State-variable method, standard-form state equations Seo Yeon Youn [Mathematical Circuit Theory and Analysis]

2 Contents I. Capacitor and inductor  C haracteristic  Approach  Simple example  State, state equation  Standard state form equation  Exercise II. State variable method III. Standard state form equation

3 Capacitor and inductor Capacitor  An electrical/electronic device that can store energy in the electric field between a pair of conductors  Protecting voltage changing .

4 Capacitor and inductor Inductor  A circuit element that consist of a conducting wire usually in form of a coil.  A Changing magnetic field produced a voltage that was proportional to the time rate of change of the current that produced the magnetic field. .

5 State variable method Approach  We want to expect time invariant circuits analysis which contain capacitor and/ or inductor  How can we expect this circuit?  Express circuit equation by the variables of voltage across the capacitor and/or current through the inductor. When we use state variable method, we can solve the problems easily. State variable method

6 Simple example : state variable characterization of first- order networks

7 State variable method Simple example : state variable characterization of first- order networks

8 Standard state form equation What is “State”? “State equation”?  A state of a lumped network at time t0 is any amount of information given at time t0 adequate to characterize completely any chosen output function from time t0 onward given any admissible input function from time t0 onward.  Choice of state is the concept of minimality.  The equations relating the current state and output of a system to its current input and past states are called the state equations.  So in State variable method, we can get next state from state equation which is formed by the variable of present state and input.

9 Standard state form equation  State equation in standard form (linear, time-invariant) (state equation) (input- state-output equation)  x : state variable (vector)  x(t) : state at time t  x(t0) : initial condition to be the state at time t0  u(t) : input function (vector)  y(t) : output function(vector)  Alternate set of standard form state equations would dictate a different choice for the state.  Standard state form equation

10 Standard state form equation  Extended form state equation (linear, time-invariant) (state equation) (input- state-output equation)  Standard state form equation

11 Standard state form equation Exercise  Solve by state variable method.

12 Standard state form equation Exercise

13 Standard state form equation Exercise

Question

Q&A Thank You  In state variable method, mathematical skill is important  Reference : -Circuit Theory : An introduction to the state variable approach, R.A. Rohrer -Basic engineering circuit analysis, J.David Irwin -