What are the voltages generated from this current source?

[zenterix]

Homework Statement

The circuit shown below contains two nonlinear devices and a current source. The characteristics of the two devices are given. Determine the voltage ##v## for

Relevant Equations

(a) ##i_S=1##A, (b) ##i_S=10##A, (c) ##i_S=1\cos{t}##.

Here is the circuit and the v-i characteristics

KCL gives us ##i_S=i_1+i_2##.

Thus, (a) and (b) are solved quickly by noting that for ##v\in [0,1]## we have ##i_1+i_2=0## so ##v## can't be in this interval for a positive current.

If we try ##v\in [1,\infty)## then we get ##i_S=1+(-2+v)=v-1##.

Thus, ##i_S=1##A gives ##v=2##V and ##i_S=10##A gives ##v=11##V.

My question is about item (c) where ##i_S=\cos{(t)}##.

What I did was consider four different cases related to the possible value of ##v##. All I did in each case was consider the KCL equation in the context of a restriction on values of ##v##. Note that only cases 2 and 3 are relevant to the solution of (c).

Case 1: ##v\in [1,\infty)##

From the KCL equation, ##i_S=-1+v##, which graphically is

and we see that ##v## satisfies our constraint when ##i_S\geq 1##.

Case 2: ##v\in [0,1)##

Here we have simply ##i_S=0##.

Case 3: ##v\in [-1,0)##

Then, ##i_S=v##

Case 4: ##v \in (-\infty, -1)##

Then, ##i_S=-1##

Putting all of this together we have

We know that ##i_S## is a sinusoid that varies between -1 and 1. Thus, ##v## takes on values between -1 and 2.

It seems that

$$v(i_S)=\begin{cases} i_S+1,\ \ \ \ \ i_S\in (0,1) \\ i_S,\ \ \ \ \ i_S\in [-1,0] \end{cases}$$

$$v(t)=\begin{cases} \cos{(t)}+1,\ \ \ \ \ t\in (-\pi/2,\pi/2) \\ \cos{(t)},\ \ \ \ \ t\in (\pi/2,3\pi/2) \end{cases}$$

My question is what happens when ##i_S=0##?

 

[zenterix]

@berkeman What difference does it make if I write v-i characteristics rather than V-I characteristics that you edited my title?

Agarwal, "Foundations of Analog and Digital Circuits":

 

[berkeman]

@berkeman What difference does it make if I write v-i characteristics rather than V-I characteristics that you edited my title?

Good question. For stand-alone symbols, it's more common in my experience to use V and I. However if you are writing the time functions explicitly, then ##v(t)## ##i(t)## would probably be more appropriate. Remember that lowercase "i" or "j" is often used for complex exponents.