# Why is the voltage across a dependent voltage source connected to an open circuit not zero?

I have the circuit shown below. I need to find the Thevenin voltage for it. If the dependent voltage source was a resistor instead, I think I could have disregarded it when using KVL (since the current equals zero). According to the solutions manual, however, I need to consider the dependent voltage source when using KVL, even though there is no current through it. Can someone explain why this is correct?

simulate this circuit – Schematic created using CircuitLab

• This is not a dependent current source, but a dependent voltage source where the voltage depends on the value of a given current. – Roger C. Mar 11 '15 at 19:11
• But is it possible to have voltage when the wire is open and there is no current in it? – Ali Mustafa Mar 11 '15 at 19:13
• Yes, it's possible. – Null Mar 11 '15 at 19:13

If the dependent voltage source was a resistor there would be no current through it since one end of it is connected to an open circuit (at node a). A resistor with no current through it has no voltage through it (since $V = IR$), so a resistor would not affect $V_{\text{TH}}$.
But the dependent voltage source has a non-zero voltage because its value is $30\times 10^3 i_0$ (where $i_0$ is the current through the upper resistor), and $i_0$ is non-zero.
There is still no current through the dependent source because it is connected to an open circuit at node a. That means the current through both resistors ($i_0$) is the same: $$i_0 = -\frac{100\text{V}}{20\text{k}\Omega + 80\text{k}\Omega} = -1\text{mA}$$
$i_0$ is negative based on the direction indicated in the circuit. The dependent voltage source's voltage is therefore $$V_D = 30\times 10^3 i_0 = -30\text{V}$$
This needs to be added to the voltage at the node between the resistors (which is a simple voltage divider) to calculate $V_{\text{TH}}$.
You can have voltage without current. That's what an open circuit is, actually. $i_o$ is non-zero due to the independent voltage source, so $30000*i_o$ is also non-zero.