1/31
Passive components and circuits - CCP
Lecture 12
2/31
Content Quartz resonators
Structure History Piezoelectric effect Equivalent circuit Quartz resonators parameters Quartz oscillators
Nonlinear passive electronic components Nonlinear resistors - thermistors Nonlinearity phenomenon
3/31
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.
5/31
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.
6/31
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
8/31
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.
9/31
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.
10/31
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:
11/31
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
12/31
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
13/31
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.
14/31
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.
sR
LQ 12
15/31
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: Damping regime Rl+Rs>0 Amplified regime Rl+Rs<0 Self-oscillating regime Rl+Rs=0
QRl
Rs
16/31
Quartz oscillators – case I, Rl+Rs>0
17/31
Quartz oscillators – case II, Rl+Rs<0
18/31
Quartz oscillators – case III, Rl+Rs=0
19/31
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 :
Tth
Tth
eR
eRB
B
0
A
AR
20/31
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
21/31
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
22/31
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.
23/31
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
24/31
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.
25/31
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
26/31
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.
27/31
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
28/31
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
29/31
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
30/31
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].
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
31/31
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
Top Related