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 contacts Quartz 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 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.quartz oscillator Walter Guyton Cady 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. crystalsceramicselectric potentialstress 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 movement Electrical energy -- Voltage and current Rs : (ESR) Equivalent series resistance Co : (Shunt Capacitance) Electrodes capacitance C1 : (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 C 0 :

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 (F s and F p ). 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, F s, 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, F a the real part can be neglected. At this frequency, the impedance has the maximum value.

10/31 Resonant frequencies calculus The imaginary part must be zero (real impedance) In the brackets, the term with R s can be neglected:

11/31 Resonant frequencies calculus The solution are:

12/31 Impedance value at resonant frequency

13/31 Remarks The series resonant frequency depends only by L 1 and C 1 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 C ech =C 0 +C p. 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 10 4 and 10 6.

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

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 :

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.

21/31 Analyzing nonlinear circuits

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

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 Chord methodTangent methodSecant 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

28/31 Linearization – nonlinearities reducing process

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.

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].

31/31 Problems  Propose a method to obtain the following characteristic starting from the mentioned nonlinear element.