Are My State Equations Correct for a Multi-Input System Using KVL and KCL?

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Discussion Overview

The discussion revolves around the correctness of state equations derived for a multi-input system using Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL). Participants are exploring the application of these laws in the context of circuit analysis, particularly focusing on state variable representations and the implications of combining circuit components.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant expresses uncertainty about whether their state equations are correctly derived, noting that the professor only covered Single Input Single Output (SISO) examples in class.
  • Another participant suggests that instead of writing differential equations, it may be more effective to work directly with state variable values for impedance or admittance in the circuit.
  • A participant questions the validity of transforming a current source into a series resistor and voltage source, expressing concern about potentially losing a variable of interest (i1) in the process.
  • There is a discussion about whether the equations for dv/dt and di/dt must be expressed solely in terms of state variables (Vc and iL) or if they can include input variables as well.
  • One participant indicates that it is acceptable to have multiple functions of state variables and emphasizes the need for enough simultaneous equations to solve for all variables involved.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best approach to derive the state equations. There are competing views on whether to combine components and how to structure the equations in terms of state variables versus input variables.

Contextual Notes

Participants have noted the importance of maintaining the integrity of variables when transforming circuit components and the necessity of formulating equations that adequately represent the system's dynamics.

Who May Find This Useful

This discussion may be useful for students studying circuit analysis, particularly those preparing for exams involving KVL, KCL, and state variable methods in multi-input systems.

Derill03
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what I've posted is a question from a study guide for our test next week, on the attachment is my work.

The professor only did SISO examples in class and moved really fast, I am not really sure if i found state equations correctly using KVL and KCL. Any help is appreciated
 

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Rather than first writing the differential equations for the time domain, you should be able to work directly with the state variable values for impedance (or admittance) for the circuit. Do you know the expressions for the impedances of L and C components? What about unit step functions (to convert the current sources)?
 
well I do see that you could convert left hand current source to series resistor and volt source, but wouldn't i lose i1? Weve been taught in class don't combine components or transform them if they hold a value your looking for.

One thing I am stuck on and this may help me greatly, does each equation dv/dt and di/dt have to be solely in terms of Vc and iL? or can i have two equations with say one has v and an input variable and other has i and an input variable? Cause right now I am trying to make substitutions to make each equation be solely in terms of state variables
 
Derill03 said:
well I do see that you could convert left hand current source to series resistor and volt source, but wouldn't i lose i1? Weve been taught in class don't combine components or transform them if they hold a value your looking for.
No need to transform or combine any components. You can write node (or mesh) equations for the circuit as-is, directly with state variable expressions. Can you write the state variable expressions for component impedances and current sources?
One thing I am stuck on and this may help me greatly, does each equation dv/dt and di/dt have to be solely in terms of Vc and iL? or can i have two equations with say one has v and an input variable and other has i and an input variable? Cause right now I am trying to make substitutions to make each equation be solely in terms of state variables

You can have more than one function of the state variable, such as V1(s) and V2(s) and I(s) and so on. Just as in the time domain you'll need enough simultaneous equations to solve for them all.
 

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