What is the density function of the current?

Question

For a JFET the V ,i characteristic is parabolic. A simplified model of this is given by
i=V² . In a circuit application the voltage to the JFET is random with a density function
f V (v)=2(1−v) for 0?v?1 . What is the density function of the current?

Answer 1

Let f (i) be the density of i. Then f (i) is "like" f (v) with an appropriate scaling (sometimes called Jacobian or Jacobian determinant) to ensure f (i) integrates to 1 over the transformed domain.

 

Let V = g (i) = sqrt (i). The scaling factor is g'(i),

g'(i) = 1/(2 × sqrt (i))

 

Then substituting g (i) for V into f (V) and multiplying by the scaling factor,

f (i) = f (g (i)) × g'(i)

= 2 (1-sqrt (i)) × 1/(2 × sqrt (i))

= 1/sqrt (i) - 1

 

The original domain was Df = {v: 0 < v < 1} so the new domain is

Di = {i: sqrt (0) < i < sqrt (1)} = 0 < i < 1.

 

You can check that f (i) integrates to 1 over the interval (0, 1).