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Lecture 05 Friday, Sept. 142.004 Fall 07

Summary from previous lecture

Electrical dynamical variables and elements

Electrical networks

Charge q(t), Q(s).Current i(t) = q(t), I(s) = sQ(s).

Voltage v(t), V(s) = Z(s)I(s).

Resistor v(t) = Ri(t), ZR(s) = R.Capacitor i(t) = Cv(t), ZC(s) = 1/Cs.Inductor v(t) = Ldi(t)/dt, ZL(s) = Ls.

+

++

v1

vk

Pvk(t) = 0

PVk(s) = 0

i1

ik

P ik(t) = 0PIk(s) = 0

Z1 Z2

Impedances in series

Z = Z1 + Z21

G =

1

G1 +

1

G2 .

Z1 Z2 Impedances in parallel

1Z

= 1Z1

+ 1Z2

G = G1 + G2.

+

V2Vi

Z1

Z2

+

+

I

Vi V2Z2Z1 + Z2

Voltage divider

KCL

KVL

OpAmp in feedbackconfiguration:

Vo(s)

Vi(s)=

Z2(s)

Z1(s).

V1(s)I2(s)

Ia(s)I1(s)

Z1(s)

Z2(s)

Vi(s)

Vo(s)

+

-

Figure by MIT OpenCourseWare.

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Lecture 05 Friday, Sept. 142.004 Fall 07

Goals for today

The DC motor: basic physics & modeling,

equation of motion,

transfer function. Next week:

Properties of 1st and 2nd order systems

Working in the s-domain: poles, zeros, and their significance

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Lecture 05 Friday, Sept. 142.004 Fall 07

Power dissipation in electrical systems

+ v(t)

i(t)

Instantaneous power dissipation

P(t) = i(t) v(t).

Unit of power: 1 Watt=1 A 1 V.NOTE: P(s) 6= I(s) V(s). Why?

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4/12Lecture 05 Friday, Sept. 142.004 Fall 07

DC Motor as a system

Transducer:

Converts energy from one domain (electrical)

to another (mechanical)

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5/12Lecture 05 Friday, Sept. 142.004 Fall 07

Physical laws applicable to the DC motor

F = (i B) l = iBl (i

B) ve = V B l = V B l (V

B)

Lorentz law:magnetic field applies force to a current

(Lorentz force)

Faraday law:moving in a magnetic field results

in potential (back EMF)

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6/12Lecture 05 Friday, Sept. 142.004 Fall 07

DC motor: principle and simplified equations of motion

multiple windings N:continuity of torque

T = 2F r = 2(iBNl)r

ve = 2V BNl = 2(r)BN l

or

T = Kmi

ve = Kv

where

Km 2BNlr torque constant

Kv 2BNlr back-emf constant

(Lorentz law)

(Faraday law)

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7/12Lecture 05 Friday, Sept. 142.004 Fall 07

DC motor: equations of motion in matrix form

ve

i

=

"2BNlr 0

01

2BNlr

#

T

orve

i

=

"Kv 0

01

Km

#

T

multiple windings N:continuity of torque

8/8/2019 MIT Lecture05

8/12Lecture 05 Friday, Sept. 142.004 Fall 07

DC motor: why is Km=Kv?

Pin = Pout

ive = T

Kvi = Kmi

Kv = Km

(power conservation)

QED.

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9/12Lecture 05 Friday, Sept. 142.004 Fall 07

DC motor with mechanical load and

realistic electrical properties (R, L)

Equation of motion Electrical

KCL: vs vL vR ve = 0

vs Ldi

dtRiKv = 0

Equation of motion Mechanical

Torque Balance: T = Tb

+ TJ

Kmi b = Jd

dt

Combined equations of motion

Ldidt

+ Ri + Kv = vs

Jd

dt+ b = Kmi

inductance

(due to windings)

load

(inertia)

dissipation

(resistance of windings)

dissipation

(viscous friction

in motor bearings)

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Lecture 05 Friday, Sept. 142.004 Fall 07

Review: step response of 1st order systems weve seen

Inertia with bearings (viscous friction)

RC circuit (charging of a capacitor)

DC motor with inertia load, bearings and negligible inductance

+

vi

+

C vC

R

Step input Ts(t) = T0u(t) Step response

(t) =T0b 1 e

t/

, where =J

b.

Step input vi(t) = V0u(t) Step response

vC(t) = V0

1 et/

, where = RC.

Step input vs(t) = V0u(t) Step response

(t) =Km

RV0

1 et/

,

where =J

b +

KmKv

R

!.