TheTopGun said:
Homework Statement
Hi, I have an electrical circuit, from which I have derived 4 equations to work out the current I of the circuit. To solve I need to put the equations into a matrix and find the eigenvalues & vectors, great I can do that. However there is an additional matrix on the end for voltage values. Previously it was assumed the circuit was DC so all voltages would = 0, as you can see in the picture below. Now its AC and I don't know what to do with the 'extra' matrix as its now a function and has to be considered.
This probably makes little sense as I admit a lot is going over my head.
Homework Equations
This picture best describes the situation.
https://www.dropbox.com/s/5y2oe359a84f7za/IMG_20141005_202343113.jpg?dl=0
The Attempt at a Solution
I can find the eigenvalues for matrix 'A' but I don't know where the extra bit comes into play...
Thankyou for any guidance.
If you know the eigenvalues of ##A##, you can solve the DE ##\dot{x} = Ax + b## using a matrix exponential:
x(t) = e^{At} x(0) + \int_0^t e^{A(t-s)} b(s) \, ds
The matrix exponential can be computed from the eigenvalues and their multiplicities:
(1) If all eigenvalues ##r_1, r_2, \ldots, r_n## of ##A## are distinct, there exist matrices ##E_1, E_2, \ldots, E_n## such that
e^{At} = \sum_{i=1}^n E_i e^{r_i t}
More generally, for any analytic function ##f## we have
f(A) = \sum_{i=1}^n E_i f(r_i)
where the ##E_i## are the same for any function. One way to find the ##E_i## is to apply this to various ##f## for which ##f(A)## is readily computable:
f(x) = 1 = x^0 \Rightarrow f(A) = I = \sum_{i=1}^n r^0 E_i = \sum_{i=1}^n E_i\\<br />
f(x) = x \Rightarrow f(A) = A = \sum_{i=1}^n r_i^1 E_i = \sum_{i=1}^n r_i E_i \\<br />
f(x) = x^2 \Rightarrow f(A) = A^2 = \sum_{i=1|}^n r_i^2 E_i,\\<br />
\vdots<br />
Here ##I## is the identity matrix.
For example, for a 4x4 matrix with four distinct eigenvalues ##r_1,r_2, r_3,r_4## we have
I = E_1 + E_2 + E_3 + E_4\\<br />
A = r_1 E_1 + r_2 E_2 + r_3 E_3 + r_4 E_4\\<br />
A^2 = r_1^2 E_1 + r_2^2 E_2 + r_3^2 E_3 + r_4^2 E_4\\<br />
A^3 = r_1^3 E_1 + r_2^3 E_2 + r_3^3 E_3 + r_4^3 E_4<br />
After solving for the ##E_i## we have
e^{Aw}= E_1 e^{r_1 w} + E_2 e^{r_2 w} + E_3 e^{r_3 w} + E_4 e^{r_4 w}
(2) If not all eigenvalues of ##A## are distinct, it is a bit more complicated, but the first order of business would be to check if, indeed, your eigenvalues are all distinct. After that, complications can be dealt with if needed.