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Passive components and circuits - CCP

Lecture 12

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Content Quartz resonators

Structure History Piezoelectric effect Equivalent circuit Quartz resonators parameters Quartz oscillators

Nonlinear passive electronic components Nonlinear resistors - thermistors Nonlinearity phenomenon

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Quartz structure

Housing

Bed-plate

Ag electrodes,on both sides

Ag contactsQuartz crystal

Inert gas, dry

4/31

History Coulomb is the first that discover the piezoelectric

phenomenon Currie brothers are the first that emphasize this

phenomenon in 1880. In the first world war, the quartz resonators were used in

equipments for submarines detection (sonar). The quartz oscillator or resonator was first developed by

Walter Guyton Cady in 1921 . In 1926 the first radio station (NY) uses quartz for frequency

control. During the second World War, USA uses Quartz resonators

for frequency control in all the communication equipments.

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Piezoelectric effect Piezoelectricity is the ability of some materials (notably

crystals and certain ceramics) to generate an electric potential in response to applied mechanical stress .

If the oscillation frequency have a certain value, the mechanical vibration maintain the electrical field.

The resonant piezoelectric frequency depends by the quartz dimensions.

This effect can be used for generating of a very stable frequencies, or in measuring of forces that applied upon quartz crystal, modifying the resonance frequency.

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Equivalent circuit

RS :Energy losses

Co :Electrodes capacitance

L1, C1 :Mechanical energy – pressure and movementElectrical energy --Voltage and current

Rs : (ESR) Equivalent series resistanceCo : (Shunt Capacitance) Electrodes capacitanceC1 : (Cm) Capacitance that modeling the movement L1 : (Lm) Inductance that modeling the movement

7/31

Equivalent impedance

The equivalent electrical circuit consist in a RLC series circuit connected in parallel with C0 :

)(

11

0112

1001

1112

CCLCCjCCR

CRjCLZ

s

sech

2

0112

102222

2222

112

)(

11

01

1

CCLCCCCR

CRCLZ

s

s

ech

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Modulus variation

In the figure is presented the variation of reactance versus frequency (imaginary part)

Can be noticed that are two frequencies for that the reactance become zero (Fs and Fp). At these frequencies, the quartz impedance is pure real.

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Resonant frequencies significance

At these frequencies, the equivalent impedance have resistive behavior (the phase between voltage and current is zero).

The series resonant frequency, Fs, is given by the series LC circuit. At this frequency, the impedance have the minimum value. The series resonance is a few kilohertz lower than the parallel one .

At the parallel resonant frequency, Fa the real part can be neglected. At this frequency, the impedance has the maximum value.

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Resonant frequencies calculus

2

0112

102222

0222

0112

10112

0112

101112

01

)(

111

01

1

CCLCCCCR

CCRCCLCCCLjCCLCCCRCLCCRZ

s

sssech

The imaginary part must be zero (real impedance)

02

0

0102

12

0112

12

02

121

4

0102

12

01101112

02

121

4

1

CCCCRCCLCLCCL

CCCCRCCLCCCLCCL

s

s

41

2110

20

21

21

2110

20

21

21

2 4444 CLCCCCLCCCCCLacb

In the brackets, the term with Rs can be neglected:

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Resonant frequencies calculusThe solution are:

10

011

2011

1022

11

111

21

02

121

211

2110112

2,1

1

2

1

1

2

11

2

2

2

CCCC

L

ffCCL

CC

CLff

CL

CCL

CLCLCCL

a

b

p

s

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Impedance value at resonant frequency

ss

s

s

ech RCCL

CRCL

CLCCL

CCjCL

CCR

CL

CRj

CLZ

111

111

11

01110

11

01

11

1

111

111

1)(

1

0

1

01

1

1

01

1

111

011

1001110

1

01

1

1

011

1011

12

1

11

))(

(

1)(

)(

CC

CCCR

L

CL

CCRCLCL

CL

CCLCCCCL

CCjCL

CCRCL

CRj

CCLCCCL

CLZ

s

s

s

ss

s

s

s

s

sech

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Remarks

The series resonant frequency depends only by L1 and C1 parameters, (crystal geometrical parameters). Can be modified only by mechanical action.

The parallel resonant frequency can be adjusted, in small limits, connecting in parallel a capacitance. Results an equivalent capacitance Cech=C0+Cp.

The adjustment limits are very low because the parallel resonant frequency is near the series resonant frequency.

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Quartz resonator parameters Nominal frequency, is the fundamental frequency and is marked

on the body. Load resonance frequency, is the oscillation frequency with a

specific capacitance connected in parallel. Adjustment tolerance, is the maximum deviation from the nominal

frequency. Temperature domain tolerance, is the maximum frequency

deviation, while the temperature is modified on the certain domain. The series resonant equivalent resistance, is resistance measured

at series resonant frequency (between 25 and 100 ohms for the majority of crystals).

Quality factor, have the same significance as RLC circuit but have high values: between 104 and 106.

sR

LQ 12

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Quartz oscillators The load circuit is equivalent

with a load resistor Rl. Depending by the relation

between Rl and Rs we have three operation regimes: Damping regime Rl+Rs>0 Amplified regime Rl+Rs<0 Self-oscillating regime Rl+Rs=0

QRl

Rs

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Quartz oscillators – case I, Rl+Rs>0

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Quartz oscillators – case II, Rl+Rs<0

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Quartz oscillators – case III, Rl+Rs=0

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Thermistors They are resistors with very high speed variation of

resistance versus temperature. The temperature variation coefficient can be negative -

NTC (components made starting with 1930) or positive PTC (components made starting with 1950).

Both types of thermistors are nonlinear, the variation law being :

Tth

Tth

eR

eRB

B

0

A

AR

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NTC and PTC thermistors The temperature coefficient is defined as:

If the material constant B is positive, than the thermistor is NTC, if the material constant B is negative, the thermistor is PTC.

2

1

T

B

dT

dR

Rth

thT

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Analyzing nonlinear circuits

E vO 1

R

Rth v

O 2R

Rth

E

1

1

:

1

1

O

O

vRth

RRthTPTC

E

RthR

ERthR

Rthv

2

2

:

1

1

O

O

vR

RthRthTNTC

E

RRth

ERthR

Rv

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Condition for using thermistors as transducers The dissipated power on the thermistor must be

small enough such that supplementary heating in the structure can be neglected.

This condition is assured by connecting a resistor in series. This resistor will limit the current through the thermistor.

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The performances obtained with a NTC divider

RT0 T0b R E

10000 25 3450 10000 5T RT Vout

0 28868.95 1.2863742 26333.94 1.3761244 24053.43 1.4682816 21998.96 1.5625518 20145.56 1.65861910 18471.27 1.75615612 16956.77 1.85482214 15585.01 1.95426916 14340.97 2.0541518 13211.32 2.15412120 12184.3 2.25384722 11249.45 2.35300224 10397.5 2.45128126 9620.204 2.54839428 8910.211 2.64407430 8260.974 2.7380832 7666.646 2.83019234 7122.002 2.92021936 6622.364 3.00799638 6163.541 3.09338240 5741.773 3.176262

Resistance vs. Temperature for NTC Thermistors

5000

10000

15000

20000

25000

30000

0 5 10 15 20 25 30 35 40

Temperature (C)

Re

sis

tan

ce

(O

hm

s)

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Vo

ut

(V)

RT

Vout

TT

TT

TT eRR 0

0

0

b

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Nonlinearity phenomena Most variation laws of physical quantities are nonlinear. Consequently, the characteristics of electronic

components based on such dependencies are nonlinear. Analysis of nonlinear systems using methods specific for

linear systems introduce errors. These methods can be applied only on small variation domains, keeping in this way the errors bellow at a imposed limits.

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Linearization – approximation of characteristics with segments

A

x

y

0

B

A

x

y

0 B

A

x

y

0

B

Chord method

Tangent method Secant method

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Linearization – approximation of characteristics with segments

Imposing the number of linearization intervals, results different errors from one interval to other.

Imposing the error, results a number of linearization intervals, and dimensions for each interval.

In both situation, the continuity condition must be assured on the ends of linearization intervals.

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Linearization – nonlinearities reducing process

R1 R2

Rs R1 R2

i1

v1

i2

v2

is

vs

is

vs

v1 v2

Rp

ip

vp

R1

i1

R2

i2

vp

ip ip

0

v

i

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Linearization – nonlinearities reducing process

0

v

i

R1 R2

Rs R1 R2

i1

v1

i2

v2

is

vs

is

vs

v1 v2

Rp

ip

vp

R1

i1

R2

i2

vp

ip ip

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Linearization – exercises

Determine the voltage-current characteristic for the situations of connecting the components with the characteristics from the figure, in series or parallel.

0

v

i

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Problems A nonlinear element with the

voltage-current characteristics from the figure is considered.

Determine the resistance connected in series/parallel with the nonlinear element in order to extend the linear characteristic in the domain of [-5V; 5V].

Determine the resistance connected in series/parallel with the nonlinear element in order to extend the linear characteristic in the domain of [-3mA; 3mA].

0

v [V]

i [m A]

1

2

3

4

5

1 2 3 4 5-1

-2

-3

-4

-5

-1-2-3-4-5

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Problems

Propose a method to obtain the following characteristic starting from the mentioned nonlinear element.

0

v [V]

i [m A]

1

2

3

4

5

1 2 3 4 5-1

-2

-3

-4

-5

-1-2-3-4-5