Find the state space solution for a circuit

[mt1200]

Homework Statement

Find the state space solution, the output is Vo

Homework Equations

VL = L*di/dt
IC = 1/C*(integral of current)
KCL
KVL

The Attempt at a Solution

I tried to apply KCL and KVL but still can't figure it out. This one has been driving me nuts for 4 days.

 

[gneill]

Can you show an attempt?

 

[mt1200]

Yeah, this is what I've been trying:

There are two meshes and one node in that circuit.

So for Node 1:

IC1 = IL + IC2
C1dV/dt = IL + C2dV/dt

That's the first equation.

For Mesh #1:

Vi = VC1 + VL
Vi = 1/C ∫ic1 + LdiL/dt
VC1 = Vi - LdiL/dt

For Mesh # 2:

VL = Ri2 + VC2
VC2 = VL - Ri2
VC2 = LdiL/dt - Ri2

I don't know what else to do, I can't put those equations into a state space matrix, since there are integral terms.

 

[gneill]

You can determine the transfer function in terms of the state space variable/operator s, which behaves in the same fashion as in the Laplace Transform.

To do so, write the impedance of an inductor as sL and that of a capacitor as 1/(sC). Solve for Vo(s) in terms of Vi(s) by whatever method you wish.

 

[mt1200]

Aren't you suppoused to use derivatives and integrals to find the space state solutions?, I'm not sure if I can use Laplace to solve this problem.

last night I realized that it's really confusing, if I try to create the space state matrixes I get something like this:

Vc1'
Vc2'
IL'

But I can't express only three state variables since I have like three currents ( It, IL and IC2).

 

[gneill]

If you can first determine the transfer function you can then use its various coefficients to write the state space model (see, for example, the wikipedia entry for "state space representation").

 

[uart]

Yeah, this is what I've been trying:

There are two meshes and one node in that circuit.

So for Node 1:

IC1 = IL + IC2
C1dV/dt = IL + C2dV/dt

That's the first equation.

For Mesh #1:

Vi = VC1 + VL
Vi = 1/C ∫ic1 + LdiL/dt
VC1 = Vi - LdiL/dt

For Mesh # 2:

VL = Ri2 + VC2
VC2 = VL - Ri2
VC2 = LdiL/dt - Ri2

I don't know what else to do, I can't put those equations into a state space matrix, since there are integral terms.

The first two equations are basically good - after you fix up the subscripts V1 and V2 in that first equation - which I assume you know but just got sloppy :).

The problem with the third equation "VC2 = LdiL/dt - Ri2" is that "i2" is not a state variable (use energy storage quantities Vc1, Vc2 and IL for your states). Can you see how to substitute for "i2" in terms of your state variable. Do that and you are almost there.