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

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SUMMARY

The discussion centers on the correct formulation of state equations for a multi-input system using Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL). Participants emphasize the importance of directly using state variable values for impedance and current sources without combining components that hold critical values. It is established that while writing equations for the system, it is permissible to have multiple equations involving different state variables, such as Vc and iL, rather than restricting each equation to a single state variable. This approach allows for a more flexible and accurate representation of the system dynamics.

PREREQUISITES
  • Understanding of state variable representation in circuit analysis
  • Familiarity with Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL)
  • Knowledge of impedance and admittance for circuit components, specifically inductors (L) and capacitors (C)
  • Ability to work with differential equations in the context of electrical circuits
NEXT STEPS
  • Study state variable analysis techniques for multi-input systems
  • Learn how to derive state equations from circuit diagrams using KVL and KCL
  • Explore the concept of impedance and admittance in more depth, particularly for L and C components
  • Investigate the use of unit step functions in circuit analysis for current sources
USEFUL FOR

Electrical engineering students, circuit designers, and professionals involved in control systems who need to understand the formulation of state equations 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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