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I just started reading state space representation from book by (Norman s.) and in this book all examples of state space representation are either of series RLC with voltage source as an input or parllel RLC with current source as an input.

But

How to represent a state space model for a parallel RLC circuit with voltage source as an input ?

I tried but in this case

1.I confused about what should be order of system (because here different system variable as an output give different order of differential equation )

2.how many minimum system variables require to write the state equations?

Or

3.its state representation is not possible ?

user215805
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1 Answers1

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There are several possible state variable presentations for the same circuit, but I guess you want state variables which are measurable quantities of the circuit. In that case you should use inductor currents and capacitor voltages as the state variables.

Capacitors have stale law dUc/dt = I/C where Uc is the cumulated voltage over the capacitor and I is the charging current. Inductors have dI/dt = U/L where I is the cumulated current through the inductor and U is the voltage over the inductor.

The series RLC circuit:

You can write the wanted equations with these laws. Express the voltage over the inductor with your V(t), I, R and Uc. Divide it by L and there's your expression for dI/dt, the right side of the first state equation.

The second state equation is simply (as already written) dUc/dt = I/C.

Circuit analysis programs which do transient analysis generate the equations automatically and solve them numerically. You must enter the initial values of the state variables i.e. inductor currents and capacitor voltages or let them be zero.

  • Thanks for answer ! But I asked about parllel RLC with voltage source as input and your answer is about series RLC , so I'll be thankful if you make required changes in your answer and suggest a method for parllel RLC with voltage source as an input – user215805 Aug 21 '20 at 20:27
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    If you have R, L and C in parallel with an ideal voltage source the case is simple, the voltage source prevents any interaction between R,L and C, everyone takes separately its own current, integration is needed only for inductor's current. R needs a zeroth order equation (=no differentation nor integration) , L and C need both 1st order equations. If your circuit is something else than R,L,C and an ideal voltage source in parallel you should draw it. –  Aug 21 '20 at 20:42
  • @user215805 user287001's point is important. If you are driving this with an ideal voltage source, the energy storage of the capacitor is determined (you don't need a state variable for the capacitor energy as it is "determined" by the input source.) You do need a state variable for the inductor current as that determines the inductor's energy and the voltage source value has to be integrated over time to keep track of the Webers. -- By the way, I recommend applying an ideal Weierstrass source. ;) Hard to come by, but fun anyway. – jonk Aug 21 '20 at 20:54
  • @user215805 (Continued) there's an ambiquity. You talk of "the order of a system" without defining the term. I used term order for equations. I would call the parallel version "1st order system" because it splits to separate non-interacting parts which everyone needs 0...1 order differential equation. But that's only an opinion, there's no law which states the terms. –  Aug 21 '20 at 20:56
  • @user287001 ,'order of system ' to refer minimum number of State variable require to write it's State equations – user215805 Aug 22 '20 at 05:22
  • @jonk, thanks for suggestion ! But as you pointed out that capacitor voltage is not needed for state variable but if let's say there was some initial condition (some charge on capacitor prior to I input applied) due to capacitor and if I don't consider voltage across capacitor a state variable then , isn't we have one less condition to determine all other possible output variables after input is applied – user215805 Aug 22 '20 at 05:31
  • @user215805 I tried to parse your writing and maybe I managed. I'll admit a little uncertainty, though. So if I don't write something useful, just blame me for my lack in reading skills. An initial condition on the capacitor would set a voltage other than zero. (Or, if you applied a voltage source that doesn't start with zero, I suppose we have a similar problem.) Either way, infinite current would be required to instantly resolve (correct) the capacitor voltage. And this fact alone would present a problem to analysis. – jonk Aug 22 '20 at 05:47
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    @user215805 However, if you want to dig into something I wrote a while back and feel like I should have written better on, you could see this answer about a parallel RLC with initial conditions. In that case, you would in fact require two state variables. – jonk Aug 22 '20 at 05:48
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    Thanks jonk ! I got your point i.e as I applied ideal voltage source , voltage across capacitor will be equal to applied voltage and initial condition of capicitor does not effect any system variable at all after input applied ,so by knowing input voltage and initial condition due to inductor current we can calculate any system variable after input applied ,and so it is just a 1st order system , but if there is a current source instead of voltage source as an input then it require both initial conditions as you written in your answer ,and so in that case it would be second order system – user215805 Aug 22 '20 at 07:13
  • @user215805 Yes. I think you got it! – jonk Aug 23 '20 at 15:47
  • @user287001 +1 After reading more about circuits, I'll return to this question again, and now your answer makes sense, so can we say talking about order of circuit (system) without specifying output is useless? Because even a simple R-L circuit with voltage source as an input can be considered as second order if we are measuring charge? Hence by default if order of system is given without specifying output ,we generally by default consider the "measurable output " with highest order as the order of the circuit? Is my explanation correct or not? – user215805 Sep 30 '21 at 18:42
  • The order of the system is = the number of independent state variables when the system is presented as a group of 1st order differential equations which give the time derivatives of the state variables as functions of the cumulated state variable values, inputs and time. Charge in RL circuit needs a device which integrates current. That introduces a new state variable - the cumulated charge. Without it the charge cannot be instantly observable in the system. It's a second order system. The other state variable is the inductor current. –  Oct 01 '21 at 22:33