Ch1 Rlc Load

7/31/2019 Ch1 Rlc Load

1/127

Bridging Theory in

PracticeTransferring Technical Knowledgeto Practical Applications


2/127

Characteristics andModeling


3/127

Intended Audience: Engineers with a basic knowledge of resistive circuits

Engineers desiring a more intuitive understanding of capacitiveand inductive circuits

Topics Covered:

Introduction to Load Modeling

Introduction to Capacitors and RC networks

Introduction to Inductors and RL networks

Example Load Models

Expected Time:

Approximately 120 minutes

RLC LoadCharacteristics and Modeling


4/127


Introduction to Capacitors and RC Networks

Introduction to Inductors and RL Networks

Example Load Models:

Turning on an Incandescent Lamp

Switching a Relay



5/127



Introduction to Capacitors and RC

Networks Introduction to Inductors and RL

Networks

Example Load Models:Turning on an Incandescent Lamp

Switching a Relay


6/127

ElectromechanicalPower Conversion

Electrical power can be converted tomechanical power

Electrical power can turn-on a motor

Electrical power can drive a Solenoid

Electrical power can be converted to heat

Electrical power can a light a LED

(=

)


7/127

Load Modeling

Power converters (the loads) can be modeled byequivalent circuits composed of simple RLC passivecomponents


8/127





Networks


Switching a Relay


9/127

Capacitors

Physical object with the ability to store electriccharge(i.e. electric voltage)

Consists of two electrically isolated metalelectrodes, typically two conductive parallelplates

Is mostly used to store energy or for filtering

purposes The isolating material the dielectric defines

the type of capacitor: e.g. tantalum or ceramiccapacitor

Circuit s mbol:

C

C i


10/127

The capacitance of a parallel plate capacitor isproportional to:

C = Capacitance;a = Area of each parallel plate;d = Distance between parallel plates;

Larger value capacitors have larger plate areas and lessspacing between plates

They can store more energy (and are more expensive)

Capacitors:Physical Properties

C ~a

d

d

a

C i


11/127

The capacitance of a parallel plate capacitor is given by:

C = Capacitance

Units of: F = A s / V

= Permittivity =0

r

Units of: A s / V m = F / m

0 = Permittivity of vacuum = 8.854x10-12

Units of: A s / V m = F / m

r = Relative permittivity = 1 (free air)

Units of: (dimensionless)

Permittivity1) : the ability of a dielectric to store electrical potentialenergy under the influence of an

electric field1) Websters 9th edition

C = a

d

Capacitors:Physical Properties

d

a


12/127

Relative Size ofCapacitance Capacitance of a free air (r = 1) parallel plate

capacitor with the dimensions of A=1m2 andd=1mm is:

Typically, capacitance values in the 1F range areuncommon

Capacitances typically range from microFarads topicoFarads

1 microFarad = 1F = 10-6F

1 nanoFarad = 1nF = 10-9F

1 icoFarad = 1 F = 10-12F

( ) ( )

= = =

11 1

1r 1

1

. x F/m ( m )1 111111 1AC . x F111111

d x m111

C it El t i l


13/127

Capacitors ElectricalProperties

The stored electrical charge Q in a capacitor isproportional to the voltage V across the capacitor:Q ~ V

The proportional factor between stored electrical

charge and voltage difference is the capacitancevalue of the capacitor:Q = C V

Q = 8 As = 8 CoulombsV = 16V

C = Q/V = 8 A s / 16V = 0.5 Farad (F)

Unit [C] = A s / V = F


14/127

ara e an er aCapacitance

Parallel capacitors Serial capacitors

C1

C2

C

C = C1 + C2 1C

1+

1C

1=

C

1

C1 C2

C


15/127

Capacitor Experiment #1

An ideal current source is connected to acapacitor

V, I

t

IC

VC

tON

IIDEAL

CIIDEAL

tON IC

VC+

- The constant current

causes the voltage

to linearly rise across

the capacitor.

Constant current source supplies

the current regardless of the

voltage drop across the load.


16/127


An ideal current source is disconnected from acapacitor

V, I

t

tON

IIDEALVC

IC

tOFF

CIIDEAL

tOFF IC

VC+

-

If the constant current

source is removed,

the voltage across the

capacitor remains

constant.


17/127

An ideal current source is connected to acapacitor


The rate ofvoltage

change is proportional

to the current.

CIIDEAL

tON IC

VC+

-

IC1

VC1

tON

V, I

t


18/127

A variable ideal current source is connected toa capacitor


The rate ofvoltage

change is proportional

to the current.

CIIDEAL

tON IC

VC+

-

IC1

VC1

tON

V, I

t

IC2 VC2


19/127

A voltage source is connected to a capacitorthrough a resistor


CVIDEAL

tON RIC

VC+

-+-

The peak current

in the capacitor is

limited by theresistor.

The voltage across

the capacitor willreach VIDEAL

Ideal voltage source

supplies the voltage

regardless of the current

load.tON

V, I

t

VIDEAL/R ICVC

VIDEAL


20/127


A voltage source is connected through avariable resistor

CVIDEAL

tON R IC

VC

+

-VC

+

-

+-


21/127



tON t

R = R1

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

IC1

VC1

V, I

VIDEAL

R1


22/127



tON

V, I

t

IC1

VC1

R1

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

VIDEAL

R1 > R2

VIDEAL

R1


23/127



tON

V, I

t

IC1

VC1

R1

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

IC2

VIDEAL

R1 > R2

VIDEAL

R1

VIDEAL

R2


24/127



tON

V, I

t

IC1

VC1

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

VC2

IC2

VIDEAL

R1 > R2

VIDEAL

R1

VIDEAL

R2

Capacitors are

charged faster

through smaller

resistors


25/127



tON t

R1 < R3

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

IC1V

C1

V, I

VIDEAL

R1


26/127



tON t

R1 < R3

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

IC1VC1

V, I

VIDEAL

R1

IC3

VIDEAL

R3


27/127



tON t

R1 < R3

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

IC1VC1

V, I

VIDEAL

R1

IC3

VIDEAL

R3

VC3

Capacitors are

charged faster

through smaller

resistors

Capacitor Experiment


28/127

Capacitor Experiment#6

0 t

VC

VIDEAL

tC

R1 < R3

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

tC= RC

The rise time of the capacitor's voltage ismonitored:

0.63VIDEAL


29/127



0 t

VC

0.63VIDEAL

3tC

R1 < R3

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

tC= RC

tC

0.95VIDEAL0.87VIDEAL

2tC

De elopment of Mathematical


30/127

Current is defined as the amount of chargewhich is transferred in a certain period of time:I = Q / t

Development of MathematicalCapacitor Model: IC vs. VC

dqi or dq i dt

dt= = (1)

The relations above are derivatives for very small changesdifferentials can be used for quasi linear changes:

i=q/t or q=i.t (1a)

Development of Mathematical


31/127


Capacitance is defined as the stored charge ona capacitor vs. the voltage across the capacitor,

C = Q / V


dqi or dq i dt

dt= =

dqC or dq C dv

dv= =

(1)

(2)

In differential form:

C=q/t or q=C.v (2a)



32/127


Capacitance is defined as the stored charge ona capacitor vs. the voltage across the capacitor,

C = Q / V

Setting (2) equal to (1) results in:


dqi or dq i dt

dt= =

dqC or dq C dv

dv= =

(1)

(2)

dvi dt C dv or i C

dt

= =


33/127

rs

CVINVC across

plates

time

Voltage

across

Capacitor

Current

through

CapacitorVC

IC

R

VIN

Capacitor & Resistor


34/127

Capacitor & ResistorNetworks

In general, there are two basic options forcapacitor placement:

C in Series with Signal Path C from Signal Path to Ground

VINC

R VOUTC

RVIN VOUT



35/127

Capacitor & ResistorNetworks

Initially a DC voltage is applied at thesignal input IN.

Current passes through the capacitorand the voltage across the capacitorincreases


VIN

C

R

VOUT+ VC -

I

VINC

RVOUT

+VC

-

I

a ac o e o


36/127

Initially a DC voltage is applied at the signalinput IN.

Current passes through the capacitor and thevoltage across the capacitor increases

When the voltage across the capacitor is equalto the input voltage the current stops


VIN

C

R

VOUT+ VIN -

I=0AVIN

C

RVOUT

+VIN

-

I=0A

a ac o e oNetworks


37/127


Current passes through the capacitor and the

voltage across the capacitor increases

When the voltage across the capacitor is equalto the input voltage the current stops

Depending on the capacitors placement, the


VIN

C

R

0V

+ VIN -

I=0A VINC

RVIN

+VIN-

I=0A

Networks

Capacitance in Series with


38/127

Capacitance in Series withSignal Path

VX

I

VOUT

t1 t2

VIN

C

R

VOUT

+ VC -

I

t1

t2

VX


39/127

Signal Path

VX

I

VOUT

t1 t2

VIN

VIN/R

VIN

VIN

C

R

VOUT

+ VC -

I

t1

t2

VX


40/127

Signal Path

VX

I

VOUT

t1 t2

VIN

VIN/R

VIN

VIN

C

R

VOUT

+ VC -

I

t1

t2

VX

-VIN/R

-VIN

Path


41/127

Pathto Ground

VIN C

RVOUT

+VC

-

I

t1

t2

VX VX

I

VOUT

t1 t2

Path


42/127

Pathto Ground

VIN C

RVOUT

+VC

-

I

t1

t2

VX VX

I

VOUT

t1 t2

VIN

VIN/R

VIN

-VIN/R

Path


43/127

Pathto Ground

VIN C

RVOUT

+VC

-

I

t1

t2

VX VX

I

VOUT

t1 t2

VIN

VIN/R

VIN

-VIN/R


44/127

RC Networks - AC Signals

What happens when an AC input signal isapplied?C in Series with Signal Path C from Signal Path to Ground

VINC

R

VOUT

C

R

VOUT

t VIN t? ?


45/127

Capacitors and AC signals

Capacitors act like frequency dependentresistor (capacitive reactance, XC)

Xc~1/(fC)

Instead of reactance, impedance (Z) is

used to characterize circuit elements:

Z=1/(2fC)


46/127


Act like frequency dependent resistor(capacitive reactance, XC)

Instead of reactance, impedance (Z) usedfor circuit elements.

Impedance1): The apparent opposition inan electrical circuit to the flow ofalternating current that is analogous to

the actual electrical resistance to a direct

C1

XCf


47/127


Act like frequency dependent resistor(capacitive reactance, XC)

Instead of reactance, impedance (Z) used forcircuit elements.

Impedance1): The apparent opposition in anelectrical circuit to the flow of alternatingcurrent that is analogous to the actual electricalresistance to a direct current.

The impedance of a circuit element represents

C1

XCf


48/127


Act like frequency dependent resistor (capacitivereactance, XC)

Instead of reactance, impedance (Z) used for circuitelements.

Impedance1): The apparent opposition in an electricalcircuit to the flow of alternating current that is analogous

to the actual electrical resistance to a direct current. The impedance of a circuit element represents its

resistive and/or reactive components

Besides the magnitude dependency between voltage

and current the impedance, Z, gives also information

C1

XCf

Capacitors Impedance


49/127

Capacitor s ImpedanceMagnitude |ZC| vs. Frequency

C1

ZCf

|Z|= /(11fC) C= uF1

.11

.11

.11

.11

.11

.111

.111

.111

.111

.111

1 111 111 111 111 111

FREQUENCY (Hz)

|Z|(kohm)


50/127


VC,Max

iC,Max

t

iC,Max = VC,Max / |ZC|

= + /2 = + 90o The current leads the

volta e


51/127

RC networks AC Signals

The capacitor acts as a frequency dependentresistor

It determines the current magnitude at a givenvoltage

It causes a 90 degree phase shift between thecapacitor current and voltage across theca acitor


VINC

R

VOUT

C

R

VOUT

t VIN t


52/127

RC networks AC Signals

For high frequency signals:

The capacitor is low impedance

Signals can pass the capacitor For low frequency signals:

The capacitor is high impedance

Signals are blocked by the capacitor


VINC

R

VOUT

C

R

VOUT

t VIN t

|ZC|=1/(2fC)

C in Series with Signal Path


53/127

C in Series with Signal PathHigh Pass Configuration

-1

-1

-1

1

1

1

1

1 .000 .111 .000 .111 .000

-1

-1

-1

1

1

1

1

1 .111 .111 .111 .111 .111

VIN

VOUT

VOUT/VINMAX

Low f 0.32

Medium f 0.76

High f 0.90

CRVIN

VOUT

|ZC|=1/(2fC)

C from Signal Path to Ground


54/127

C from Signal Path to GroundLow Pass Configuration

VIN

VOUT

VOUT/VINMAX

Low f 0.96

Medium f 0.74

High f 0.39

C

RVIN

VOUT

-1

-1

-1

1

1

1

1

1 .000 .111 .000 .111 .000

-1

-1

-1

1

1

1

1

1 .111 .111 .111 .111 .111

|ZC|=1/(2fC)



55/127

Capacitor & ResistorNetworks Summary

Connected to DC voltages:

Capacitors will allow current to flow only until they arecharged

Once charged, they block future current flowFor AC signals:

Capacitors act similar to frequency dependent resistors

Low impedance at high frequencies

High impedance at low frequencies.


VINC

R

VOUT

C

R

VOUT

VIN

Characteristics and


56/127





Networks


Switching a Relay


57/127

Inductors

Physical object which can store a magnetic field (electriccurrent)

Consists of a conductive wire

Wire is typically a tightly wound coil around a center core

(toroid) Usually used for energy conversion and for filtering

purposes

The inductor type is usually defined by its core material

for example, air coil or ferrite coil inductors) Circuit symbol

Lor

Physical


58/127

PhysicalProperties of Inductors

The inductance of a toroid, for instance, is given by:

L = Inductance;N = Number of turns of the coil;a = Coil cross section;

= Average field length;

0 = permeability of vacuum =410-7 V.s/(A.M)

r =relative permeability

Larger value inductors have more turns and bigger cross sectionin less volume. They can store more energy (and may be moreexpensive).

A

l

Wire

Core

L=0.rN2.a/l

l


59/127

Inductance of a ToroidL = Inductance

Units of: H = V

s/AN = Number of turns of the coila = Coil cross section

Units of: m2

= Average field lengthUnits of: m

= Permeability = 0r ;Units of: H/m = V s/A m

0 = Permeability of free space = 410-7

Units of: H/m = V s/A mr = Relative permeability

Permeabilty1) : the property of a ferro-magnetic substance thatdetermines the degree in which it modifies the magnetic flux in theregion occupied by it in a magnetic field

1) acc. to Websters 9th edition

a

l

l

L=N2a/l

d


60/127

Inductance

Inductance of a free air toroid (r = 1) with the crosssection of a=5cm2, average field length of =10cm, andN=100 turns is

Inductors in the H range are used in switchingregulators

Small relays, solenoids usually have mH values ofinductance

Inductors in general typically range from a few Henries(H) to micro Henries (H):

1 microHenry = 1H = 10-6H

l

( ) ( )11 1 1

1

1

( )( H / m) x m1 111 1111 11L ~ . x H11111

x m1111

=

Inductors -


61/127

The change of magnetic field or coil flux ( ) in aninductor is proportional to the change of electriccurrent (I) flowing through the inductors windings:

~ I

The proportional factor between coil flux andcurrent is given by the inductance of the coil: =L I

I

N

Inductors Electrical Properties

Inductors -


62/127

Inductors Electrical Properties

The change of magnetic field or coil flux ( ) inan inductor is proportional to the change ofelectric current (I) flowing through theinductors windings: ~ I

The proportional factor between coil flux andcurrent is given by the inductance of the coil: = L I

L = /I = 1 Vs / 2 A = 0.5 Henry (H)

Unit [L] = Vs/A = H

I = 2A

N

1Vs

I d t


63/127

Inductance

Serial inductors Parallel inductors

L = L1 + L2 1L

1+

1L

1=

L

1

L1

L2

L

L1 L2

L

I d t E i t #1


64/127

An ideal voltage source is connected toan inductor

V, I

t

Inductor Experiment #1

VIDEAL

IL

VL

tON

tON

L

IL

VIDEAL

VL

+

-

The constant voltage

causes the current

to increase through

the inductor.

+

-


65/127

I d t E i t #3


66/127



V, I

t

VIDEAL

IL1

VL1

tON

tON

L

IL

VL

+

-

+

- The rate ofcurrentchange is proportional

to the voltage.

+

-

I d t E i t #3


67/127



V, I

t

VIDEAL

IL1

VL1

tON

tON

L

IL

VL

+

-

+

- The rate ofcurrentchange is proportional

to the voltage.

VL2

IL2

+

-

I d t E i t #4


68/127

A voltage source is connected to an inductorthrough a resistor


LVIDEAL

tON RIL

VL

+

-

+

-

tONt

VIDEALV

L

VIDEAL/RIL

The peak voltage

across the inductor

is VIDEAL.

The current through

the inductor will

reach VIDEAL/R.

I d t E i t #5


69/127



CVIDEAL

tON RIC

VC

+

-VC

+

-

+-

IL

VLL

I d t E i t #5


70/127

CVIDEAL

tON R

IC

VC

+

-VC

+

-

+-

IL

VL L



tON t

R1 > R2

IL1

V, I

VIDEAL

VIDEAL/R1

VL1

I d t E i t #5


71/127

CVIDEAL

tON R

IC

VC

+

-VC

+

-

+- L



tON t

R1 > R2

VL2

IL1

V, I

VIDEAL

VIDEAL/R1

IL2VIDEAL/R2

VL1

The smaller the

resistor, the longer

it takes the current

to become steady

IL

VL L

Ind ctor E periment #5


72/127

CVIDEAL

tON R

IC

VC

+

-V

C

+

-

+

-

ILL



tON t

R1 < R3

VL3

IL1

V, I

VIDEAL

VIDEAL/R1

IL3VIDEAL/R3

VL1

The smaller the

resistor, the longer

it takes the current

to become steady

IL

VL L

Ind ctor E periment #6


73/127


0 t

VL

VIDEAL

tC


tC

= L/R

CVIDEAL

tON RIC

VC

+

-VC

+

-

+

-

ILL

IL

VL L

0.37VIDEAL



74/127


0 t

VL

VIDEAL

tC


0.37VIDEAL

0.05VIDEAL

3tC

tC= L/R

CVIDEAL

tON RIC

VC

+

-VC

+

-

+

-

ILL

IL

VL L

2tC

0.14VIDEAL



75/127

The self induced coil voltage when exposed toan alternating magnetic field is proportional tothe change of coil flux vs. time:

Inductor Model: IL vs. VL

ind

d d

v N dt dt

= =



76/127

The self induced coil voltage when exposed to analternating magnetic field is proportional to the changeof coil flux vs. time:

The voltage v applied across an inductor is alwaysdirectly opposed to the self induced voltage vind:

v = -vind = Nd /dt = d/dt (=> d = vdt)


ind

d dv N

dt dt

= =

ind

d dv v N or d v dt

dt dt

= = = =

(1)

Development of Mathematicald d l


77/127

The self induced coil voltage when exposed to analternating magnetic field is proportional to the changeof coil flux vs. time:

The voltage v applied across an inductor is alwaysdirectly opposed to the self induced voltage vind:

v = -vind = Nd /dt = d/dt (=> d = vdt)

The inductance is defined as coil flux vs. coil current,L= / IL, differentially expressed as:


ind

d dv N

dt dt

= =

ind

d dv v N or d v dt

dt dt

= = = = (1)

dL or d Ldi

di

= =

(2)

Development of Mathematicald d l


78/127


Setting (1) equal to (2), the voltage -current relation for an inductor equalscan be found:

indd dv v N or d v dtdt dt

= = = = (1)

dL or d Ldi

di

= = (2)

div L

dt=

Inductors


79/127

Inductors

time

Voltage

across

Inductor

Current

through

Inductor

VL

IL

IL,max=VIN/R

VIN RVL

VIN

Inductor & Resistor


80/127

Networks

L in Series with Signal Path L from Signal Path to Ground

VINL

R VOUTL

RVIN VOUT

In general, there are two basic options forinductor placement:

ResistorNetworks


81/127

ResistorNetworks


VIN

L

R

VOUT

L

RVIN

VOUT

Initially a DC voltage is applied at thesignal input IN.

A voltage drops across the inductor andthe current through the inductorincreases

+ VL -

I

+VL-

I

ResistorNetworks


82/127

ResistorNetworks


VIN

L

R

VOUT

L

RVIN

VOUT


A voltage occurs across the inductor and thecurrent through the inductor increases

When the current through the inductor is at itsmaximum and remains constant, the voltage

across the inductor e uals zero

+ 0V -

I

+0V-

I

ResistorNetworks


83/127

ResistorNetworks


VIN

L

R

VIN

L

RVIN

0V

Initially a DC voltage is applied at the signal input IN.

A voltage drops across the inductor and the currentthrough the inductor increases

When the current through the inductor is at its maximumand remains constant, the voltage across the inductorequals zero

Depending on the inductors placement the steady state

final voltages are VOUT = VINor VOUT = 0V

+ 0V -

I

+0V-

I

Inductance in Series with


84/127

Signal PathV

X

I

VOUT

t1 t2

VIN

L

R

VOUT

+ VL -

I

t1

t2

VX

Inductance in Series withi l h


85/127

Signal PathV

X

I

VOUT

t1 t2

VIN

VIN/R

VIN

VIN

L

R

VOUT

+ VL -

I

t1

t2

VX

Inductance in Series withi l h


86/127

Signal PathV

X

I

VOUT

t1 t2

VIN

VIN/R

VIN

VIN

L

R

VOUT

+ VL -

I

t1

t2

VX

Inductance From Signalh G d


87/127

Path to Ground

VINL

R VOUT

+VL-

I

t1

t2

VXV

X

I

VOUT

t1

t2

Capacitance FromSi l P h G d


88/127

Signal Path to Ground

VIN

R VOUT

I

t1

t2

VX

VX

I

VOUT

t1 t2

VIN

VIN/R

VIN

L

+VL-

Capacitance FromSi l P th t G d


89/127

Signal Path to Ground

VIN

R VOUT

I

t1

t2

VX

VX

I

VOUT

t1 t2

VIN

VIN/R

VIN

L

+VL-

-VIN

RL Networks - AC Signals


90/127

RL Networks - AC Signals

What happens when an AC input signal isapplied?


VIN L R

VOUT

L

R

VOUT

t VIN t? ?

Inductors and AC signals


91/127


Act like frequency dependent resistor(inductive reactance, XL)


XL=2fL



92/127


Act like frequency dependent resistor(inductive reactance, XL)


Impedance: The apparent opposition in

an electrical circuit to the flow ofalternating current that is analogous tothe actual electrical resistance to a direct

current.

XL=2fL



93/127

Inductors and AC signals Act like frequency dependent resistor (inductive

reactance, XL)

Instead of reactance, impedance (Z) used for

circuit elements. Impedance: The apparent opposition in an

electrical circuit to the flow of alternating

current that is analogous to the actual electricalresistance to a direct current.

The impedance of a circuit element representsits resistive and/or reactive components

XL=2fL



94/127


Act like frequency dependent resistor (inductivereactance, XL)

Instead of reactance, impedance (Z) used for circuit

elements. Impedance: The apparent opposition in an electrical

circuit to the flow of alternating current that is analogousto the actual electrical resistance to a direct current.

The impedance of a circuit element represents itsresistive and/or reactive components

Besides the magnitude dependency between voltageand current the impedance Z gives also information

about the phase shift between the two.

XL=2fL

Inductors ImpedanceMagnitude |Z | vs Frequency


95/127

Magnitude |ZL| vs. Frequency

|ZL|(ohm)

frequency (Hz)

1

1

11

11

11

11

11

11

1 1111 1111 1111 1111 1111

|ZL|=2..f.L



96/127


VL,Max

iL,Max

t

iL,Max = VL,Max / |ZL|

= - /2 = -90o The current lags the

volta e

RL networks AC signals


97/127

RL networks AC signals

The inductor acts as a frequencydependent resistor

It determines the current magnitude at agiven voltage

It causes a 90 degree phase shift

between the inductor current and volta e


VINL

R

VOUT

L

R

VOUT

t VIN t

RC networks AC signals


98/127

RC networks AC signals

For low frequency signals:

The inductor is low impedance

Signals can pass the inductor For high frequency signals:

The inductor is high impedance

Signals are blocked by the inductor


VINL

R

VOUT

L

R

VOUT

t VIN t

|ZL

|=2fL

L in Series with Signal PathL P C fi i


99/127

-1

-1

-1

1

1

1

1

1 .000 .111 .000 .111 .000

-1

-1

-1

1

1

1

1

1 .000 .111 .000 .111 .000

Low Pass ConfigurationV

IN

VOUT

VOUT/VINMAX

Low f 0.96

Medium f 0.76

High f 0.38

LRVIN

VOUT

Z=2..f.L

L from Signal Path to GroundHi h P C fi ti


100/127

High Pass ConfigurationV

IN

VOUT

VOUT/VINMAX

Low f 0.32

Medium f 0.74

High f 0.92

L

RVIN

VOUT

-1

-1

-1

1

1

1

1

1 .000 .111 .000 .111 .000

-1

-1

-1

1

1

1

1

1 .111 .111 .111 .111 .111

|ZL|=2..f.L

Inductor & ResistorNetworks Summary


101/127

Networks Summary

Connected to DC voltages:

The voltage across an inductor changes as currentincreases

The voltage across inductor is 0V when current is constant

For AC signals:

Inductors act similar to frequency dependent resistors

Low impedance at low frequencies

High impedance at high frequencies.


VINL

R VOUTL

RVIN VOUT


102/127



103/127

Modeling




Networks


Switching a Relay

Lamp Experiment


104/127

Lamp Experiment

Turn on an incandescent light bulb andmeasure the current

1

2

14V

Iton

Lamp Experiment


105/127

p p

Turn on an incandescent light bulb andmeasure the current

Result:

~ 600mA

~ 5.6A

ton

1

2

14V

Iton

Model For an Incandescent LightBulb


106/127

Bulb

600mA

5.6A

ton

14V Light

Bulb



107/127

Bulb

600mA

5.6A

ton

14V

R1

V V11R = =1

I . A11

R = .0 000



108/127

Bulb

600mA

5.6A

ton

14V

23.3

[ ]1I =I exp -t/RC



109/127

Bulb

600mA

5.6A

ton

14V

23.3

R2

V. R =111 1I

R = .0 000



110/127

Bulb

600mA

5.6A

ton

14V

23.3

2.8

C

C= . mF11

Simulation of Lamp RC


111/127

Model

Time (ms)

InputC

urrent(A)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

500 150100 250200 350300ton

1

1

.000

.000

. mF00

V00

ton

Model


112/127

Time (ms)

InputC

urrent(A)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

500 150100 250200 350300ton

1

1

.000

.000

. mF00

V00

ton

A RC Load Model forIncandescent Light Bulbs


113/127

Incandescent Light Bulbs

The model for this lamps is represented by the networkbelow

When a lamp initially turns on, the filament is cold andhas a relatively low resistance BUT as the filamentwarms up, the resistance increases dramatically1

2

23.3

2.80

3.6mF

f(T)

Lamp Experiment


114/127

When a lamp initially turns on, the filament is cold andhas a relatively low resistance

As the filament warms up, the resistance increasesdramatically

~ 600mA

~ 5.6A



115/127

Modeling




Networks


Switching a Relay

Switching a Relay


116/127

Switching a Relay

To the right a highside switchingapplication is shown

The switch itself ismodeled as a simplemechanical switch

The relay can bemodeled as a lowohmic resistor andinductor connected in

VBattery

VR

VL

IL

S

Relay

Switching On a Relay


117/127

g y

timeVR

time

S

timeVL

timeIL

VBattery

+

VR

-

+

VL-

IL

S

open closed

VL decays over time

IL = (VR-VL) / R

Switching Off a Relay (1)


118/127

time

S

timeIL

time

time

g y

VBattery

IL

S

closedopen

+

VR

-

+

VL-



119/127

time

S

timeILIL cannot become

zero instantaneously!

timeVR

For VL < 0V,

VR < 0V

timeVL

VL becomes negative

to force the current to 0A

(VL

= -L*di/dt)

g y

VBattery

IL

S

closedopen

+

VR

-

+

VL-



120/127

time

S

timeIL

timeVR

timeVL

g y

VBattery

IL

S

closedopen

Arcing

+

VR

-

+

VL-

IL cannot go to

zero instantaneously!

For VL < 0V,

VR < 0V (R~0)

VL goes far below ground

to force the current to 0A

Switching Off aRelay No Arcing (1)


121/127

time

S

time

IL

timeVR

timeVL

Relay No Arcing (1)V

Battery

IL

S

closed open

ID +

VL-

+

VRtime

ID



122/127

time

S

time

IL

timeVR

timeVL

Relay No Arcing (2)V

Battery

IL

S

closed open

ID

Diode turns on andprovides a current path

+

VL-

+

VRtime

ID

Switching Off a Relay NoArcing (3)


123/127

time

S

time

IL

timeVR

timeVL

Arcing (3)VBattery

IL

S

closed open

ID +

VL-

+

VRtime

ID

If R~0, VL ~ VD

If R~0, VR ~ -VD



124/127

time

S

time

IL

timeVR

timeVL

Relay No Arcing (4)closed open

timeID

VBattery

IL

S

ID +

VL-

+

VR



125/127

time

S

time

IL

timeVR

timeVL

Relay No Arcing (5)closed open

timeID

If R~0, VL ~ 3VD

If R~0, VR ~ -3VD

VBattery

IL

S

ID +

VL-

+

VR

diL/dt = VL / L



126/127

Modeling




Networks


Switching a Relay


127/127

Thank you!

www.btipnow.com
http://www.btipnow.com/http://www.btipnow.com/

Date post:	05-Apr-2018
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