From the above expression for Q factor obtained from Maxwell's inductance capacitance bridge. The value of the Q factor is proportional to the product of R_{4} and C_{4}. We know that, usually, the value of C_{4} will be in μF or pF.

In order to have a greater value of Q factor, R_{4} must be in megaohms or greater than that, so that the product is high. Resistance of such a high value is very difficult to obtain and also very costly. Hence, the measurement of inductance using Maxwell's inductance-capacitance bridge is limited to low range values of Q i.e., between 1 and 10. Hay's bridge is used for high Q factor coils.

The below shows the phasor diagram for Maxwell's inductance capacitance bridge.

Also, from the equations of L_{1}, it is clear that L_{1} is the product of R_{2}, R_{3}, and C_{4}. The value of C_{4} is usually in μF. Now if the value of R_{2} and R_{3} is selected such that R_{2} R_{3} is 10^{6} then L_{1} = 10^{6} × C_{4} × 10^{-6} = C_{4}. Hence, the dial of the variable capacitor C_{4} will directly give the value of inductance.

### Advantages of Maxwell's Bridge :

- The calculated values of R
_{1} and L_{1} are independent of C_{4} and R_{4} respectively. Hence, there is no effect on R_{1} and L_{1} by choosing R_{4} and C_{4} as variable elements. - The stray magnetic fields do not disturb the balance, as the variable element used here is the capacitor.
- The balance equations do not contain the frequency terms and hence the fluctuations in frequency do not disturb the operation or balancing of bridge.
- As both R
_{1} and L_{1} depend upon the product of R_{2} and R_{3}, their product can be adjusted conveniently for any inductance value. - Calculation of Q-factor is very simple and easy.
- Very wide range of inductance can be measured not only at power frequencies but also at audio frequencies.

_{1}and L_{1}are independent of C_{4}and R_{4}respectively. Hence, there is no effect on R_{1}and L_{1}by choosing R_{4}and C_{4}as variable elements._{1}and L_{1}depend upon the product of R_{2}and R_{3}, their product can be adjusted conveniently for any inductance value.### Disadvantages of Maxwell's Bridge :

- The use of a standard variable capacitor makes the bridge highly expensive. However, this can be overcome by using a fixed capacitor and varying R
_{4} and an additional resistor in series with the inductance to be measured. - The bridge is limited for the measurement of medium Q coils only i.e., Q value between 1 and 10.

_{4}and an additional resistor in series with the inductance to be measured.