Multiloop Circuits — Question 8

Question 8

Resistance matching eliminates a bridge branch without changing terminal current

A bridge has R1 from A to M, R2 from M to B, R3 from A to N, R4 from N to B, and any positive Rb from M to N. A voltage E is applied from B to A.

1. Derive the resistance-ratio condition for zero current through Rb.

2. At balance, derive the equivalent resistance seen by the source and show why it is independent of Rb.

Question 8 - Solution

Resistance matching eliminates a bridge branch without changing terminal current

(a) Method and calculation

$$V_M-V_B=E\frac{R_2}{R_1+R_2},\quad V_N-V_B=E\frac{R_4}{R_3+R_4}.$$$$V_M=V_N\Longleftrightarrow R_2R_3=R_1R_4\Longleftrightarrow R_1/R_2=R_3/R_4.$$

(b) Interpretation and further calculation

$$I_b=(V_M-V_N)/R_b=0,\quad I_s=\frac E{R_1+R_2}+\frac E{R_3+R_4}.$$$$R_{\mathrm{eq}}=\frac{(R_1+R_2)(R_3+R_4)}{R_1+R_2+R_3+R_4}.$$

Physical check

At balance, varying a finite positive bridge resistance changes neither node potential nor source current.