Initial Conditions of Resistor, Inductor & Capacitor
Download as PPT, PDF•28 likes•53,572 views
Initial conditions provide the values of arbitrary constants that appear in differential equation solutions for circuits. They describe the circuit variables like current and voltage in inductors and capacitors immediately before and after a switch is opened or closed. Determining initial conditions involves finding the inductor current and capacitor voltage at times t=0- and t=0+ around the instant when the switch position changes. Examples show how to calculate inductor current, its derivatives, and capacitor voltage at t=0+ using KVL and component equations. Initial conditions capture the past history of energy storing elements in the circuit.
Initial Conditions of Resistor, Inductor & Capacitor
- 1. INITIAL CONDITIONS : WHY TO STUDY • Differential Equations written for a network may contain arbitrary constants equal to the order of the differential equations. • The reason for studying initial conditions is to find the value of arbitrary constants that appear in the general solution of differential equations written for a given network.
- 2. INITIAL CONDITIONS • In Initial conditions, we find the change in selected variables in a circuit when one or more switches are moved from open to closed positions or vice versa. t=0- indicates the time just before changing the position of the switch t=0 indicates the time when the position of switch is changed t=0+ indicates the time immediately after changing the position of switch
- 3. INITIAL CONDITIONS • Initial condition focuses solely on the current and voltages of energy storing elements (inductor and capacitor) as they will determine the circuit behavior at t>0. • PAST HISTORY OF THE CIRCUIT WILL SHOW UP AS THE CAPACITOR VOLTAGES AND INDUCTOR CURRENTS
- 4. INITIAL CONDITIONS 1. RESISTOR The voltage current relation of an ideal resistance is V=R*I From this equation it can be concluded that the instantaneous current flowing through the resistor changes if the instantaneous voltage across it changes & vice versa The past voltage or current values have no effect on the present or future working of the resistor i.e.. It’s resistance remains the same irrespective of the past conditions
- 5. INITIAL CONDITIONS 2. INDUCTOR The expression for current through the inductor is given by
- 6. INITIAL CONDITIONS Hence if i(0- )=0A , then i(0+ )=0A So we can visualize inductor as a open circuit at t=0+
- 7. INITIAL CONDITIONS • If i(0- )=I0 , then i(0+ )=I0 i.e. the inductor can be thought as a current source of I0 as shown
- 8. INITIAL CONDITIONS FINAL CONDITIONS : From the basic relationship V= L*(di/dt) We can state that V=0 in steady state conditions at t= as (di/dt)=0 due to constant current
- 9. INITIAL CONDITIONS 3. CAPACITOR The expression for voltage across the capacitor is given by
- 10. INITIAL CONDITIONS If V(0- )=0V , then V(0+ )=0V indicating the capacitor as a short circuit
- 11. INITIAL CONDITIONS If V(0- )= V volts, then the capacitor can be visualized as a voltage source of V volts
- 12. INITIAL CONDITIONS • Final Conditions The current across the capacitor is given by the equation i=C*(dv/dt) which indicates that i=0A in steady state at t= due to capacitor being fully charged.
- 13. INITIAL CONDITION EXAMPLE-1 : In the network shown in the figure the switch is closed at t=0. Determine i, (di/dt) and (d2 i/dt2 ) at t=0+ . At t=0- , the switch is Closed. Due to which il(0- )=0A Vc(0- )=0V
- 14. INITIAL CONDITION At t=0+ the circuit is From the circuit il(0+ )=0A Vc(0+ )=0V
- 15. INITIAL CONDITION • Writing KVL clockwise for the circuit Putting t=0+ in equation (2)
- 16. INITIAL CONDITION • Differentiating equation (1) with respect to time
- 17. INITIAL CONDITION Example 2: The position of switch was changed from 1 to 2 at t=0. Steady State was achieved when the switch was at position 1. Find i, (di/dt) & (d2 i/dt2 ) at t=0+
- 18. INITIAL CONDITION At t=0- , the circuit is shown in figure The inductor is in steady state so it is assumed to be shorted. So the current through it is il(0- )=20/10=2A Vc(0- )=0V
- 19. INITIAL CONDITION So at t=0+ , the switch is at position 2 Here the Inductor behaves as a current source of 2A. The circuit is shown below il(0+ )=2A Vc(0+ )=0V
- 22. THANK YOU