# Hay′s Bridge Circuit Theory Phasor Diagram Advantages Applications

Published on 24/2/2012 & updated on 16/8/2018## Hay's Bridge Applications

Before we introduce**Hay's bridge**let us recall the limitations of Maxwell bridge, in order to understand what is the necessity of

**Hay's Bridge Applications**. Maxwell bridge is only suitable for measuring medium quality factor coils however it is not suitable for measuring high quality factor (Q > 10). In order to to overcome from this limitation we need to do modification in Maxwell bridge so that it will become suitable for measuring Q factor over a wide range. This modified Maxwell bridge is known as Hay's bridge.

## Hay's Bridge Theory

As I said earlier that**Hay's bridge**is modified Maxwell bridge, now question arises here in our mind that where weneed to do modification. In order to to understand this, let us consider the connection diagram given below:

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In this bridge the electrical resistance is connected in series with the standard capacitor. Here l_{1} is unknown inductor connected in series with resistance r_{1}. c_{4} is standard capacitor and r_{2}, r_{3}, r_{4} are pure electrical resistance forming other arms of the bridge.
From the theory of ac bridge we can write at balance point,

Substituting the values of z_{1}, z_{2}, z_{3} and z_{4} in equation (1) we get,

Now, Q factor of a coil is given by
The equations (4) and (5) are dependent on the source frequency hence, in order to find the accurate value of l_{1} and r_{1} we should know the correct value of source frequency.
Let us rewrite the expression for l_{1},
Now if we substitute Q >10 then 1/Q^{2} = 1 / 100 and hence we can neglect this value, thus neglecting 1/Q^{2} we get r_{2}r_{3}c_{4} which is same as we have obtained in Maxwell bridge hence **Hay's bridge circuit** is most suitable for high inductor measurement.
Let us know more about **Hay's bridge circuit** diagram of Hay's bridge that will be very useful in understanding the **Hay's bridge phasor diagram**. A meter is connected between points b and d of the bridge. The arm ab consists of resistance r_{1} and inductor, l_{1} (total drop across this is e_{1}) and arm ad consists of pure resistance r_{2} (total drop across this is e_{2}). The arm bc consists of pure resistance making a drop of e_{3} while the arm cd consists of resistor r_{4} and a capacitor making the total drop of e_{4}. Now let us draw **phasor diagram of Hay's bridge**, at null point e_{1} must be equal to e_{2} and also e_{3} must be equal to e_{4} as the current flow through bd is zero. Let us take i_{1} as the reference axis and thus current i_{2} leads by i_{1} by some angle (as shown in **Hay's bridge phasor diagram** below) because a capacitor is connected in branch cd making current i_{2} lead by i_{1}. Let us mark e_{1} and e_{2} and the resultant of e_{1} and e_{2} of course equal to e. The phase difference between the voltage drop across the electrical resistance r_{4} and >capacitor c_{4} is 90^{o} (measured in degrees) is clearly shown in the phasor diagram of Hay's Bridge.
In many practical situations bridge circuit that uses ac source, an electron ray indicating is utilized to find out the balance condition by opening and closing the shadow area of the tube. For audible balance headsets are used, but accuracy is reduces due to this method. In order to control the operating power to the bridge and to complete the detector circuit various switches are used. Many times the two switching functions are merge into single key, known bridge key this is to ensure that power applied to the bridge before to the detector circuit, due this sequence the effect of inductor and capacitance is reduced.

## Advantages of Hay's Bridge

The bridge gives very simple expression for the calculation of unknown inductor of high value. The Hay's bridge require low value of r_{4}while Maxwell bridge requires high value of r

_{4}. Now let us analyse why should put low value of r

_{4}in this bridge: Consider the expression of quality factor, As r

_{4}presents in the denominator hence for high quality factor, r

_{4}must be small.

### Disadvantages of Hay's Bridge

**Hay's bridge**is not suitable for measurement of quality factor (Q<10) for Q<10 we should use Maxwell bridge.

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