The CH32V003 reference manual does not explain some key information for using the I2C peripheral. The issue is mainly with the I2C1_CKCFGR register, but there is also confusion around the FREQ field of the I2C1_CTLR2 register.
Digging a bit, it turns out that WCH appears to be using the same I2C peripheral IP as Gigadevices and Puya use in many of their microcontrollers. Both of the other vendors have a bit more documentation in their manuals.
The explanation of the I2C1_CKCFGR register turns out to be fairly simple, and the values used by the I2C examples going around the net appear to be correct. Its documentation in the reference manual says the following
However, what is left unsaid is the following
Mode
Duty
T_high clock cycles
T_low clock cycles
F/S=0
Either
CCR
CCR
F/S=1
DUTY=0
CCR
2*CCR
F/S=1
DUTY=1
9*CCR
16*CCR
The bitrate is just BR = F_APB1 / (T_high + T_low), and thus the magical factors of 2, 3 and 25 used in the examples are explained. This also explains where the quoted duty cycles of 33% and 36% come from.
This still leaves the FREQ field in I2C1_CTLR2 to be explained. It is described as
The Gigadevices and Puya documentation agree with WCH's documentation on this: you're expected to program the field with a value, which is the peripheral clock frequency given in integer megahertz.
All examples for the CH32V003 I've seen on the net, however, program it as F_APB1 / 2000000, which is half of the correct value. Also, many of the examples document the value 2000000 as the frequency of the peripheral, which is not correct at all. WCH's documentation is worded in less obvious terms than the other vendors, which may be the source of the confusion.
But what does this register do? If it's possible to write only half of the correct value to the field and still have the peripheral seemingly work, it can't be very critical. This is only covered in the Puya documentation, and still with not enough detail. The English translation states:
This register must be configured with the value of the APB clock frequency to generate data setup and hold times that are compatible with the I2C protocol.
Not sure why the timings given in the I2C1_CKCFGR are not enough to guarantee proper setup and hold, but apparently there is additional signal conditioning which this field affects.
The best (overall) frequency reference I have is in my Racal-Dana 1992 frequency counter, which has the ovenized oscillator option 04A. It's great, but calibrating it with the equipment I have is an annoying task. I calibrate it against GPS. However, as GPS time is stable enough only in the long term, it requires measuring a lot of GPS PPS samples. This takes a long time, and even longer if I want to adjust the time base. This got me thinking if I could automate the process, and after some thought I realized the best approach would be a GPS disciplined oscillator.
While there are commercial GPSDO devices for sale, I can't justify the cost for buying one. I could however design and build one. Turns out I can justify much higher cost for such a project, as there's a a lot of benefit in the learning experience.
My goal would be to produce 10MHz +- 1ppb. I also want to keep the phase noise low, but that's somewhat secondary and without quantitative requirements.
The GPS modules I've used are really old Fastrax parts. They are specified to 50ns RMS jitter on the PPS output. Not great, but workable. This pushes the integration time to around 1 hour to get down to 1 ns jitter. I would not trust the PPS pulse below a 1 hour window anyway (more on that later). Anyway, this puts a requirement on the stability of the oscillator: it would need to be stable (better than +-1ppb) for much longer than the GPS integration period - say 10 hours. This puts me well into the OCXO domain.
DIY OCXO - fail
I happen to have some old VCTCXOs (voltage controlled temperature compensated crystal oscillator), which I found thrown out years ago. They seemed high quality and expensive (from Rakon), so I always wanted to find a use for them. I first thought to build temperature control around one of those and convert it to an OCXO. Using a DS18B20 and PWM controlling a heater I managed to get the temperature to be controlled well within 0.1 degrees Celsius. However, I could not get the crystal oscillator to behave though. I kept the control voltage grounded to make sure any noise on the control voltage wasn't causing the issue. In the end it may have been due to the supply voltage sensitivity of the part, which is not too good at +-300 ppb for +-5% supply voltage. This would require about 1 mV stability for the supply voltage for my application.
Rakon VCTCXO frequency and temperature, moving average over 5 minutes Measured with Racal-Dana 1992 and DS18B20
Looking around the internets, I found that all the cool cats are playing with OSC5A2B02 OCXOs, which are available on Aliexpress for very cheap. So I ordered some. At about 3€ a piece it didn't seem like a too big investment.
OSC5A2B02 testing
The OSC5A2B02 is much less sensitive to supply variations (+-2ppb for +-5% supply voltage), so just about any regulator suffices. What is important to note however is the control voltage sensitivity, which is about 1000 ppb per volt, or equivalently 1ppb per millivolt. For the initial tests, I simply grounded the control voltage to eliminate it's contribution.
OSC5A2B02 frequency (insulated in PE foam), moving average over 5 minutes Measured with Racal-Dana 1992
The variation was much larger than the datasheet of the oscillator promised. This is when I realized, that I actually didn't have a clear idea of what the stability of my frequency counter was, and that I could just be seeing it's variation.
To get a better idea of the frequency counter variation, and thus the true stability of the OCXO, I connected a GPS PPS on the B channel of the frequency counter.
OSC5A2B02 frequency (top) and GPS PPS (bottom), moving average over 20 minutes Measured with Racal-Dana 1992
Bingo! Turns out most of the variation observed in the OSC5A2B02 frequency is actually due to variation of my frequency counter! Who would have guessed that a 3€ Aliexpress OCXO today is so much better than an instrument worth thousands in the 1980s.
Normalizing the OCXO frequency with the PPS frequency (after heavy filtering), the OCXO appears to be within +-1ppb over a day as long as there is no control voltage variation.
Phase noise and jitter
The OCXO datasheet gives some spot values for the phase noise:
-80 dBc @ 1 Hz
-120 dBc @ 10 Hz
-140 dBc @ 100 Hz
-145 dBc @ 1 kHz
-150 dBc @ 10 kHz
Integrating over this range gives ~0.3 mrad RMS of jitter, or about 4.8 ps.
Control voltage generation
As said earlier, the control voltage is about 1 ppb per millivolt over a range of 4 volts total. Using 16 bit discretization for the range leads to about 0.06 ppb resolution. To get a ballpark figure of how least significant bit transitions affect the phase noise, let's consider the case of modulating the 10 MHz carrier by +-0.03 ppb ( = +-0.3 mHz) at 1 Hz frequency (assume single tone sinusoidal modulation for simplicity). This is given by
\[ y(t) = \cos(2 \pi f_c t + A \sin(2 \pi f_m t)) \]
In which \( A = \frac{\Delta f}{f_m} = \frac{0.3 \text{mHz}}{1 \text{Hz}} = 3 \cdot 10^{-4} \).
Using the sum of angles identity for cosine, we get
Since \( A << 1 \) we can approximate \( \cos(A \sin(2 \pi f_m t)) \approx 1 \) and \( \sin(A \sin(2 \pi f_m t)) \approx A \sin(2 \pi f_m t) \). This gives
Thus considering the single sideband phase noise, we see that the modulating frequency is attenuated by \( \frac{A}{2} \) with respect to the carrier, which is -76.5 dBc with our numbers - or an additive 3.3 ps. So not great, but not terrible - especially if we minimize the occurrence of transitions by adding some hysteresis. Anyway, we shouldn't use any less than 16 bits of resolution.
The next issue is how to implement a 16 bit DAC cheaply. Here a pretty obvious candidate is to use PWM. Proper DAC chips with 16 bits cost close to 10 euros, while PWM and a lot of filtering can be achieved with less than a euro. Much less than 1 Hz of bandwidth is perfectly fine, so the question is just how much filtering is needed. Assume the PWM repetition period is 400 Hz and amplitude is 4V. Take then the attenuation of the filter as \(G\). Recalling that the sensitivity of the OCXO is 1 ppm per volt, the power of such a modulation relative to the carrier is given by \( 0.05 G \) (Later edit. Reading back this text, I was as confused as you are about where I got this the number. The factor comes from the above calculation, where we see that modulating the phase with amplitude A is seen as a spur of relative amplitude A/2 at an offset frequency. A 4V control modulation causes a 4 ppm change in the frequency, i.e. 40 Hz. The phase modulation amplitude is thus A = 40 Hz / 400 Hz = 0.1. Half of this is seen in the output, thus 0.05 or about -26 dBc). To push the additive jitter down to the same scale as the intrinsic jitter of the oscillator, the attenuation needs to be at least 44 dB. This should be easily achievable with simple RC filters.
The main issue with the control voltage is thus the stability requirement. The control voltage reference should remain within 1 mV over several hours. This could be combated with a high stability voltage reference. Problem is that those are expensive, and I would like to keep everything as cheap as possible. As we already have a temperature control loop (in the OCXO itself that is), we might as well use that to keep a voltage reference at constant temperature also. Turns out a TL431C has a typical stability of 4 mV over the entire temperature range, and hopefully better than 1 mV when kept at constant voltage. Also turns out TL431s are really cheap.
Microcontroller
The plan is to implement the PLL using a microcontroller. The uC would be clocked from the OCXO and it times the interval between PPS pulses. This allows determining the OCXO frequency.
The OCXO works on 5V. It would be useful if the microcontroller would also operate on a 5V supply. It also needs to have hardware input capture features, allowing precise measurement of the PPS period against the OCXO frequency. And like everything else, it must be cheap. As luck would have it, I recently ordered 50 units of CH32V003 controllers for 0.20€ per piece (including shipping). They check all the boxes for this project. The even have an internal PLL to allow doubling the 10 MHz OCXO clock for improved timing resolution.
Prototype
I implemented a very quick hack of the system on a CH32V003. It is clocked from the OSC5A2B02 with an internal PLL configured to double the frequency. This clock is used to drive a timer peripheral, which is configured for input capture from the PPS pulse. Timer resolution is increased in software from the HW provided 16 bits to 32 bits - otherwise a full PPS period could not be counted.
After each captured PPS pulse, a simple validation is performed to try to ignore erroneous pulses. A simple PI controller then controls to minimize the frequency error (frequency lock loop). The controller is just parametrized by the control bandwidth and was designed to have critically damped dynamics. Controlling just the frequency still leaves an uncontrolled phase error. I'll be looking into that too, but so far it is of no concern.
The DAC is implemented with a 16 bit PWM running at 305 Hz and filtered with a ~0.2 Hz (-3 dB) first order lowpass filter. The filtering leaves a lot to desire, and this will have to be improved for the final product. The voltage reference is provided by a jellybean TL431C, which is not temperature controlled.
Crude schematic of the prototype voltage control
Operating the control voltage as shown in the figure above allows reducing the full swing range, which increases resolution.
Experiments with the prototype have shown better than expected performance. Though I'm still lacking a data channel for the GPS data, which means that I don't know when the PPS is valid, nor can I keep absolute phase. This leads to some bad samples passing validation and causing trouble.
Closed loop controlled frequency, locked to GPS PPS GPS PPS moving average over 5 minutes
The figure above shows closed loop control of the OCXO locking on to the GPS PPS frequency. The initial condition was deliberately set about 100ppb off to see the dynamics of the control. The bandwidth of the controller was chosen as ~4 mHz for this experiment to perform the experiment at a reasonable speed. This is too wide a bandwidth for proper operation though. From the plots, we see that the dynamics are nearly critically damped, with only very minor overshoot. This is good enough for me!
75 hour long term stability experiment GPS PPS moving average over 1 hour
The next experiment was a long term stability test. Here I set the controller bandwidth to ~0.1 mHz to better reject noise in the PPS signal. During the experiment, there were a few moments at which the PPS became invalid. This causes the sudden spikes seen in the error graph. Regardless, the error remained well within +-1 ppb, which was the design goal. Also, the prototype only implements very simple control voltage filtering and the voltage reference is still just at room temperature. The design goal thus seems very much achievable, and possibly going down to +-0.25 ppb is possible with more care taken. The control voltage graph is computational, based on an assumed ideal voltage. I don't know for sure if the drift seen in it is due to the OCXO really needing to be adjusted or the voltage reference drifting and needing compensation, but my guess is on the voltage reference.
Nils Berglund, who is the author of many amazing simulation videos on Youtube, recently posted videos showing visualizations of waves on a sphere. In particular, this video caught my eye, depicting tsunami waves of the 2011 Japan earthquake travelling across the Pacific ocean.
In fact, back in 2011 I had made a very similar visualization inspired by the events, and later uploaded it to Youtube - see video below. The video was made as promotional material to get new students to major in mathematics at Aalto University. At that time I was a graduate student at the maths department working toward my PhD.
Anyway, it was as an undegrad in 2007 when I first got interested in solving the wave equation on the sphere. I had been studying the finite element method with great interest, and had programmed my own simple implementation of nodal P1 elements on triangles. I had succesfully used that to solve waves in planar domains - see video below. I thought that in order to solve the wave equation on a sphere, I would just use a mesh approximating the sphere instead. Turns out it's almost as simple as that, but there are a couple nontrivial details.
My undergrad studies included a mandatory "special assignment". These assignments were often to write a research paper on a subject - though they were published only internally at the department. I suggested to my professor that as my assignment I could study solving the wave equation on spheres using FEM.
Now, finally, I thought I'd revisit the topic, and have something generic written down on the subject a bit more publicly.
Wave equation and the finite element method
To get up to speed, let's look at the wave equation and the finite element method in a quick and non-rigorous fashion. The wave equation (in strong form) is
in which \( \ddot{u} \) is the second derivative with respect to time, \( c \) is the wave speed, which we'll just take as 1 from here on, and \( \Omega \) is the domain in which we're solving the equation at. In addition, there are initial conditions and boundary conditions that need to be described, but let's skip those for now.
FEM works with the weak formulation of the problem. To get the weak form from the strong form, both sides of the equation are multiplied by a test function \( v\) and integrated over the domain \( \Omega \). Integration by parts is applied to reduce the regularity requirement of \( u \) from 2 derivatives down to 1. The weak form is then: find \( u \in V \) such that
\[ \int_\Omega \ddot{u} v\ \mathrm{d}x + \int_\Omega \nabla u \cdot \nabla v\ \mathrm{d}x = 0 \quad \forall v \in V \]
With FEM, as with other Galerkin methods, we replace the space \(V\) with a finite dimensional approximation \(V_h\). We come up with a basis \( \{v_1,\ldots,v_N\} \) of \(V_h\), and write the finite dimensional approximate solution \( u_h \) as
\[ u_h = \sum_{i=1}^N U_i v_i \]
The problem then becomes: find \( U_i \) such that
The idea in the finite element method specifically is to define the basis functions \( v_i \) using a mesh, which fills the domain \(\Omega\). Assuming now a triangular mesh, a simple approach is to take each basis function as a piecewise first order polynomial, each piece being one triangle. Then, assign \(v_i = 1\) when evaluated at vertex \(p_i\), and \(v_i = 0\) when evaluated at any other vertex.
The integrals can be written as the sum of the separate contributions from each element
For any given \(i\) and \(j\), on the vast majority of the elements, \(v_i\) or \(v_j\) (or both) is identially zero on the element, and thus they don't contribute to the integral. Contributions come from only the triangles, that contain both vertex \(p_i\) and vertex \(p_j\).
Now, to actually compute the integrals over a triangle, a standard trick is to do a change of variables and perform the integration over a reference triangle instead. A coordinate on the reference triangle \( \hat{x} \) is easily mapped to \( x \) on the target triangle \(T\) via an affine transform \( x = A_T \hat{x} + b_T \). Let's take the triangle \(\hat{T}\) with vertices \( \hat{p}_1 = (0,0) \), \( \hat{p}_2 = (1,0) \), \( \hat{p}_3 = (0,1) \) as our reference triangle. With this choice, and the fact that affine transforms map first order polynomials back to first order polynomials, the basis functions can be expressed on the reference element in a simple way.
The 3 basis functions on the reference element are such that they get the value 1 in one vertex and 0 in the others. Written explicitly, they are
Here's where things get at bit non-standard from the usual FEM implementation. Our reference element lives in \(\mathbb{R}^2\) while the global element lives in \(\mathbb{R}^3\). This means that the matrix in our affine mapping is non-square, and we need to be a bit more careful with considering how the area scales and how the functions change under the change of variables.
Usually, with the change of variables \(x = f(\hat{x})\), we see the differential area element change as \( \mathrm{d}x = |\det J_f(\hat{x})| \), where \( J_f(\hat{x}) \) is the Jacobian of \( f(\hat{x}) \). This does not however make sense for a non-square \(J_f\) and thus something more general is needed. Turns out, that a slightly more general expression for the differential area element is \( \mathrm{d}x = \sqrt{\det J_f(\hat{x})^t J_f(\hat{x}) }\ \mathrm{d}\hat{x} \). Since \(J_f^t J_f\) is always square, this is well-defined in our case. Note, that in the case where \(J_f\) is square, this reduces to exactly the usual form. In our case, the Jacobian of the mapping is simply the matrix \(A_T\) of the affine transform, and we have \(\mathrm{d}x = \sqrt{\det A_T^t A_T}\ \mathrm{d}\hat{x} \).
The contribution of element \(T\) to \(M_{ij}\) can be computed as
While that's all for the mass matrix, for the stiffness matrix we have the additional headache of how the gradient will transform under the mapping. For basis function \(v_i\) we have \(v_i(x) = v_i(A_T \hat{x} + b_T) = \hat{v}_\hat{i}(\hat{x})\). Skipping a bit of detail for brevity, it follows from the chain rule that \( (\nabla \hat{v}_\hat{i})(\hat{x}) = A_T^t (\nabla v_i)(x) \), which by itself is enough for the typical case as \(A_T^t\) is invertible. For our case, however, we must consider additionally that the gradient \((\nabla v_i)(x)\) is actually a vector in the tangent space of \(\Omega\) at \(x\). Since the tangent space of \(\Omega\) within element \(T\) is spanned by the range of \(A_T\), there must exist some \(g\) such that \((\nabla v_i)(x) = A_T g\). Thus
\[ A_T^t A_T g = (\nabla \hat{v}_\hat{i})(\hat{x}) \]
from which it follows that
\[ g = (A_T^t A_T)^{-1} (\nabla \hat{v}_\hat{i})(\hat{x}) \]
After spatial discretization with FEM we're left with the ordinary differential equation
\[ M \ddot{U} + K U = 0 \]
Without going into detail, the problem is particularly well suited for e.g. the Crank-Nicolson method or the leapfrog integration method. These methods are stable for undamped oscillating problems and they do not produce decaying solutions. The key word for such methods is symplectic.
Crank-Nicolson
For Crank-Nicolson, we'll first convert the equation into first order form as
\[ \begin{pmatrix} I & 0 \\ 0 & M \end{pmatrix} \begin{pmatrix} \dot{U} \\ \ddot{U} \end{pmatrix} = \begin{pmatrix} 0 & I \\ -K & 0 \end{pmatrix} \begin{pmatrix} U \\ \dot{U} \end{pmatrix} \]
Denote then
\[ B = \begin{pmatrix} I & 0 \\ 0 & M \end{pmatrix} \]
\[ A = \begin{pmatrix} 0 & I \\ -K & 0 \end{pmatrix} \]
and
\[ z = \begin{pmatrix} U \\ \dot{U} \end{pmatrix} \]
to get
\[ B \dot{z} = A z \]
For the discretization, consider the derivative on the left size replaced by the approximation \( \frac{z_{n+1} - z_n}{\Delta t} \), and consider the function evaluation on the right side being performed at the point \( \frac{z_{n+1} + z_n}{2} \). With this, we'll arrive at
\[ B \frac{z_{n+1}-z_n}{\Delta t} = A \frac{z_{n+1} + z_n}{2} \]
The larger down-side of the Crank-Nicolson method is the requirement for solving a large non-symmetric system of equations for every time step.
Leapfrog integration
Leapfrog integration is an explicit symplectic method particularly well suited for mechanical problems, where the accelerations are due to position only, i.e. of the form
\[ \ddot{u}(t) = f(u(t)) \]
which our problem is an example of, as for our problem we have
This might look like Euler's method, but it is subtly different. The position update at \(n+1\) uses the velocity at \(n+1\) and not at \(n\) as for Euler's method.
This still requires solving a system of equations, but now the coefficient matrix is half the size as for the Crank-Nicolson method, and is also symmetric.
Turns out, however, that the mass matrix \(M\) is strongly diagonally dominant and spectrally equivalent to the identity operator. Thus, for large systems \(M\) is quite accurately approximated by a diagonal matrix that has the row (or column) sums of \(M\) as the diagonal elements. The system of equations thus becomes trivial.
