Expressing dependent current source voltage using KVL

I am reading about mesh and node equations, and the book says that the voltage between $V_1$ and $V_2$ can be expressed as: $$10\cdot I = V_2 - V_1$$

How is this possible? And is there a general formula for expressing the voltage across the dependent current source? It says that the controlling current is expressed as: $$I = \dfrac{V_s - V_1}{R_1}$$

(which I do understand as per ohm's law). But for the dependent source, there is no resistance. I know if you are using mesh analysis, you can assign an arbitrary resistance, say Ro, which will factor out of both mesh equations that contain a current source and it is therefore not an issue.

Also, if I applied KCL to the supernode, would it look like:

$$I = \dfrac{V_1}{Z_1} + \dfrac{V_2}{Z_2}$$

• I don't follow what your issue is? Are you confused that 10*I could be a voltage (because it's just 10 times a current)? If you're confused about that, just understand that a current controlled voltage device takes a current and outputs a voltage. You shouldn't think of controlled voltage/current devices as resistors as they usually act like voltage/current sources usually rather than sinks. Sep 25, 2014 at 18:46
• @horta: I was confused and thought that it was a current source, but actually it's a voltage source and then the voltage is obviously V2-V1. And your latter statement helps that I should not think of it as a resistor. Sep 26, 2014 at 13:06

How is this possible?

(1) it's a voltage source so the voltage $V_2 - V_1$ is fixed by this source.

(2) it's a current controlled voltage source so the 'gain' is a transresistance - the gain has units of ohms.

Transresistance is essentially from transfer resistance. If the controlling current were in fact the current through the source, the source would be indistinguishable from resistor.

Since the controlling current is elsewhere, the voltage across the source is proportional to a current elsewhere thus, transfer resistance.

And is there a general formula for expressing the voltage across the dependent current source?

To be sure, this isn't a current source (see point 1 above), it's a voltage source controlled by a current.

If it were a dependent current source, the voltage across is not fixed by the source but, rather, by the attached circuit. In other words, there is no general formula to express the voltage across a current source in terms of the current through.

• So if this circuit was existing, it could have the current controlled voltage source which has a gain that is a transresistance? If you could explain, how physically does that dependent source have it's voltage controlled by the current elsewhere in the circuit? Or is it just a consequence of the relationships existing between the voltage and current in that specific circuit? Sep 26, 2014 at 13:08
• @Prevost, first realize that these ideal circuit elements are combined to form mathematical models for physical circuit elements. Thus, a physical transresistance amplifier can be modelled with a CCVS along with some resistors and a capacitor or two. Sep 26, 2014 at 16:08

Call the mesh on the left mesh 1 and the one on the right mesh 2, with directions both clockwise.

When writing KVL equations, you need to express the voltage in each branch in terms of the mesh currents.

Now your I variable is just the same as I1, the current in mesh 1. So it's easy to write the voltage of the dependent source in terms of mesh currents.

$$V_{\mathrm{CCVS}} = 10 I_1$$

Your KVL equation for mesh 2 is

$$I_2 Z_2 + (I_2 - I_1) Z_1 - 10 I_1 = 0$$