# State Equation of negative resistance from current source

I have to find the state equation from the circuit, the problem is, on the node with red marker. Here is what I've done

$$\u = I_C+I_R\$$

$$\u = C\frac{dV}{dt} + \frac{V}{R}\$$

and if the value from components is included, then

$$\u = \frac{dV}{dt} - V\$$ (because of negative resistance)

is this right? based on the solution given, it should be

$$\u = \frac{dV}{dt} + V\$$

is the negative in resistance is ignored?

## 2 Answers

The steady-state for a CC and this network supplies a voltage of I/2ohms divided in half. But the step response of switching on this constant current is broad spectrum so a frequency domain impedance ratio is needed with the DC and Step input to get a response with 0 initial conditions.

So the State before CC is applied must be defined to yield the State Equations. If not then the result is SS of 1/(1+1) Vinput=Voutput. Where Vinput=I*2Ohms

No component, when placed in series with an ideal current source, affects the rest of the circuit. This includes positive resistances, nonlinear resistances, voltage sources, inductors, capacitors, or the combination of a negative resistance and a capacitor like you have here. Regardless of what you put in series with it, the ideal current source figures out how to force its designed current through it, so the total source (ideal current source plus series element) is just equivalent to the ideal current source alone.

In the real world, of course there are no truly ideal current sources. But you haven't shared (and maybe don't know) enough about what the actual current source is in your circuit to predict whether the real current source will behave like an ideal source in this circuit or not.

• Current sources are only limited by voltage headroom and frequency bandwidth generally. In this example it does not apply. – Tony Stewart EE75 Sep 27 '18 at 1:58