Starting with We can derive the current as We see the solution typically has a

# Starting with we can derive the current as we see the

• 23

This preview shows page 7 - 16 out of 23 pages.

¨ Starting with ¨ We can derive the current as ¨ We see the solution typically has a TRANSIENT which dies out eventually, and as t tends to ∞, the solution settles to a steady state. time constant Prof. C.K. Tse: Dynamic circuits—Transient 8 A simple first-order RL circuit ¨ Consider a RL circuit. ¨ Before t = 0, the switch is closed (turned on). Current goes through the switch and nothing goes to R and L. Initially, i L (0 ) = 0. ¨ At t = 0, the switch is opened. Current goes to R and L. ¨ We know from KCL that I o = i R + i L for t > 0, i.e., ¨ The constitutive relations give ¨ Hence, ¨ Þ ¨ The solution is From the initial condition, we have i L (0 ) = 0. Continuity of the inductor current means that i L (0 + ) = i L (0 ) = 0. Hence, A = – I o Thus, Prof. C.K. Tse: Dynamic circuits—Transient 9 Transient response of the RL circuit ¨ Starting with ¨ We can find v L ( t ): time constant Prof. C.K. Tse: Dynamic circuits—Transient 10 Observation — first-order transients ¨ First order transients are always like these: Prof. C.K. Tse: Dynamic circuits—Transient 11 Let’s do some math 0 5 x ( t ) t x ( t ) = 5(1 – e t/ t ) 0 5 x ( t ) t x ( t ) = 5 e t/ t 0 6 x ( t ) t 1 x ( t ) = 1 + 5(1 – e t/ t ) x ( t ) = 1 + 5 e t/ t 0 5 x ( t ) t –2 x ( t ) = –2 + 7(1 – e t/ t ) 0 4 x ( t ) t –3 0 6 x ( t ) t 1 x ( t ) = –3 + 7 e t/ t Prof. C.K. Tse: Dynamic circuits—Transient 12 General first-order solution NO NEED TO SOLVE ANY EQUATION, just find 1. the starting point of capacitor voltage or inductor current 2. the ending point of ………… ………. ……. ………. ………. 3. the time constant t Prof. C.K. Tse: Dynamic circuits—Transient 13 Finding t For the simple first-order RC circuit, t = C R. For the simple first-order RL circuit, t = L / R. The problem is Given a first-order circuit (which may look complicated), how to find the equivalent simple RC or RL circuit. Prof. C.K. Tse: Dynamic circuits—Transient 14 A quick way to find t Since the time constant is independent of the sources, we first of all set all sources to zero. That means, short-circuit all voltage sources and open- circuit all current sources. Then, reduce the circuit to + R 1 R 2 C R 1 R 2 C C R 1 || R 2 C eq R eq R eq L eq either or Example: t = C ( R 1 || R 2 ) Prof. C.K. Tse: Dynamic circuits—Transient  #### You've reached the end of your free preview.

Want to read all 23 pages?

• Summer '16
• Martin Chow
• RC circuit, RL circuit, Prof. C.K. Tse

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern  