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## Q. 1.14

Consider the following circuit involving a total of eight components.Find $v_{x}$.

## Verified Solution

We again label the nodes, from 1 to 4, and define the current directions. Node 2 is defined as the combination of three intersection points

Applying KCL at nodes 3 and 2, we obtain
• KCL(3): $2 + i_{w} − 4 = 0 \longrightarrow i_{w} = 2$,
• KCL(2):$i_{s} − i_{z} − i_{w} + 4 + i_{y} = 0 \longrightarrow i_{s} + i_{y} − i_{z} = −2$.
Then, using KVL, we derive
• KVL(1 → 2 → 4 → 1): $4i_{s} + 7i_{z} = 10$,
• KVL(2 → 3 → 4 → 2): $20i_{w} − 2 × 6 − 7i_{z} = 0 \longrightarrow i_{z}= 4A$.

Using the updated information, we can also find $i_{s}$, as well as $i_{y}$ as
$4i_{s} = 10 − 7i_{z} = −18 \longrightarrow i_{s} = −9∕2 A$,
$i_{y} = −2 − i_{s} + i_{z} = −2 + 9∕2 + 4 = 13∕2 A$.
Finally, we apply KVL(1 → 2 → 1) to find $v_{x}$ as
$−v_{x} + 2i_{y} − 4i_{s} = 0 \longrightarrow v_{x} = 2i_{y} − 4i_{s} = 13 + 18 = 31 V$.