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 inductors1. 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:t

)t(Li

2

21

Series combinations of inductors

+ -v1(t) + -v2(t)

+

-vN(t)

Series combinations of inductors• A series combination of inductors can be

represented as a single equivalent inductance

Þ

Parallel combinations of inductors

i1(t) i2(t) iN(t)

Þ

Example

• Determine the equivalent inductance, Leq

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:

• 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

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 Þ u(t) = 0• Also called the unforced response, natural response, or

zero input response• All physical systems dissipate energy Þ yh(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 yh(t)0 as t, and y (t) = yp(t) + yh(t):

• y (t) yp(t) as t

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

• 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) = V0

• 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-) = V0

• 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.