
Theory of OperationQuartz Crystal Theory of Operation and Design Notes Oscillator Theory of Operation and Design Notes
Quartz crystal units serve as the controlling element of oscillator circuits
by conversion of mechanical vibrations to electrical current at a specific
frequency. This is accomplished by means of the "Piezoelectric"
effect. Piezoelectricity is electricity created by pressure. In a piezoelectric
material, the application of mechanical pressure along an axis will result in
the creation of an electrical charge along an axis at right angles to the first.
In some materials, the obverse piezoelectric effect is found, which means that
the imposition of an electric field on the ends of an axis will result in a
mechanical deflection along an axis at right angles to the first. Quartz is
uniquely suited, in terms of mechanical, electrical and chemical properties, for
the manufacture of frequency control devices. Quartz crystal units which
oscillate within certain frequency and temperature ranges have been developed
over the years. Figure 1 shows the location of specific elements within a quartz
stone. The elements as shown above vibrate in various modes, the most important of
which are the extensional, flexural and shear. The mode of vibration determines
the maximum frequency, stability vs. temperature, and resistance of a given
element. The various modes of vibration are shown in Figure 2, while a
comparison of the various frequency stabilities vs temperature are shown in
Figure 3. Of the various elements, the "AT" cut has become the most popular
as it is available at relatively high frequencies, exhibits excellent frequency
vs temperature stability and is widely available at reasonable cost. The
frequency vs. temperature capabilities of the "AT" cut crystal unit
are illustrated in Figure 4. Fundamental vs. Overtone Drive Level Series vs. Parallel Load Capacitance Frequency Tolerance Frequency Stability Aging Pullability Equivalent Circuit Impedance/Reactance Curve Quality Factor (Q) Calculation of Load Capacitance
Cstray includes the pin to pin input and output capacitance of the microprocessor chip at the Crystal 1 and Crystal 2 pins, plus any parasitic capacitances. As a rule of thumb, Cstray may be assumed to equal 5.0 pF. Therefore, if CL1 = CL2 = 50pF, CL = 30pF. Trim Sensitivity
Where (Ct) is the sum of Co and CL. Solder Reflow of Surface Mount Devices Useful Crystal Equations
Oscillator Theory of Operation Crystal controlled oscillators may be considered as consisting of an amplifier and a feedback network that selects a part of the amplifier output and returns it to the amplifier input. A generalized depiction of such a circuit is shown below. Figure 1.0 In order for an oscillator circuit to operate, two (2) conditions must be met: (A) The loop power gain must be equal to unity. (B) The loop phase shift must be equal to 0, 2Pi, 4Pi, etc. radians. The power fed back to the input of the amplifier must be adequate to supply the oscillator output, the amplifier input, and to overcome circuit losses. The exact frequency at which an oscillator will operate is dependent on the loop phase angle shifts within the oscillator circuit. Any net change in phase angle will result in a change in the output frequency. As the usual goal of an oscillator is to provide a frequency that is essentially independent of variables, some means of minimizing the net phase shift must be employed. Perhaps the best, and certainly the most common means of minimizing the net phase shift is to use a quartz crystal unit in the feedback loop. The impedance of a quartz crystal changes so dramatically with changes in the applied frequency that all other circuit components can be considered as being of essentially constant reactance. Therefore, when a crystal unit is used in the feedback loop of an oscillator, the frequency of the crystal unit will adjust itself so that the crystal unit presents a reactance which satisfies the loop phase requirements. A depiction of the reactance vs. frequency of a quartz crystal unit is shown below. Figure 2.0 As is apparent from Figure 2.0, a quartz crystal unit has two frequencies of zero phase. The first, or lower of the two, is the series resonant frequency, usually abbreviated as Fs. The second, or higher of the two frequencies of zero phase is the parallel, or antiresonant frequency, usually abbreviated as Fa. Both the series and parallel resonant frequencies appear resistive in an oscillator circuit. At the series resonant point, the resistance is minimal and the current flow is maximal. At the parallel point, the resistance is maximal and the current flow is minimal. Therefore, the parallel resonant frequency, Fa, should never be used as the controlling frequency of an oscillator circuit. A quartz crystal unit can be made to oscillate at any point along the line between the series and parallel resonant points by the inclusion of reactive components (usually capacitors) in the feedback loop of the oscillator circuit. In such a case, the frequency of oscillation will be higher than the series resonant frequency but lower than the parallel resonant frequency. Because of the fact that the frequency resulting from the addition of capacitance is higher than the series resonant frequency, it is usually called the parallel frequency, though it is lower than the true parallel frequency. Just as there are two frequencies of zero phase associated with a quartz crystal unit, there are two primary oscillator circuits. These circuits are generally described by the type of crystal unit to be used, namely ìseriesî or ìparallel.î SERIES CIRCUIT: Figure 3.0 As is apparent from Figure 3.0, a series resonant oscillator circuit provides no means of adjusting the output frequency, should adjustment be required. In the above circuit, resistor R1 is used to bias the inverter and to cause it to operate in its linear region. This resistor also provides negative feedback to the inverter. Capacitor C1 is a coupling capacitor, used to block DC voltage. Resistor R2 is used to bias the crystal unit. This resistor strongly influences the drive current seen by the crystal unit, therefore care must be taken that too small a value is not chosen. Crystal unit Y1 is a series resonant crystal unit, specified to operate at the desired frequency and with the desired frequency tolerance and stability. PARALLEL CIRCUIT: This circuit uses a single inverter, with two capacitors in the feedback loop. These capacitors comprise the ìload capacitanceî and together with the crystal unit, establish the frequency at which the oscillator will operate. As the value of the load capacitance is changed, so is the output frequency of the oscillator. Therefore, this circuit does provide a convenient means of adjusting the output frequency, should adjustment be required. The resistors R1 and R2 serve the same functions as detailed for the series resonant circuit shown in Figure 3.0. The two load capacitors, CL1 and CL2, serve to establish the frequency at which the crystal unit and therefore the oscillator will operate. Crystal unit Y1 is a parallel resonant crystal unit, specified to operate with a specified value of load capacitance, at the desired frequency and with the desired frequency tolerance and stability. LOAD CAPACITANCE:
Where CL1 and CL2 are the load capacitors and CS is the circuit stray capacitance, usually 3.0 to 5.0 pF. It must be noted that changes in the value of the load capacitance will result in changes in the output frequency of the oscillator. Therefore, if precise frequency control is required, then a precise specification of load capacitance is required. To illustrate, assume that a crystal unit is specified to operate at a frequency of 20.000MHz with a load capacitance of 20.0 pF. Assume that the crystal unit is then placed in a circuit which presents a value of 30.0 pF. The frequency of the crystal unit will then be lower than the specified value. Conversely, should the circuit in question present a value of 10.0 pF, the frequency will be higher than the specified value. The relationship between frequency and load capacitance is shown below. Figure 5.0 DRIVE LEVEL: POWER = (Irms^{2} * R) (2) Where I is the rms current through the crystal unit and R is the maximum resistance value of the specific crystal unit in question. Equation (2) is simply ìOhms lawî for power. Measurement of the actual drive level in an operating oscillator circuit may be accomplished by temporarily inserting a resistor in series with the crystal unit. The resistor must be of the same ohmic value as the crystal unit. The voltage drop across the resistor may then be read and the current and power dissipation calculated. The resistor must then be removed. As an alternative means of measuring the drive level, a current probe may be used at the output lead of the crystal unit, if space permits. FREQUENCY VS MODE: Should it be desired to develop an oscillator at a frequency higher than the limiting frequency, advantage must be taken of the fact that quartz crystal units will oscillate at odd integer multiples of their ìfundamentalî frequency. We may define the ìfundamentalî frequency as ìthat frequency which naturally occurs at a given set of mechanical dimensions.î Therefore, if a crystal unit has a fundamental frequency of 10.0MHz, it can also be made to oscillate at 3, 5, 7, etc. times the fundamental. That is, the unit will oscillate at 30.0, 50.0, 70.0, etc. MHz. These multiples of the fundamental frequency are called ìovertonesî and are identified by the integer of multiplication, as in the ìthird overtoneî, the ìfifth overtoneî, etc. When use at an overtone frequency is required, the crystal unit must be specified to operate at the desired frequency and on the desired overtone. One should never attempt to order a fundamental mode crystal unit and then operate it at an overtone frequency. This is so due to the fact that the crystal manufacturing processes differ for fundamental and overtone crystal units. In many cases, the characteristics of the integrated circuit used in a particular oscillator design dictate that the fundamental frequency of the crystal unit be suppressed in order to ensure operation at the desired frequency and on the desired overtone. In such cases, it is usually necessary to modify the oscillator circuit. One method of modification is to add a ìtankî circuit, consisting of an inductor and a capacitor. These modifications are shown in Figure 6.0 and 7.0 Figure 6.0 Figure 6.0 depicts the modification of a series resonant circuit while Figure 7.0 depicts the modification of a parallel resonant circuit. Figure 7.0 In both cases, the tank circuit is tuned to resonate at some frequency between the fundamental and the desired frequency. This results in the unwanted frequency being shunted to ground, leaving only the desired frequency being present at the output of the oscillator. DESIGN CONSIDERATIONS: NEGATIVE RESISTANCE: Values of negative resistance exceeding five times the maximum resistance of the crystal unit are better yet. As negative resistance tends to decrease at elevated temperatures, it is recommended that the test be performed at the highest temperature of the operating range. © 2008 Fox Electronics. All rights reserved. Revised:
