I have the following 4 governing equations
i) V1(t) = a.Id(t)^2 + Rd.Id(t)
ii) V1(t) = V(t) - R.I(t)
iii) Ic(t)= C.dV(t)/dt
iv) I(t) = Ic(t) + Id(t)
The values of a, Rd, R and C are constant.
And I need to find an equation in terms of V on the LHS and I on the RHS, but no matter how I look at it I can't find a solution. I've been two days looking at this and I must be missing something. everytime I try to solve it I start to get terms which are the product of I or dI/dt and V or dV/dt
I'm just going to rewrite this in a way I think is a little clearer, dump the constant terms (really not that important to this solution) and use the following terms
X, Y, Z, J and K
Where everything is a function of t
I want to combine these equations into a single equation where the X, Y, and Z terms are no longer in the equation and there are only terms of J on the RHS and K on the LHS or visa versa
i) X = Y^2 + Y
ii) Z = dX/dt
iii) J = Z + Y
iv) K = X + J
This should be possible but not matter which way I combine these I get something along the lines of
K = J^2 + ( K'(t) + J'(t) )^2 - 2*J ( K'(t) + J'(t) ) - ( K'(t)+J'(t) )
and obviously this is going to leave you with products of J and Ks that can't be seperated (I need all the K'(t) terms to be over with it's buddy on the LHS). No matter what stratagy I use to avoid this I can't help but get these terms combining.
Quite a challenging thingy!
Tried : k = K'(t) , j = J'(t)
to get:
K^2 - 4k^2 + 8k(J - j) = 4J^2 + 4j^2 - 8Jj - 4J - 1
Close, but no ceegar...
For curiosity, tried K = K'(t) , J = J'(t) ; so:
K = J^2 + (K + J)^2 - 2J(K + J) - (K + J) ; leads to:
J = K(K - 2)
And that has endless integer solutions:
J....K
24,-4
15,-3
8,-2
3,-1
0,0
-1,1
0,2
3,3
8,4
Hope that helps....probably not!
Lots of others at this site more advanced than me...maybe one will step in