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Passive components and circuits - CCP
Lecture 12
1/31
Content
 Quartz resonators






Structure
History
Piezoelectric effect
Equivalent circuit
Quartz resonators parameters
Quartz oscillators
 Nonlinear passive electronic components
 Nonlinear resistors - thermistors
 Nonlinearity phenomenon
2/31
Quartz structure
Housing
Quartz crystal
Bed-plate
Ag electrodes,
on both sides
Ag contacts
Inert gas,
dry
3/31
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.
4/31
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.
5/31
Equivalent circuit
L1, C1 :
Mechanical energy –
pressure and
movement
Electrical energy -Voltage and current
RS :
Energy
losses
Co :
Electrodes capacitance
Rs : (ESR) Equivalent series resistance
Co : (Shunt Capacitance) Electrodes capacitance
C1 : (Cm) Capacitance that modeling the movement
L1 : (Lm) Inductance that modeling the movement
6/31
Equivalent impedance
 The equivalent electrical circuit consist in a RLC series
circuit connected in parallel with C0 :
Z ech
 2 L1C1  1  jRs C1
 
 Rs C1C0  j (C0  C1   2 L1C1C0 )
Z ech 
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
7/31
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.
8/31
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.
9/31
Resonant frequencies calculus
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
The imaginary part must be zero (real 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 brackets, the term with Rs can be neglected:


  b 2  4ac  L12C12 4C02  4C0C1  C12  4 L12C12C02 C0  C1   L12C14
10/31

Resonant frequencies calculus
The solution are:

2
1, 2
 b   2 L1C1C0  L1C12  L1C12


2a
2 L12C12C0

12 
1
1
 f1  f s 
L1C1
2
1
L1C1
C0  C1
1
 
 f2  f p 
L1C1C0
2
2
2
1
C1C0
L1
C0  C1
11/31
Impedance value at resonant frequency
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 
12/31
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.
13/31
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.
Q
2 L1
Rs
14/31
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:
Rs
Q
Rl
 Damping regime Rl+Rs>0
 Amplified regime Rl+Rs<0
 Self-oscillating regime Rl+Rs=0
15/31
Quartz oscillators – case I, Rl+Rs>0
16/31
Quartz oscillators – case II, Rl+Rs<0
17/31
Quartz oscillators – case III, Rl+Rs=0
18/31
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 :
Rth  R 0  A  e
Rth  A  e
B
T
B
T
19/31
NTC and PTC thermistors
 The temperature coefficient is defined as:
1 dRth
B
T 

 2
Rth dT
T
 If the material constant B is positive, than the
thermistor is NTC, if the material constant B is
negative, the thermistor is PTC.
20/31
Analyzing nonlinear circuits
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 
21/31
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.
22/31
The performances obtained with a NTC divider
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
30
35
40
Temperature (C)
RT  RT0 e
 b T0 T  


 T0T 
23/31
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
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.
24/31
Linearization – approximation of
characteristics with segments
y
y
y
B
B
0
0
x

A
Chord method
B
x

0
x

A

Tangent method
A
Secant method
25/31
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.
26/31
Linearization – nonlinearities reducing process
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
27/31
Linearization – nonlinearities reducing process
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
28/31
Linearization – exercises
v
 Determine the voltagecurrent characteristic for
the situations of
connecting the
components with the
characteristics from the
figure, in series or
parallel.
0
i
29/31
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].
v [V]
5
4
3
2
1
0
-5
-4
-3
-2
-1
-1
1
2
3
4
i [mA]
5
-2
-3
-4
-5
30/31
Problems
v [V]
 Propose a method to
obtain the following
characteristic starting from
the mentioned nonlinear
element.
5
4
3
2
1
0
-5
-4
-3
-2
-1
-1
1
2
3
4
5
-2
-3
-4
-5
31/31
i [mA]