# series RLC circuit

Discussion in 'Homework Help' started by gusmas, Mar 4, 2011.

1. ### gusmas Thread Starter Active Member

Sep 27, 2008
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0
Hey Guys/Girls

I have a turorial that ask me a question about a RLC network.(SYN400S Tut2.pdf). Now for the the first question I can find the differential equation. Eventhough the question asks for differential equationS I can only find one?

Question 1.a: ( I coped the eqation from wiki , because I dont have a scanner its difficult to show my own work)

Now Question 1.b asks for two first order differential equations, using the state space method. Now I have never heard of state space I am only studying it later this year and all the books and notes I could find could not explain to me on how to choose what my state variables will be?

Can some1 help

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2. ### steveb Senior Member

Jul 3, 2008
2,433
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This weekend I can try to put together some basic info in state-space approach. For now, I'll just focus on your question. Keep in mind that there are sometimes different choices that can be made for the state variables.

Your equation is a second order differential equation that is analogous to a mechanical equation that is in terms of velocity. Just as velocity is the time rate of change of positon, current is the time rate of change of charge, hence you can easily generate two first order equations by using the definiton of velocity (which is current in this case) as one of the state equaitons.

Here you end up with the following.

${{dQ}\over{dt}}=i(t)$
${{di}\over{dt}}=-{{R i(t)}\over{L}}-{{Q(t)}\over{LC}}+{{v(t)}\over{L}}$

Note that I wrote the state equations as the derivative of the state variable as a funciton of other state variables, input variables and circuit parameters. This is the standard form. It's not always easy to get the standard form, particularly with nonlinear equations, but with linear equations it is always possible using a trick I'll show you later on. Note that the above type of state-space system is a linear system. Linear state space systems can be represented using matrix notation which brings in the standard form with A, B, C and D matrices. ... One step at a time though.

Now, you may notice that Q/C is also the capacitor voltage Vo, so you can reformulate the state-space equation in terms of Vo rather than Q. Try writing the SS equations in terms of the states i(t) and Vo(t). The top equation becomes the formula for a capacitor current as follows:

$i(t)=C{{dV_o}\over{dt}}$

So, you might have come up with the state equations using this approach directly, and many people never even try to use or think about charge Q in circuit equations. As rule of thumb, every capacitor and inductor will generate a state equation, but there are exceptions, because it's really the number of independent states that matters. Often capacitor voltage (or charge) is a state and coil current (or flux) is a state, but again, there are exceptions. With experience, you will spot these things instantly.

Last edited: Mar 4, 2011
3. ### gusmas Thread Starter Active Member

Sep 27, 2008
239
0
ok cool I am starting to understand

Here is my equations for 1.b
for some kind of reason i cant copy from word over so i took a screen shot.

I hope my equations are correct

• ###### Question1b.jpg
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Last edited: Mar 5, 2011
4. ### steveb Senior Member

Jul 3, 2008
2,433
469
OK, you are on the right track, but there are a few comments I can make here.

First of all, what you wrote is basically correct, but it's not quite in the accepted standard form for a state space representation. The main issue is that you converted the charge Q into output voltage in one spot, but not in all spots. Once you convert to a new state variable, you want to express all equations in terms of the new variable, not the old one.

Second, you made the decision to covert current i(t) into a flux by using the relation Vo2=L i(t), where the variable V02 is the coil flux, You can certainly do this because it is analogous to my use of charge Q instead of voltage. However, if you chose to use flux in one spot, then you need to use it in all spots.

Third, it's not a good idea to use the variable Vo2 for flux because it can be confused with a voltage. Note that voltage is time rate of change of flux. Actually, I blame the author of the book, and not you, for this because the problem tells you to find the states V01 and V02, but that circuit does not have two voltage states. Typically people use the variable X, rather than V for general variables that might be currents, voltages, fluxes or charges etc.

To help make this more concrete, I'll give you several different versions of the state-space representation. The key thing to note is that every state space system of equations is always in terms of input variables and the chosen state variables, and you do not mix up the choice of state variable once chosen. Another thing to note is that all 4 representations below require you to use all circuit parameters R, L and C. You may be able to express one state equations without parameters, but the full system of equations must make use of all relevant parameters.

Option 1:

States: Q(t) for the capacitor charge; i(t) for the coil current

${{dQ}\over{dt}}=i(t)$

${{di}\over{dt}}=-{{R i(t)}\over{L}}-{{Q(t)}\over{LC}}+{{v_i(t)}\over{L}}$

Option 2:

States: Q(t) for the capacitor charge; F(t) for the coil flux

${{dQ}\over{dt}}={{F(t)}\over{L}}$

${{dF}\over{dt}}=-{{R F(t)}\over{L}}-{{Q(t)}\over{C}}+{{v_i(t)}}$

Option 3:

States: vo(t) for the capacitor voltage; F(t) for the coil flux

${{dv_o}\over{dt}}={{F(t)}\over{LC}}$

${{dF}\over{dt}}=-{{R F(t)}\over{L}}-{v_o(t)}+{{v_i(t)}}$

Option 4:

States: vo(t) for the capacitor voltage; i(t) for the coil current

${{dv_o}\over{dt}}={{i(t)}\over{C}}$

${{di}\over{dt}}=-{{R i(t)}\over{L}}-{{v_o(t)}\over{L}}+{{v_i(t)}\over{L}}$

Note that this last option #4 is the one typically used in circuit analysis because it is in terms of voltages, currents and circuit parameters. But, knowing all forms, and being able to think in terms of charge and flux is important sometimes.

Last edited: Mar 6, 2011
5. ### gusmas Thread Starter Active Member

Sep 27, 2008
239
0
ok i understand what you are saying just for option 4, why is the capacitor voltage Vo not devided by L? Because option 1 and 4 are almost the same except for option1 is using the capacitor charge and option 4 is using the capacitor voltage. q= Vc*C so the C will cancel out but there will still be L as the denominator?

6. ### steveb Senior Member

Jul 3, 2008
2,433
469
Yes, you are correct. That was a typo on my part. I corrected it above.

7. ### gusmas Thread Starter Active Member

Sep 27, 2008
239
0
Hey again, i actually got 70% for that tut, thanks alot for helping me to understand state space!!

8. ### gusmas Thread Starter Active Member

Sep 27, 2008
239
0
steveb do you have any experience working in simulink?

9. ### steveb Senior Member

Jul 3, 2008
2,433
469
Yes, I use it often.