Download Lecture 12

Lecture 12
Quartz resonators. Non-linear passive
electronic components
Passive Electronic Components and Circuits (PECC)
V. Bande, Applied Electronics Department
www.ael.utcluj.ro (English version)-> Information for students
1
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Quartz resonators
 Structure
Casing
Quartz
crystal
Socket
Silver electrodes
(on both sides)
Silver contacts
Dry inert
gas
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Quartz resonators
 Short history
• Coulomb is the first that scratches the surface in respect with the
piezoelectric effect.
• Currie brothers are the first scientists that reveal the phenomenon - in
1880.
• During first World War, quartzes are being for submarines detection
– SONAR sensors.
• 1920 – Walter Cady – discovers how to control frequency with the
help of a quartz.
• 1926 – the first radio station (NY) is broadcasting on a quartz
controlled frequency.
• During World War II, the US Army modifies all its communication
equipment in order to generate quartz controlled frequencies.
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Quartz resonators
 Piezoelectric effect
• Under the effect of a variable electrical field, the quartz crystal is
mechanically vibrating on the same frequency as the electrical field.
• If the oscillating frequency has a certain value, the mechanical
vibration maintain as well the electrical field.
• The frequency at which this phenomenon occurs is called
piezoelectric resonance and is strongly dependent by the quartz
crystal’s dimensions.
• The piezoelectric effect can be used to generate very stable electrical
frequencies (quartz controlled oscillators), force measurement
(piezoelectric sensors) by acting on the quartz dimensions and
modifying its resonance frequency.
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Equivalent circuit
 Mechanical – Electrical Analogy
Cm
Lm
Rs
C0
Mechanical energy
Electrical energy
Pressure
and
displacement
Voltage
and
current
(Lm,Cm)
Rs – ESR – Equivalent Series Resistance – models the quartz energy losses
C0 – Shunt Capacitance – the electrodes parasitic capacitance
Cm, Lm– the LC circuit that models the movement (displacement)
Equivalent circuit
 The equivalent electrical impedance
• The equivalent electrical circuit is basically a series RLC circuit
connected in parallel with a C0 capacitance:
Z ech
Z ech 
 2 L1C1  1  jRs C1
 
 Rs C1C0  j (C0  C1   2 L1C1C0 )
1
1

 L C  1
2
2

1
1
  2 R s2C12
 2 R 2C 2C 2  (C0  C1   2 L1C1C0 ) 2
s
1
0
Equivalent circuit
 The variation of the impedance module
• In the adjacent picture, the
reactance
(imaginary
part)
variation is presented.
• There are two frequencies at
which the reactance becomes
zero: Fs and Fa. Thus, in this
situation, the quartz impedance
has only real part.
Equivalent circuit
 The electrical meaning of the resonance frequencies
• At this two resonance frequencies, the equivalent impedance has a
purely resistive behavior  the phase-shift between voltage and
current is zero.
• The series resonance frequency – Fs – is the series LC circuit
resonance frequency. At this value, the impedance has minimum
value.
• The parallel resonance frequency – Fa – is the frequency at which the
real part can be neglected. At this value, the impedance has maximum
value.
Equivalent circuit
 Fs and Fa determination
Z ech



 


2
2
2
2
2 2 2
1 Rs C1C0  L1C1  1  Rs C1 C0  C1   L1C1C0  j  L1C1  1 C0  C1   L1C1C0   R s C 1 C0
 

 2 R s2C 12C 02  (C0  C1   2 L1C1C0 ) 2
• If we impose the condition that the imaginary part to be zero (purely
resistive impedance:
 4 L12C12C0   2 L1C1 C1  C0   L1C1C0  Rs2C12C0  C1  C0   0
 4 L12C12C0   2 L1C 2  2 L1C1C0  Rs2C12C0  C1  C0   0
1
• In the parenthesis, the term that contains the Rs can be neglected –
very low value, almost zero:


  b 2  4ac  L12C12 4C02  4C0C1  C12  4 L12C12C02 C0  C1   L12C14

Equivalent circuit
 Fs and Fa determination
• The solutions are:
12, 2
 b   2 L1C1C0  L1C12  L1C12


2a
2 L12C12C0

1
1
 
 f1  f s 
L1C1
2
2
1
1
L1C1
C0  C1
1
 
 f2  f p 
L1C1C0
2
2
2
1
CC
L1 1 0
C0  C1
Equivalent circuit
 The impedance value at both resonance frequencies:
1 1  j
Z ech (1 ) 
Rs C1
L1C1
L1C1
RC 1
 L1C1 s 1
 Rs
1 Rs C1C0

L1C1 C1
LCC 
 j  C0  C1  1 1 0 
L1C1 
L1C1

L1C1 (C0  C1 )
Rs C1
1 j
L1C1C0
L1Cs
Z ech (2 )  L1Cs

Rs C1C0
L1C1C0 (C0  C1 )
 j (C0  C1 
)
L1C1C0
L1Cs

 L1C1 
L1Cs 
 1
1
 L1Cs   L1
Rs C1C0
Rs C1 C0  C0 
1  
L1Cs
C1  C1 
Equivalent circuit
 Conclusions:
• The series resonance frequency is dependent by the L1 and C1
parameters, thus is dependent by the quartz geometrical parameters.
This frequency can be adjusted only through mechanical actions!
• The parallel resonance frequency can be adjusted in a small domain
by connecting a Cp capacitance in parallel with the quartz crystal. This
capacitance will be connected in parallel with the C0 – electrodes
capacitance, resulting an equivalent capacitance: Cech = Co + Cp. The
boundaries between which the adjustment can be made are very close,
because growing the Cech, you can reach the series resonance
frequency value.
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Quartz resonators parameters
• The nominal frequency – is the resonator’s assigned frequency during
fabrication and its being printed on the resonator’s casing.
• The load resonance frequency – is the oscillating frequency for the case
in which a certain specified capacitance is connected in parallel.
• The adjustment tolerance – is the maximum possible deviation of the
oscillating frequency in respect with the nominal frequency.
• The temperature domain tolerance – is the maximum possible
deviation of the oscillating frequency when the temperature varies
between minimum and maximum admitted values.
• The resonance equivalent series resistance – is the resistance
measured at the series resonance frequency (between 25 and 100 ohms
for the most common crystals).
• The quality factor – values between 104 and 106:
2 L1
Q
Rs
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Quartz oscillators
• Inside electronic circuits that contain
quartzes, the load connected at its
terminals can be viewed as a Rl
impedance.
• Depending on the relationship
between Rl and Rs, there can be three
different regimes:
Rs
 Damped regime – oscillation attenuation
 Amplified regime – oscillation amplification
 Auto-oscillating regime – oscillation sustaining
Q
Rl
Quartz oscillators
 Damped regime
Quartz oscillators
 Amplified regime
Quartz oscillators
 Auto-oscillating regime
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Thermistors
• The thermistors are variable resistors that have a very fast resistance
variation when the temperature is changing.
• The temperature variation coefficient can be negative – NTC (negative
temperature coefficient – components fabricated since 1930) or
positive – PTC (positive temperature coefficient – components
fabricated since 1950).
• Both thermistor types are non-linear, the resistance variation law with
the temperature being:
Rth  R 0  A  e
Rth  A  e
B
T
B
T
Thermistors
 NTC’s and PTC’s thermistors
• The temperature variation coefficient is defined as follows:
1 dRth
B
T 

 2
Rth dT
T
• If the material constant “B” is positive, then we will have an NTC
thermistor, if “B” is positive we will have a PTC thermistor.
Thermistors
 Non-linear circuits analysis
R
Rth
E
Rth
v
O1
Rth
1
E 
E
R
R  Rth
1
Rth
R
PTC : T  Rth 
 vO1 
Rth
vO1 
E
R
v
O2
R
1
E 
E
Rth
R  Rth
1
R
Rth
NTC : T  Rth 
 vO 2 
R
vO 2 
Thermistors
 Using thermistors as transducers
• The thermistors dissipated power must be lower enough in order that
the supplementary heating produced inside the thermistor body to be
negligible.
• This condition can be assured by connecting high value resistances in
series with the thermistor which will lead to a smaller current that
passes through the thermistor.
Thermistors
 Example: A voltage divider with a NTC thermistor
b
3450
Vout
1.286374
1.376124
1.468281
1.562551
1.658619
1.756156
1.854822
1.954269
2.05415
2.154121
2.253847
2.353002
2.451281
2.548394
2.644074
2.73808
2.830192
2.920219
3.007996
3.093382
3.176262
R
10000
E
5
Resistance vs. Temperature for NTC Thermistors
30000
3.2
3
RT
25000
2.8
Vout
2.6
20000
2.4
2.2
15000
2
1.8
10000
1.6
1.4
5000
1.2
0
5
10
15
20
25
Temperature (C)
RT  RT0 e
 b T0 T  


 T0T 
30
35
40
Vout (V)
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
T0
25
RT
28868.95
26333.94
24053.43
21998.96
20145.56
18471.27
16956.77
15585.01
14340.97
13211.32
12184.3
11249.45
10397.5
9620.204
8910.211
8260.974
7666.646
7122.002
6622.364
6163.541
5741.773
Resistance (Ohms)
RT0
10000
T
Quartz resonators. Non-linear passive electronic components
Quartz resonators
Structure
Short history
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
Non-linear passive electronic components
Non-linear resistors – thermistors
Nonlinearity phenomena
Nonlinearity phenomena
• Almost al physical quantities variation laws are non-linear!
• As a consequence, the electronic components characteristics which are
based on those laws are also non-linear.
• The non-linear systems analysis using linear methods, specific to linear
systems introduces errors. Those methods can be applied only on
restrictive small intervals of quantities variations. In this way, the
errors are being kept under the maximum allowed errors.
Nonlinearity phenomena
 Linearization – Piece by piece linearization
y
y
y
B
B
0
0
x

A
Chord (ro. Coarda)
method
B
x

0
x

A

Tangent method
A
Secant method
Nonlinearity phenomena
 Linearization – Piece by piece linearization
• You can either impose the number of the intervals on which the
linearization is being made and different errors will occur from an
interval to another.
• Or you can impose the maximum acceptable error during the
linearization procedure, thus resulting the number on the interval on
which the linearization can be made and also the interval maximum
length.
• For both condition, at the end of each interval, respectively at the
beginning of the following interval, the continuity condition must be
assured.
Nonlinearity phenomena
 Linearization – nonlinearities elimination procedure
v
i1
R1
i2
R2
v1
v2
is
Rs
is
R1
vs
R2
v1
v2
vs
ip
Rp
0
vp
i
i1
R1
ip
ip
i2
R2
vp
Nonlinearity phenomena
 Linearization – nonlinearities elimination procedure
v
i1
R1
i2
R2
v1
v2
is
Rs
is
R1
vs
R2
v1
v2
vs
ip
Rp
0
i
vp
i1
R1
ip
ip
i2
R2
vp
Nonlinearity phenomena
 Linearization – an example:
v
• Please determine the voltagecurrent characteristic for the
situations
in
which
the
components with the two
characteristics revealed in the
picture are connected in series,
respectively in parallel.
0
i