Question:
Published on: 1 October, 2023

Explain the worming of Anderson’s bridge with a neat sketch. Derive the required expression for obtaining the unknown inductance.

By using this bridge, the unknown inductance is measured in terms of a known capacitance and resistance.

Let, $$L_1$$ = Self-inductance to be measured

$$R_1$$ = Resistance of self-inductor

$$C$$ = Fixed Capacitor

$$R_1$$ = Resistance connected in series with self-inductor

$$R_4, R_3, R_2, r$$ = Known non-inductive resistances.

At balance, $$I_1 = I_3$$ and $$I_2 = I_C+I_4$$

$$I_1R_3=I_C\times\frac{1}{j\omega C}$$

$$I_C=I_1R_3j\omega C$$

Writing other balance equations,

$$I_1\left(r_1+R_1+j\omega L_1\right)=I_2R_2+I_Cr$$ ........(1)

and, $$I_C\left(r+\frac{1}{j\omega C}\right)=\left(I_2-I_C\right)R_4$$ .........(2)

Putting the value of $${I}_C$$ in equation $$\left(1\right)$$, we get,

$$I_1\left(r_1+R_1+j\omega L_1\right)=I_2R_2+I_1R_3j\omega Cr$$

$$I_1\left(r_1+R_1+j\omega L_1-R_3j\omega Cr\right)=I_2R_2$$ ........(3)

Putting the value of $$I_C$$ in equation (2)

$$I_1R_3j\omega C\left(r+\frac{1}{j\omega C}\right)=\left(I_2-I_1R_3j\omega C\right)R_4$$

$$I_1\left(R_3j\omega Cr+R_3j\omega C R_4+R_3\ \right)=I_2R_4$$ .........(4)

From equation (3) & (4) we get,

$$I_1\left(r_1+R_1+j\omega L_1-R_3j\omega Cr\right)=I_1\left(\frac{R_2R_3}{R_4}+\frac{j\omega C R_2R_3r}{R_4}+j\omega C R_2R_3\right)$$

Equating the real and imaginary parts $$R_1=\frac{R_2R_3}{R_4}-r_1$$

$$L_1=c\frac{R_3}{R_4}[r\left(\ R_4+R_2\right)+R_2R_4]$$

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