CRYSTAL AND OSCILLATORS
Crystal And Oscillators
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1. Overview
The page also includes discrete oscillator elements such are crystals.
2. Crystals
Crystals are piezoelectric components which can be used to build an oscillator. Combined with driving circuitry, they form an oscillator which can be made to output a periodic waveform to be used as a clock source for digital logic (e.g. flipflops, microcontrollers, FPGAs, e.t.c.). They can also be called piezoelectric resonators.
2.1. Equivalent Circuit
A piezoelectric crystal resonator can be modelled as a series LCR circuit in parallel with a capacitor:
The series components \(C_1\), \(L_1\), and \(R_1\) model the physical properties of the piezoelectric crystal, and are called the motional arm[8]. They are not real physical electronic components inside the crystal. The parallel component \(C_0\) is the lead capacitance[2].
\(L_1\): This models the mechanical mass of the quartz in motion. Lower frequency crystals have a value of \(12H\) (yes, that’s whole Henries, much larger than micro/milliHenries of most real inductors!). This value can drop down to \(1100mH\) for the higher frequency crystals, which are smaller and therefore less mass.
\(C_1\): This models a number of mechanical properties of the quartz crystal: the stiffness, the area of the electrodes, and the thickness/shape of the wafer. The value for fundamental mode crystals ranges between \(0.005pF\) and \(0.030pF\).
\(R_1\): This models the impedance of the crystal when it is oscillating at it’s series resonant frequency. When a series LC circuit is at resonant frequency, it’s impedance is \(0\Omega\), therefore the impedance (and therefore current) is purely determined by this \(R_1\). \(R_1\) is inversely proportional to the active area of the crystal, therefore smaller crystals have a larger \(R_1\).
\(C_0\): This models the parallel capacitance (a.k.a. shunt capacitance) between the two leads of a crystal. It is the measured capacitance between the two leads when the crystal is not excited (i.e. not vibrating). \(C_0\) typically ranges from \(17pF\).
2.2. Series And Parallel Resonant Frequencies
The series resonant frequency \(f_S\) only depends on the motional arm (physical properties of the crystal) as shown in \(Eq.\ \ref{eq:seriesresfreq}\)[8].
The parallel resonant frequency \(f_P\) also depends on the parallel parasitic capacitance \(C_0\), as shown in \(Eq.\ \ref{eq:parallelresfreq}\)[8].
The series and parallel resonant frequencies are usually very close together. Crystals below 30MHz are operated at a frequency somewhere between the series and parallel resonant frequencies[8].
2.3. Quality Factor
The quality factor for crystal oscillators is extremely large, typically 10,000 or greater. This is due to the very low series resistance (typically around \(5\Omega\).
The quality factor is determined by the following equation:
where:
\(X_L\) is the impedance of the inductor.
2.4. 32.678kHz Crystals
32.678kHz is a popular frequency for crystals (also just shortened to 32kHz crystals) because it is exactly \(2^{15}\). This means you can use one with a 15bit binary counter to get a precise 1second (1Hz) clock or "tick". It is also a good tradeoff in terms of power consumption (lower frequency = lower power consumption, great for battery powered devices) and size (higher frequency means smaller crystal package). 32.678kHz crystal are very common in any embedded circuit design which needs a real time clock (RTC). As such, they are also "dirt cheap"!
Many microcontrollers have pins which you can connect a 32.678kHz crystal to, with pin names such as XTAL32
, 32K_XP/32K_XN
(ESP32). Sometimes the 32
is added to distinguish it from the "main" higher frequency oscillator pins, which typically support crystals in the frequency range from 148MHz. The microcontroller has an internal oscillator for driving the crystal. A Pierce oscillator is a popular oscillator topology used in microcontrollers to drive these crystals. For example, all of the 32.678kHz oscillators in the MSP430 range of microcontrollers use Pierce oscillators[7].
As shown in Figure 3, 32.678kHz crystals are also called tuning fork crystals, as the crystal is usually cut into the shape of a tuning fork[8] (a vibrates in a similar manner to a larger, metal one).
2.4.1. Turnover Temperature
Turnover temperature (\(T_O\)) is a term used with 32.768kHz crystals to describe the temperature at which the crystal is at it’s highest oscillation frequency. 32.768kHz crystals have a negative parabolic frequency response to temperature (frequency drops proportionally to the square of the temperature change) at the turnover temperature is at the maxima (at lower or higher temperatures, the frequency begins to drop). Most 32.678kHz crystals have a turnover temperature \(T_O\) between 20 and 30°C and \(\alpha\) of approx. \(0.034ppm^{\circ}C^2\)[6].
\(Eq.\ \ref{eq:32khzdrift}\) shows how to calculate the drift from the current operating temperature of the crystal.
where:
\(ppm\) is the drift from \(f_O\), the oscillation frequency at the turnover point, in partspermillion
\(T\) is the operating temperature of the crystal, in \(^{\circ}C\)
\(T_O\) is the temperature at the turnover point, in \(^{\circ}C\)
\(\alpha\) is a part specific coefficient, specified in the datasheet, in \(ppm^{\circ}C^{2}\). If no coefficient is listed, \(\alpha=0.034ppm^{\circ}C^{2}\) is a good assumption
Rather than using \(ppm\), \(Eq.\ \ref{eq:32khzdriftasratio}\) shows how you can instead write is a ratio of \(\frac{f}{f_O}\).
where:
\(f\) is the actual oscillation frequency
\(f_O\) is the oscillation frequency at the turnover point, typically \(32.678kHz\)
2.5. The Negative Resistance Test
The negative resistance test can be used to find the oscillator load safety margin present on your circuit design. The test is performed by inserting a potentiometer in series between the crystal and the oscillator (which may be inside a microcontroller). You then slowly increase the resistance until you find the point at which the oscillator fails to startup correctly[8].
2.6. OvenControlled Crystal Oscillators (OCXOs)
Highperformance crystal oscillators are kept with temperaturecontrolled environments to increase the stability of the oscillator. They are called ovencontrolled crystal oscillators (OCXOs).
The crystals are designed to have a turningpoint, a point of greatest stability, close to the oven temperature. OCXOs, rather than having a temperature stability in the ppm (partspermillion) range like normal crystals, have a stability in the ppb (partsperbillion) range (20ppb would be a viable stability).
Peltier devices can be used as the "oven" to keep the crystal’s temperature constant.
2.7. Popular Crystal Packages
The HC49/U package is a popular choice for older throughhole crystals.
Newer crystals come in small, custom SMD packages, with typically either 2 or 4 pins (with the 4pin packages usually have two GND pins).
2.8. Simulation
Crystal oscillators can be difficult to simulate accurately in most SPICEbased programs[3]. Most SPICE programs use the NewtonRaphson algorithm for converging to a solution. Unfortunately, the NewtonRaphson algorithm is not suitable for very high Q circuits, of which a crystal resonantor is definitely one (Q values of \(10,000\) or more!). The time step has to be set so small for accurate simulation of crystal resonantor circuits that it can take days of simulation to "startup" the ceramic resonantor (i.e. reach steadystate oscillation from poweron).
2.9. Crystal Component Packages
For info on crystal component packages, see the Crystal Packages page.
3. Oscillators
This site uses the word oscillator to represent a component with an selfcontained oscillating feature that has power, ground, and signal out pins. This site uses the word crystal to represent an component which contains a oscillating element (in the form of a crystal), which requires an external oscillation circuit before it useful.
3.1. Designators
A common designator prefix to use for oscillators is \(Y\) (e.g. \(Y1\)). I do not recommend using the prefix \(XC\) as this should be reserved for crystal oscillators.
3.2. Important Parameters
3.2.1. Phase Noise
Phase noise is a way of describing the stability of the crystal in the frequency domain.
3.2.2. StartUp Time
Symbol: \(T_{SU}\)
The startup time for most oscillators is within the range 220ms. This startup time can be important in lowpower designs when the start/stop time of the crystal results in wasted energy.
4. MEMS Oscillators
MEMS oscillators are built using small mechanical structures (less than 0.1mm in any dimension) that vibrate at set frequencies when electrostatic forces are applied. This mechanical vibratory part of a MEMS oscillator is called the MEMS resonator. This is etched into a silicon die, and surrounding electronics contain both the driving, measuring, and compensation circuitry.
They use less power than a crystalbased oscillator, making them suitable for batterypowered devices. They are manufactured using standard IC manufacturing processes, so they are also more durable. They typically have better frequency stability over their operating temperature range, with common values being 10ppm at room temperature and 100pm over their entire operating temperature range.
MEMS oscillators do not like ultrasonic cleaning baths. Ultrasonic baths may permanently damage the oscillator or cause long term reliability issue[1].
4.1. Packaging
MEMS oscillators have been made in packages which are also commonly used for crystal packages, such as the 2012 SMD package.
Some common industry sizes for oscillators include:

1612: 1.6 mm × 1.2 mm

2016: 2.0 mm × 1.6 mm

2520: 2.5 mm × 2.0 mm

3225: 3.2 mm × 2.5 mm

5032: 5.0 mm × 3.2 mm

7050: 7.0 mm × 5.0 mm
5. Wien Bridge Oscillator
The Wien bridge oscillator is a relatively simple oscillator that can generate reasonably accurate sine waves. It is named after a bridge circuit designed by Max Wien in 1891 for the measurement of impedances. William R. Hewlett (of HewlettPackard fame) designed the Wein bridge oscillator using the Wein bridge circuit and the differential amplifier.
However the modern way to draw this is to split up the noninverting and inverting feedback circuits like this:
In my opinion this is a clearer way of drawing the circuit. Wien bridge oscillators are used in audio applications.
The series RC and parallel RC circuits form highpass and lowpass circuit elements, respectively.
5.1. Wien Bridge Equations
Let’s first look at the series and parallel RC circuits that provide the positive feedback.
The impedance \(Z_S\) of the series RC circuit is:
The impedance \(Z_P\) of the parallel RC circuit is:
We can then write an equation for the voltage at the noninverting pin of the opamp in terms of the output voltage, and then describing it as a ratio we can get the gain of the RC network, \(\beta\)
(the symbol \(\beta\)
used here is consistent with the Barkhausen stability criterion):
Now if we focus on the purely resistive feedback network to the inverting pin of the opamp, you should recognize this as the standard noninverting gain configuration, where the gain is:
In steadystate oscillation, the reduction in amplitude of \(v_{out}\) to \(v_{noninv}\) as to be exactly "countered" by the gain provided from \(v_{noninv}\) to \(v_{out}\). This is also known as the Barkhausen criterion:
Now lets aim to separate the real and imaginary terms and write it as an equation which equals 0:
For this equation to hold true, both the real and imaginary parts must be equal to 0. If we focus on the real part first we can find \(\omega\)
in terms of \(R\) and \(C\):
Or in terms of natural frequency rather than angular frequency:
We can now look at the real part of the equation, which also must be 0. This gives us criterion for the ratio of the resistors \(R_3\) and \(R_4\):
We can plug this back into the equation for the noninverting gain of the amplifier so see what gain this results in:
5.2. Realistic Wien Bridge Oscillator Circuits
There is a problem with the above Wien Bridge oscillator circuits which limits them to the realm of theory only. It all comes back to the requirement that the Wien Bridge oscillator must have a loop gain of exactly 1 to function properly (Barkhausen stability criterion). If the gain is less than this, the oscillator will not start (or will stop if already started). If it is more than 1, the oscillator output will saturate and your sine wave output will start looking more like a square wave. Wien bridge oscillators typically need a nonlinear component (a component which has a resistance which changes with applied voltage) to actively limit the loop gain and keep it at 1.
Common methods of actively limiting the gain include using:

Incandescent bulb (resistance increases as it heats up)

Diodes across in parallel with feedback resistors (resistance decreases as voltage increases)

JFETs.
Wien bridge oscillators can also be made from a single supply[5].
5.3. Diode Limited Example And SPICE Simulation
I have just used the calculated capacitance and resistance values, and not picked the nearest realistic E96 value so that it’s easier to keep track of where the values come from. 
The first thing is to choose an oscillation frequency. Let’s choose \(1kHz\). We will also choose an arbitrary capacitance of \(C = 10nF\). It’s good to keep the capacitance somewhat low and use ceramic NP0/C0G type dielectrics, as they will introduce the lowest amount of distortion. This means we need a resistance of:
Let’s use diodes as our nonlinear element to make sure the loop gain stays at 1. The trick is to put the diodes in parallel with a portion of the \(R_3\) resistance, and make the inverting feedback gain slightly larger than 3 before the diodes begin to conduct.
Ignoring the diodes for a moment, let’s aim for a inverting gain of 3.2 and pick an arbitrary value for \(R_4\) of \(1k\Omega\).
Now, we want the oscillations at \(v_{out}\) to reach a decent proportion of the supply voltage before the diode kicks in. Since we are running of \(\pm 12V\) supplies, lets make the diodes begin to conduct at \(5V\). Let’s calculate how much current is flowing through the series leg of \(R_3\) and \(R_4\) when there is \(1V\) at the output:
Assuming the diodes begin to conduct at \(0.6V\), with \(0.312mA\) of current flowing this would be across a resistance of:
So we need to split \(R_3\) into two resistors in series, one of \(1.92k\Omega\) with the diodes across it \(R_{3,diode}\) and one of:
We can then simulate the circuit using ngspice (an opensource SPICElike circuit simulator):
You can count 6 cycles in approx. 6ms, which puts the simulated frequency at:
which agrees well with what we designed for!
You can download the following assets:
5.4. JFET GainLimited Example
Using a JFET to partially switch in another resistor in parallel with the groundconnected gain resistor \(R_4\) in the Wien bridge oscillator circuit is another method for preventing the oscillator for saturating (as opposed to the diode method shown above). This JFET gainlimited approach is meant to introduce less distortion than the diodelimited approach above, as the RC circuit driving the JFET’s gate does not change much over a single cycle (assuming a suitable large RC time constant is picked).
Schematics of this technique are shown below, with the circuit setup to oscillate at the same frequency as the diode gainlimited variant mentioned above.
Note the diode and RC circuit controlling the JFET’s gate. When the circuit is first powered up, the gate is at ground and hence the gatesource voltage \(V_{GS} = 0V\). Therefore the JFET is almost fully on (remember, JFETs are depletion mode devices), and \(R_5\) is in parallel with \(R_4\), increasing the gain of the opamp. As the output voltage beings to oscillate, on the negative part of the cycle, diode \(D_1\) will conduct and charge the RC lowpass filter \(C_3\) and \(R_6\) with a negative voltage. This will decrease \(V_{GS}\) below \(0V\), which will begin to turn the JFET off. This will then increase the equivalent resistance of \(R_5\) in parallel with \(R_4\) and decrease the opamp gain. This will continue until the system reaches a steadystate and oscillates forever.
And below are the simulation results for this circuit:
6. Ring Oscillators
A ring oscillator (a.k.a. RO) is an electronic oscillator made up of a chain of an oddnumber of digital logic NOT gates. The output of the last NOT gate is fed into the input of the first. The oscillator relies on the propagation delay from the input of the first NOT gate to the output of the last NOT gate to set the oscillation frequency.
6.1. Simulation
I ran into convergence issues when using the 74HCU04
SPICE model I found floating around on the internet (located in a file called [74HCng.lib](ringoscillatorsingle/74HCng.lib)). Simulating one instance of the inverter worked fine, but I got the dreaded doAnalyses: TRAN: Timestep too small
error when connecting the second/third/e.t.c inverter in the ring. The convergence issue still occurred even when driving the first inverter instance from a slow frequency PULSE
voltage source (rather than the driving it from the output of the last inverter), indicating it wasn’t a problem with the ring structure.
I then looked harder around the internet and found the MyHCU04
SPICE model [posted on Google Groups by the late Jim Thompson](https://groups.google.com/g/sci.electronics.basics/c/k93fFgwnws?pli=1):
On popular request, 74HCU04 Spice Model rescued from 1993 archives and posted on the Device Models & Subcircuits page of my website…
This SPICE model for an inverter fixed the convergence issues I was having (if anyone else is interested in this file, I’ve saved it [here](ringoscillatortriple/MyHCU04.lib)). Hurrah!
7. Manufacturer Part Numbers

SiT1533AI: SiTime standard clock oscillators and MEMS oscillators.

SiT1533AIH4D1432.768G: MEMS clock oscillator.
References

[1]: https://www.mouser.com/datasheet/2/371/SiT1533_rev1.4_032020181324419.pdf, retrieved 20210118.

[2]: https://www.ctscorp.com/wpcontent/uploads/AppnoteCrystalBasics.pdf, retrieved 20210428.

[3]: https://designersguide.org/forum/Attachments/GEHRING__Fast_CrystalOscillatorSimulation_Methodology.pdf, retrieved 20210428.

[4]: https://www.electronicstutorials.ws/oscillator/crystal.html, retrieved 20210429.

[5]: https://www.analog.com/media/en/technicaldocumentation/applicationnotes/AN111.pdf, retrieved 20210501.

[6]: ST Microelectronics (2009, Jul). AN2971 Application note: Using the typical temperature characteristics of 32 KHz crystal to compensate the M41T83 and the M41T93 serial realtime clocks. Retrieved 20210909, from https://www.st.com/resource/en/application_note/an2971usingthetypicaltemperaturecharacteristicsof32khzcrystaltocompensatethem41t83andm41t93serialrealtimeclocks—stmicroelectronics.pdf.

[7]: Spevak, Peter and Forstner, Peter (2006, Aug). _MSP430 32kHz Crystal Oscillators _. Texas Instruments. Retrieved 20210910, from https://www.ti.com/lit/an/slaa322d/slaa322d.pdf.

[8]: Atmel (2015, Mar). AVR4100: Selecting and testing 32kHz crystal oscillators for Atmel AVR microcontrollers. Retrieved 20210912, from http://ww1.microchip.com/downloads/en/appnotes/doc8333.pdf.
Authors
Related Content:
 Silicon Controlled Rectifiers (SCRs)
 Shift Registers
 Peltiers (Thermoelectric Cooler)
 Electropermanent Magnets (EPMs)
 Neon Sign Transformers
Tags:
 electronics
 components
 oscillators
 crystals
 MEMS
 XTAL
 XC
 XO
 OCXO
 frequency
 clocks
 power consumption
 stability
 accuracy
 ultrasonic baths
 ring oscillators
 NOT gates
 turnover temperature