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amr55533
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Homework Statement
Consider the electrical circuit shown:
http://imageshack.us/a/img525/8163/p1circuit.png
Let the state variables be x1(t)=Vc(t), x2(t)=iL(t), and x3(t)=Vc(t); output is Vo(t). Write the state-space equations in matrix form and find the transfer function, T(s)=Vo(s)/Vi(s).
Homework Equations
KCL and KVL
The Attempt at a Solution
State Variables:
x1(t)=Vc(t)
x2(t)=iL(t)
x3(t)=Vo(t)
Outputs:
Vo(t)
Inputs
Vi(t)
Differential Equations for State Variables:
X1'=dV1/dt=i2
X2'=di4/dt=V2
X3'=dVo/dt=i5
Now this is the part that I am stuck at. I know that I have to solve for X1', X2', and X3' in terms of the state variables and inputs only. However, I can't seem to reduce the equations enough to get it into this format.
Basically, I am trying to solve for i2, i5, and V2 in terms of i4, V1, Vo, and Vi only (the state variables and inputs). Once I have these equations, I can easily put them into matrix form and solve using MATLAB. I solved a problem earlier that was exactly the same, only the first capacitor was replaced with an inductor. So, I think it is the capacitor that is giving me problems.
A few equations that I found:
Vi=i1+i3+i5+Vo
i3=i1-i2
i5=i3-i4
V1=Vi-i1
V2=V1-i3
Vo=V2-i5
Thanks for the help!Edit:
I looked over the problem again, and it seems that I can't solve for i3 without it containing a V2 or an i2. Is there any way to solve for i3 with a combination of only i4, V1, Vo, and Vi? Once I find this, I will be able to solve the problem.
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