Multiloop Circuits — Question 10

Question 10

A passive junction potential is a conductance-weighted mean

A central node X connects only through positive resistors R1,R2,R3 to nodes held at V1,V2,V3. No independent current is injected into X.

1. Derive VX and prove that it lies between the smallest and largest boundary potentials.

2. For boundary potentials 12.0 V, 0 and -3.00 V with resistors 2.00, 3.00 and 6.00 ohm respectively, find VX and all currents directed outward from X.

Question 10 - Solution

A passive junction potential is a conductance-weighted mean

(a) Method and calculation

$$\sum_{k=1}^3\frac{V_X-V_k}{R_k}=0\Rightarrow V_X=\frac{\sum_kG_kV_k}{\sum_kG_k},\quad G_k=1/R_k>0.$$$$V_{\min}\sum G_k\le\sum G_kV_k\le V_{\max}\sum G_k\Rightarrow V_{\min}\le V_X\le V_{\max}.$$

(b) Interpretation and further calculation

$$V_X=\frac{12/2+0/3-3/6}{1/2+1/3+1/6}=5.50\,\mathrm V.$$$$I_1=-13/4\,\mathrm A,\ I_2=11/6\,\mathrm A,\ I_3=17/12\,\mathrm A,\quad \sum I_k=0.$$

Physical check

A passive resistive node cannot exceed every neighboring potential without violating current conservation.