# Maxwell Bridge Inductance Capacitance Bridge

This bridge is used to find out the self inductor and the quality factor of the circuit. As it is based on the bridge method (i.e. works on the principle of null deflection method), it gives very accurate results. **Maxwell bridge** is an AC bridge so before going in further detail let us know more about the ac bridge.

## AC Bridges

AC Bridges consist of a source, balance detector and four arms. In AC bridges, all the four arms consists of impedance. The AC bridges are formed by replacing the DC battery with an AC source and galvanometer by detector of Wheatstone bridge.They are highly useful to find out inductance, capacitance, storage factor, dissipation factor etc.Now let us derive general expression for an AC bridge balance

Figure given below shows AC bridge network:

Here Z_{1}, Z_{2}, Z_{3} and Z_{4} are the arms of the bridge.

Now at the balance condition, the potential difference between b and d must be zero. From this, when the voltage drop from from a to d equals to drop from a to b both in magnitude and phase.

Thus, we have from figure e_{1} = e_{2}

[math] i_1\cdot Z_1\;=\;i_2\cdot Z_2\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot(1)[/math] [math] i_1\;=\;i_2\;=\frac{e}{Z_1\;+\;Z_3}\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot(2)[/math] [math] i_2\;=\;i_4\;=\frac{e}{Z_2\;+\;Z_4}\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot(3)[/math]

From equation 1, 2 and 3 we have Z_{1}.Z_{4} = Z_{2}.Z_{3} and when impedance are replaced by admittance, we have Y_{1}.Y_{4} = Y_{2}.Y_{3}.

Now consider the basic form of an AC bridge. Suppose we have bridge circuit as shown below,

In this circuit R_{3} and R_{4} are pure electrical resistances. Putting the value of Z_{1}, Z_{2}, Z_{3} and Z_{4} in the equation that we have derived above for AC bridge.

[math] We\; get\;(r_1\;+\;j\omega l_1)\cdot R_4\;=\;(r_2\;+\;j\omega l_2)\cdot R_3[/math] Now equating the real and imaginary parts we get

[math] r_1\;=\;\frac{R_3}{R_4}\cdot r_2\; and\; l_1\;=\;\frac{R_3}{R_4}\cdot l_2[/math] Following are the important conclusions that can be drawn from the above equations: (a) We get two balanced equations that are obtained by equating real and imaginary parts this means that for an ac bridge both the relation (i.e.magnitude and phase) must be satisfied at the same time. Both the equations are said to be independent if and only if both equation contain single variable element. This variable can be inductor or resistor. (b) The above equations are independent of frequency that means we do not require exact frequency of the source voltage and also the applied source voltage waveform need not to be perfectly sinusoidal.

## Maxwell's Bridge

Under this we going to study about the following (a)**Maxwell's inductor bridge**(b)

**Maxwell's inductor capacitance bridge**

### Maxwell's Inductance Bridge

Let us now discuss **Maxwell's inductor bridge**. The figure shows the circuit diagram of Maxwell's inductor bridge.
In this bridge the arms bc and cd are purely resistive while the phase balance depends on the arms ab and ad.

Here l_{1} = unknown inductor of r_{1}.
l_{2} = variable inductor of resistance R_{2}.
r_{2} = variable electrical resistance.
As we have discussed in ac bridge according to balance condition, we have at balance point

[math] l_1\;=\;\frac{r_3}{r_4}\cdot l_2\;and\;r_1\;=\;\frac{r_3}{r_4}(r_2\;+\;R_2)[/math]
We can vary R_{3} and R_{4} from 10 ohms to 10,000 ohms with the help of resistance box.

### Maxwell's Inductance Capacitance Bridge

In this**Maxwell Bridge**, the unknown inductor is measured by the standard variable capacitor. Circuit of this bridge is given below, Here, l

_{1}is unknown inductance, C

_{4}is standard capacitor. Now under balance conditions we have from ac bridge that Z

_{1}.Z

_{4}= Z

_{2}.Z

_{3}

[math] (r_1\;+\;j\omega l_1)\frac{r_4}{1\;+\;j\omega C_4r_4}\;=\;r_2\cdot r_3[/math] [math] r_1\cdot r_4\;+\;j\omega l_1\cdot r_4\;=\;r_2\cdot r_3\;+\;j\omega r_2r_3C_4r_4[/math] Let us separate the real and imaginary parts, the we have,

[math] r_1\;=\;r_2\cdot \frac{r_3}{r_4}\;and\;l_1\;=\;r_2\cdot r_3\cdot C_4[/math] Now the quality factor is given by,

[math] Q\;=\;\frac{\omega l_1}{r_1}\;=\;\omega C_4\cdot r_4[/math]

### Advantages of Maxwell's Bridge

(1) The frequency does not appear in the final expression of both equations, hence it is independent of frequency. (2)**Maxwell's inductor capacitance bridge**is very useful for the wide range of measurement of inductor at audio frequencies.

### Disadvantages of Maxwell's Bridge

(1) The variable standard capacitor is very expensive. (2) The bridge is limited to measurement of low quality coils (1 < Q < 10) and it is also unsuitable for low value of Q (i.e. Q < 1) from this we conclude that a Maxwell bridge is used suitable only for medium Q coils.The above all limitations are overcome by the modified bridge which is known as Hey's bridge which does not use an electrical resistance in parallel with the capacitor.