How to identify the state variables of a circuit?.

[mt1200]

Hi, it's me again, as you may know I've always have had a very hard time with circuit analysis.

How can you possibly know which states variables are you suppoused to use in second or third order circuits?.

Let's say you have to do the following exercise:

If you want to obtain a state space model for Vg(t) and V2, what would you do?.

I would try to apply some KVL and KCL, but then what?. What variable are you supposed to solve for?. if you derive some variables you may not end up with the circuit solution for V2

 

[rootX]

There is no unique way of picking state variables. You can write the differential equation and pick the state variables here or

In each case we adopt the strategy of associating state variables with the energy storage elements in the system.

Williams, R. L., and Lawrence, D. A., Linear State-Space Control Systems, John Wiley & Sons, 2007.

 

[mt1200]

But the system may become impossible to solve if I pick some random state variables I want.

 

[Hechima]

In addition to choosing a state variable for each element that stores energy (the two capacitors and the one inductor), you should also consider that your state space model may
look something like this:

State: X1, X2, X3

$\frac{d}{dt}X_{1}=aX_{1}+bX_{2}+cX_{3}$
$\frac{d}{dt}X_{2}=eX_{1}+fX_{2}+gX_{3}$
$\frac{d}{dt}X_{3}=hX_{1}+iX_{2}+jX_{3}$

You will need a first order differential equation for each state variable, so I would recommend assigning state variables based on how inductors and capacitors relate current and voltage:

$i_{c} = C\frac{dV}{dt}$
$v_{L} = L\frac{di}{dt}$

I would recommend that you use these relations to solve for a first derivative for each capacitor and inductor. In each equation, the voltage or current variable being differentiated will be a suitable state variable.

Once you choose your state variables, you should be able to eliminate all other variables using KVL and KCL etc.

 

[alphysicist]

To identify the state variables of a circuit, you first need to understand the concept of state variables. State variables are the minimum set of variables that can describe the behavior of a system at a given time. In the context of circuits, state variables refer to the voltages and currents at any given time.

To identify the state variables in a circuit, you can follow these steps:

1. Draw the circuit diagram: Start by drawing the circuit diagram and labeling all the components with their respective values.

2. Identify the independent sources: Look for any independent sources, such as voltage or current sources, in the circuit. These sources will be the inputs to the system.

3. Apply KVL and KCL: Apply Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) to the circuit to determine the relationships between the different components.

4. Simplify the circuit: Use any simplification techniques, such as series and parallel combinations, to reduce the complexity of the circuit.

5. Identify the state variables: Once the circuit is simplified, you can identify the state variables by looking at the voltages and currents at different points in the circuit. These variables will be the ones that are not directly related to the independent sources.

6. Write the state equations: Once you have identified the state variables, you can write the state equations using KVL and KCL. These equations will describe the behavior of the system over time.

In the exercise given, to obtain a state space model for Vg(t) and V2, you would follow the steps mentioned above. You would first identify the state variables, which could be the voltage across a specific component or the current through a specific branch. Then, using KVL and KCL, you can write the state equations for these variables.

It is important to note that the choice of state variables may vary depending on the specific circuit and the desired outcome. It is a skill that comes with practice and understanding of circuit analysis.