Hi, I'm trying to find what I(t) and V2(t). - Firstly I'm changing to impedances and then predefining a few new variables - zc1, zr1, zc2, zr2 and zc3 - then I'm going to do mesh analysis and then the current I'm trying to work out should be the answer of I2. and the voltage I'm trying to also workout will do Zc3 * I3. Is all of this correct or have a made a mistake? Thanks !!
It looks ok except that you are postmultiplying the voltage vector by the inverse of the impedance matrix. You must premultiply the voltage vector by the inverse of the impedance matrix.
The voltage V2(t) is poorly defined because there is no indication of polarity. Is V2(t) the voltage at the top of the capacitor relative to the bottom of the capacitor, or the other way around. Both are valid and completely consistent with the original diagram. So you need to define what YOU choose to interpret V2(t) to mean. A similar thing applies to the voltage source. Whoever wrote the problem was pretty sloppy. When I was a student I tended to define things the opposite of what was almost certainly meant and I would clearly label my definition and proceed to solve the problem. Occasionally I got docked because my answer didn't agree with the author's solution that the grader was working from, but I always got the points back when I challenged the grade. I made my point, I demonstrated that I knew what was going on (i.e., I was learning what I was supposed to be learning) and I think the grader and, in one case, the instructor learned a bit about the need to be explicit. But in hind sight I wish I hadn't done it because I can now more fully appreciate how much harder I made the grader's life (not so much the instructor) and they really didn't need the added stress. Today, what I would do would make whatever assumption was most likely intended, clearly label my assumption in case I were wrong (and because my own work would not be complete if I didn't), and make a note of the ambiguity. In doing it this way, I don't make the grader's life harder; in fact, I might well be saving them some grief by bringing the ambiguity to their attention so that they are better able to deal with it if it becomes an issue on other submissions. This makes the grader happy with me and more inclined to give my work the benefit of the doubt. This last part is a key principle in the proper care and feeding of homework graders. I'll take a look at your work in a few minutes.
Your mesh equations are fine, but your final matrix equation has problems. I don't do a lot with linear algebra (have never even taken a course, much to my everlasting regret), so I'm not positive what rule you are breaking. But on the RHS you have a the product of two vectors and the result will be a vector/matrix in which the number of rows is equal to the number of rows in the first operand and the number of columns will be equal to the number of columns in the second operand, thus you will end up with a 3x3 matrix instead of a 3x1 column vector. I may not be super conversant with the rules, but at least I try to understand enough to be able to sanity check results. I think your problem is that matrix multiplication is not commutative, so if you multiply the impedance matrix by the inverse of it on the left hand side of the left hand equation, then you have to multiply it on the left hand side of the right hand equation in order to maintain equality.
Oh, good. What I said was consistent with what the Electrician noted and I know he is conversant with matrix manipulation.