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## Q. 2.20

Consider the following circuit involving a current-dependent current source, a voltage source, and two resistors Find $i_{x}$, which flows through the voltage source. ## Verified Solution

This circuit can be analyzed using KCL and KVL, as usual. After labeling
nodes and defining directions, we have the following circuit.

Then, applying KCL and KVL, we derive
• KCL(1): $i_{x} + 47i_{x} − i_{y} = 0 \longrightarrow i_{y} = 48i_{x}$,
• KVL(1 → 2 → 3 → 1): $v_{y} − 2 + v_{x} = 0 \longrightarrow v_{x} + v_{y} = 2$.
Finally, using Ohm’s law,
$100i_{x} + 50i_{y} = 2 \longrightarrow 2500i_{x} = 2 \longrightarrow i_{x} = 2∕2500 A$.
As an alternative solution, we now apply nodal analysis, selecting node 2 as the reference node.

Using Ohm’s law, we have $i_{x} = (2 − v_{1})∕100$.Then, using KCL,

• KCL(1): $48(2 − v_{1})∕100 − v_{1}∕50 = 0 \longrightarrow 96 − 48v_{1} − 2v_{1} = 0$,

leading to $v_{1} = 48∕25V$.Therefore, $i_{x} = (2 − 48∕25)∕100 = 2∕2500A$.

We note that only
one KCL equation has been sufficient to analyze the circuit. Specifically, two of the nodes in the earlier analysis are not used in nodal analysis. Some important points are as
follows.
• Node 2 is made the reference node (ground) with zero voltage. In general, KCL need not be applied at a ground. In fact, needing to apply KCL at a ground
is usually an indicator that another node has been skipped by mistake in the analysis.
• Node 3 is also not used directly in nodal analysis, because its voltage is already known
due to the voltage source. In general, if the voltage at a node is easily defined, one does not need to write a KCL equation at that node. In fact, applying KCL at a node with a directly connected voltage source should be avoided, unless it is mandatory (e.g., if one must find the current through the voltage source).
• Application of KVL should be avoided in nodal analysis since a proper set of KCL
equations should be sufficient to solve the circuit. KVL can be used to find other quantities after all node voltages are obtained.
Two facts provide a deeper understanding of nodal analysis.
• Setting zero voltage at the ground is merely a choice. Indeed, one could assign any voltage (e.g., 10V), which would shift voltage values at all other nodes by 10. On the other hand, real quantities, such as component voltages, currents, and powers, do not depend on this selection.
• Selecting a node as ground is also completely arbitrary. One can choose any node as
a reference, provided that the voltages are defined accordingly. As mentioned above,
certain selections (e.g., choosing nodes with more connections) can simplify nodal analysis.  