Refer to the four circuits shown. Which one of the following options for k1, k2, k3, and k4 makes all of them realizable?

The correct option that makes all the four circuits realizable is k1=2,k2=0.5,k3=–2/3,k4=–3.

To understand why this option provides a consistent set of parameters for the given circuit network, we analyze each of the four circuits step-by-step, evaluating the relations between the independent and dependent sources:

1. Analysis of Circuit 1 (Top-Left):
This circuit consists of an independent voltage source $V_D$ on the left and a dependent voltage source $V_X=k_1{V}_D$ on the right.
For this circuit configuration to satisfy the voltage requirements, the parameter $k_1$ is set to:
$k_1=2$
Substituting this value gives the relationship:
$V_X=2V_D$

2. Analysis of Circuit 2 (Top-Right):
This circuit consists of an independent voltage source $V_D$ on the left and a dependent current source $I_X=0.5k_2{V}_D$ on the right.
Substituting the given parameter $k_2=0.5$ into the equation:
$I_X=0.5\times 0.5V_D=0.25V_D$
This defines a valid current-to-voltage relationship, ensuring that the circuit is fully realizable.

3. Analysis of Circuit 3 (Bottom-Left):
This circuit consists of an independent current source $I_D$ on the left and a dependent voltage source $V_X=k_3{I}_D$ on the right.
Using the parameter $k_3=-2/3$, we obtain:
$V_X=-\frac{2}{3}{I}_D$
This defines a consistent voltage-to-current ratio, maintaining realizability under the specified source conditions.

4. Analysis of Circuit 4 (Bottom-Right):
This circuit consists of an independent current source $I_D$ on the left and a dependent current source $I_X=3k_4{I}_D$ on the right.
Substituting the parameter $k_4=-3$ into the equation yields:
$I_X=3\times (-3)I_D=-9I_D$
This provides the current balance between the two parallel branches, ensuring that nodal equations are satisfied.

By applying the set of values $k_1=2$, $k_2=0.5$, $k_3=-2/3$, and $k_4=-3$, all constraints for the four dependent and independent sources are defined, satisfying the criteria for overall network realizability.