In a circuit containing a dependent current source of value β times a voltage V across a resistor, how would you set up nodal equations?

In a circuit containing a dependent current source of value β times a voltage V across a resistor, how would you set up nodal equations?

• A. Solve using only KVL and ignore nodal equations.
• B. Replace the dependent current source with an equivalent independent current source of βV.
• C. Include a term βV in the KCL at the node as the current contributed by the dependent source; relate V to circuit voltages accordingly.
• D. Write KCL at the ground node only and omit the dependent source.

Answer: (C)

In nodal analysis with a dependent current source, the current produced by that source must appear in the KCL equation at the node(s) it connects, just like any other current. Here the current is β times a voltage V, so you include a term βV in the KCL at the node where the dependent source injects or draws current. The controlling quantity V is not a fixed value—it’s the voltage across a particular resistor—so you express V in terms of node voltages (for example, V equals the difference between the two node potentials across that resistor). Then you substitute that expression into βV and solve the resulting set of equations for the unknown node voltages. This approach is necessary because you can’t rely on KVL alone to capture the behavior of currents through sources, and you can’t replace a dependent source with a fixed independent source since its value changes with V. Omitting the dependent source or writing equations only at ground would miss the current that the source contributes to the network and leave you unable to solve for the node voltages correctly.

In nodal analysis with a dependent current source, the current produced by that source must appear in the KCL equation at the node(s) it connects, just like any other current. Here the current is β times a voltage V, so you include a term βV in the KCL at the node where the dependent source injects or draws current. The controlling quantity V is not a fixed value—it’s the voltage across a particular resistor—so you express V in terms of node voltages (for example, V equals the difference between the two node potentials across that resistor). Then you substitute that expression into βV and solve the resulting set of equations for the unknown node voltages.

This approach is necessary because you can’t rely on KVL alone to capture the behavior of currents through sources, and you can’t replace a dependent source with a fixed independent source since its value changes with V. Omitting the dependent source or writing equations only at ground would miss the current that the source contributes to the network and leave you unable to solve for the node voltages correctly.