Refer to the four circuits shown.
Which one of the following options for \(k_1\), \(k_2\), \(k_3\), and \(k_4\) makes all of them realizable?
Show Hint
Look at which pairs of sources sit between the same two nodes: same-type sources, V-V or I-I, in parallel must match, while mixed V-I pairs never conflict.
Step 1: Redraw what "parallel" really means here.
Since the top and bottom of each rectangle are plain connecting wires, the left branch and right branch of every circuit share the same pair of nodes. That is exactly the definition of a parallel connection, even though the picture looks like a single loop.
Step 2: State the rule for each source combination.
Put two voltage sources across the same pair of nodes and their values must agree, or KVL fails. Put two current sources feeding the same node in the same direction and their sum must be zero, or KCL fails. Mix a voltage source with a current source and neither law is threatened, because the voltage source simply supplies whatever current is needed.
Step 3: Circuit 1.
$V_D$ and $V_X=k_1V_D$ are both voltage sources with the same polarity across the same nodes, so $V_D=k_1V_D$, giving $k_1=1$.
Step 4: Circuits 2 and 3.
Each of these pairs one voltage source with one current source. No equation is forced on $k_2$ or $k_3$, since the voltage source adjusts its own current freely. Both circuits work for any value of $k_2$ and $k_3$.
Step 5: Circuit 4.
$I_D$ and $I_X=3k_4I_D$ both push current into the same node in the same direction, so KCL needs $I_D+I_X=0$, that is $I_D+3k_4I_D=0$, giving $k_4=-\dfrac{1}{3}$.
Step 6: Put it together.
The only combination that keeps every circuit consistent is $k_1=1$ and $k_4=-\dfrac{1}{3}$, with $k_2$ and $k_3$ completely free.
\[ \boxed{k_1=1,\ k_4=-\tfrac{1}{3},\ \text{for all }k_2,k_3} \]
