Understanding Inductors and RC Circuits: A First-Order Circuit Overview

1. Lecture 16 Inductors Introduction to first-order circuits RC circuit natural response Related educational modules: Section 2.3, 2.4.1, 2.4.2

2. Energy storage elements - inductors • Inductors store energy in the form of a magnetic field • Commonly constructed by coiling a conductive wire around a ferrite core

3. Inductors • Circuit symbol: • L is the inductance • Units are Henries (H) • Voltage-current relation:

4. Inductor voltage-current relations • Differential form: • Integral form:

5. Annotate previous slide to show initial current, define times on integral, sketchy derivation of integration of differential form to get integral form.

6. Important notes about inductors • If current is constant, there is no voltage difference across inductor • If nothing in the circuit is changing with time, inductors act as short circuits • Sudden changes in current require infinite voltage • The current through an inductor must be a continuous function of time

7. Inductor Power and Energy • Power: • Energy:

8. Series combinations of inductors

9. Series combinations of inductors • A series combination of inductors can be represented as a single equivalent inductance

10. Parallel combinations of inductors

11. Example • Determine the equivalent inductance, Leq

12. First order systems • First order systems are governed by a first order differential equation • They have a single, first order, derivative term • They have a single (equivalent) energy storage elements • First order electrical circuits have a single (equivalent) capacitor or inductor

13. First order differential equations • General form of differential equation: • Initial condition:

14. Solutions of differential equations – overview • Solution is of the form: • yh(t) is homogeneous solution • Due to the system’s response to initial conditions • yp(t) is the particular solution • Due to the particular forcing function, u(t), applied to the system

15. Homogeneous Solution • Lecture 14: a dynamic system’s response depends upon the system’s state at previous times • The homogeneous solution is the system’s response to its initial conditions only • System response if no input is applied u(t) = 0 • Also called the unforced response, natural response, or zero input response • All physical systems dissipate energy yh(t)0 as t

16. Particular Solution • The particular solution is the system’s response to the input only • The form of the particular solution is dictated by the form of the forcing function applied to the system • Also called the forced response or zero state response • Since yh(t)0 as t, and y(t) = yp(t) + yh(t): • y(t) yp(t) as t

17. Qualitative example: heating frying pan • Natural response: • Due to pan’s initial temperature; no input • Forced response: • Due to input; if qin is constant, yp(t) is constant • Superimpose to get overall response

18. On previous slide, note steady-state response (corresponds to particular solution) and transient response (induced by initial conditions; transition from one steady-state condition to another)

19. RC circuit natural response – overview • No power sources • Circuit response is due to energy initially stored in the capacitor v(t=0) = V0 • Capacitor’s initial energy is dissipated through resistor after switch is closed

20. RC Circuit Natural Response • Find v(t), for t>0 if the voltage across the capacitor before the switch moves is v(0-) = V0

21. Derive governing first order differential equation on previous slide • Talk about initial conditions; emphasize that capacitor voltage cannot change suddenly

22. RC Circuit Natural Response – continued

23. Finish derivation on previous slide • Sketch response on previous slide

24. RC Circuit Natural Response – summary • Capacitor voltage: • Exponential function: • Write v(t) in terms of :

25. Notes: • R and C set time constant • Increase C => more energy to dissipate • Increase R => energy disspates more slowly

26. RC circuit natural response – example 1 • Find v(t), t>0

27. Example 1 – continued • Equivalent circuit, t>0. v(0) = 6V.