# State space representation of parllel RLC circuit

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 ?

• Show it for a series RLC and hopefully you'll get an answer. You need to show/demonstrate some effort. Aug 21, 2020 at 19:34
• @andy ,I added a solution for series RLC network Aug 21, 2020 at 19:42
• Most lectures I watched on Youtube were confusing. This is was the best bet youtube.com/watch?v=nz2rY79QhsQ Aug 21, 2020 at 20:28
• The order is the number of lumped L's and C's Aug 21, 2020 at 20:42
• I think you would have to write the equations with your dq/dt=v(t) for a parallel circuit. Aug 21, 2020 at 20:46

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 Aug 21, 2020 at 20:27
• 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, 2020 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, 2020 at 20:54
• @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, 2020 at 5:48
• 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 Aug 22, 2020 at 7:13