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

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

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

Inductor voltage-current relations Differential form: Integral form:

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

Important notes about inductors 1.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 2.Sudden changes in current require infinite voltage The current through an inductor must be a continuous function of time

Inductor Power and Energy Power: Energy:

Series combinations of inductors

A series combination of inductors can be represented as a single equivalent inductance

Parallel combinations of inductors

Example Determine the equivalent inductance, L eq

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

First order differential equations General form of differential equation: Initial condition:

Solutions of differential equations – overview Solution is of the form: y h (t) is homogeneous solution Due to the system’s response to initial conditions y p (t) is the particular solution Due to the particular forcing function, u(t), applied to the system

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  y h (t)  0 as t 

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 y h (t)  0 as t , and y (t) = y p (t) + y h (t): y (t)  y p (t) as t 

Qualitative example: heating frying pan Natural response: Due to pan’s initial temperature; no input Forced response: Due to input; if q in is constant, y p (t) is constant Superimpose to get overall response

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)

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

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

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

RC Circuit Natural Response – continued

Finish derivation on previous slide Sketch response on previous slide

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

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

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

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