First-Order Transient Circuits

							Transient Analysis - First Order
Circuits

     Switches, Transient Response,
      Steady-State Response, and
         Differential Equations

          Kevin D. Donohue, University of Kentucky   1
Transient Response
   DC analysis of a circuit only provides a description of
    voltages and currents in steady-state behavior.

   When the applied voltage or current changes at some
    time, say t0, a transient response is produced that dies
    out over a period of time leaving a new steady-state
    behavior.

   The circuit’s differential equation must be used to
    determine complete voltage and current responses.



                  Kevin D. Donohue, University of Kentucky     2
Examples
 Describe v0 for all t. Identify transient and steady-state
 responses.
                       VDC                t=0           R             +
                                                                  C   v0
                                                                       -
                                                     t=0
 Show:
                        t 
           VDC 1  exp
                               volts     for t  0
 v0 (t )              RC  
            0 volts                        for t  0
           

For steady-state response, let t  , for transient response subtract
out steady-state response.



                       Kevin D. Donohue, University of Kentucky            3
    Instantaneous Voltage and Current
    Changes in Capacitors and Inductors:
    What would be the required                                          ic

     current, ic , in this circuit for the         VDC             t=0            +
     voltage on the capacitor to                                              C   vC
                                                                                   -
     change instantaneously?

                                                                         iL
     What would be the required
                                                                   t=0
     voltage, vL , in this circuit for the         IDC                            +
                                                                              L   vL
     current in the inductor to change                                             -
     instantaneously?

Conclusion: If the source cannot produce infinite instantaneous
power, then neither the capacitor voltage, nor the inductor current can
change instantaneously.


                        Kevin D. Donohue, University of Kentucky                       4
Switch Notation and Initial Conditions:

 In order to denote the time right before t=0 (limit from the left as
 t0), and the time right after t=0 (limit from the right as t0),
 the following notation will be used:

                                     Let t=0+ be the moment after the switch
              t=0                    is closed and t=0- be the moment before
                                     the switch is closed.

 For circuits with practical sources,
 the voltage across a capacitor cannot                v c ( 0  )  vc ( 0  )
 change instantaneously,

 and the current in an inductor
 cannot change instantaneously                   i L (0  )  i L (0  )

                    Kevin D. Donohue, University of Kentucky                     5
 Complete Solution by the Differential
 Equation Approach
5 major steps in finding the complete solution:
    Determine initial conditions on capacitor voltages
     and/or inductor currents.
    Find the differential equation for either capacitor
     voltage or inductor current (mesh/loop/nodal ….
     analysis).
    Determine the natural solution (complementary
     solution).
    Determine the forced solution (particular solution).
    Apply initial conditions to the complete solution to
     determine the unknown coefficients in the natural
     solution.
                 Kevin D. Donohue, University of Kentucky   6
Example
 Find the complete solution for iL for vs  10V


                       t=0            25 W                +
       vs                                       0.25 H    vL
                                                          -


     Show for t  0:     iL  0.4(1  exp(100t ))


               Kevin D. Donohue, University of Kentucky        7
Example
 Find the complete solution for vc when is  1 mA


                                   t=0

              is          t=0                                   +
                                         100 W           1 mF   vc
                                                                -
                          R




        Show for t  0:       vc  0.1 exp( 10 t )


                   Kevin D. Donohue, University of Kentucky          8
Step-by-Step Method
    The solution of circuits containing energy storage elements can be divided into a
    steady-state and transient component. In addition, when only one energy storage
    element is present, the Thévenin resistance can be obtained with respect to the
    terminal of the energy storage element and used to compute the time constant for the
    transient component.

                                                         t
   Assume solution is of the form x(t )  K1  K 2 exp   
                                                         
   Assume steady-state before the switch is thrown and let either vc (0 )  vc (0 ) or
     iL (0 )  iL (0 ) , and find initial condition for quantity of interest x(0 )


   Let K1 = steady-state solution after switch is thrown, K2  x(0 )  K1
    and   CRth , or   L Rth



                           Kevin D. Donohue, University of Kentucky                         9