Bandgap Reference

Does the bandgap reference really have anything to do with the silicon bandgap voltage?

The basic idea of the bandgap reference is to obtain a low temperature coefficient voltage source by cancelling the negative temperature coefficient of the diode forward voltage with the positive temperature coefficient of differential $V_{be}$.   The difference of the $V_{be}$ of two different size but otherwise identical bipolar transistors is $\Delta V_{be} = k T /q \ln \eta$ (PTAT), where $\eta$ is the current density ratio, k=1.38e-23, q=1.6e-19.  The diode voltage tempco is about -2mV/°C; to match that, $k/q \ln \eta $ = 2m, so $\Delta V_{be}$ = 600mV at the room temp.  And the diode voltage at the room temp is also about 600mV.  So the output is about 1.2V, which they say is close to the silicon bandgap voltage (1.166V at 0K).  But the essential scheme does not really seem to depend on it.

Update:

Recently I watched a lecture by A. Paul Brokaw,  "A Transistor Voltage Reference, and What the Band-Gap Has To Do With It", 1989.  He gave the clearest explanation that I've ever seen.  PTAT ($\Delta V_{be}$) and CTAT ($V_{be}$) with the right proportion add up to the constant reference, and at 0K, PTAT is 0V and $V_{be}$ is equal to the bandgap voltage.

VG0 is the silicon bandgap voltage, Vbeo is the base-emitter voltage at temperature To. 

Logarithmic Amplifier Compensation

A logarithmic amplifier is based on the exponential relationship between the collector current and the base-emitter voltage.  It is also an opamp circuit that has a gain in the feedback path, so a unit-gain stable opamp may not be stable.  It is necessary to add phase compensation.

Here is the simple logarithmic amplifier circuit and we derive the loop gain,

We will use OP-41 as the opamp. OP-41 has a typical DC gain of 134dB and gain-bandwidth product of 500K (ω1 = 3e6).  That puts the dominant pole at about 0.1Hz.  A high order pole above 1MHz leads to a phase margin of about 77 degrees.  This is typical of an internal dominant pole compensated opamp.  As we can see the loop gain plot is shifted up, the phase margin is reduced.

We place a capacitor across the collector and the emitter as compensation.  The capacitor adds a lag network.

With the zero a decade below the cross over frequency (Rs C > 3us), this results in an improved phase margin (back to near the unit-gain phase margin).

We have ignored the junction capacitance from the npn transistor.

We compare these with simulations.  We model the opamp as a two-pole transfer function.  The input current is 1mA.  The feed back has a gain of 40 or 32dB.

The phase margin is 8° without compensation and 50° with a 22n cap.  The zero and the pole are at the estimated location of 7.2KHz and 280KHz.

The compensation can be improved, which we will discuss in a later time.

Pen Drawing Tablet

XP-Pen Star G640 drawing tablet, 6x4", 8192 pressure levels, battery free stylus with two push buttons, resolution of 5080 LPI, for $25

It works on the electromagnetic induction principle.  

From the teardowns that other people have shown on the internet, the stylus is entirely passive with a coil and capacitors, that would exclude any encoding scheme.  The working hypothesis is that the tablet will transmit a signal to excite the LC tank, the LC tank oscillates and signals get picked up by the tablet.  The working area of the tablet is a PCB with horizontal and vertical traces.  The position of the stylus is calculated perhaps by centroiding on the signal strength.

The pressure level is transmitted probably by frequency modulation.  There is a ferrite on the rod that the nib is attached and the rod is pushed on a spring.  The pressure on the nib moves the ferrite relative to the coil, changing the resonant frequency.  And pressing the button connects additional capacitor to the tank, also altering the frequency.

We will try to verify the working principle without disassembling the stylus or the tablet.  We will use a simple wire loop to pick up the signals.

Without the stylus on the tablet, we pick up some signal from the tablet.   It is possibly just a 500KHz square wave ac-coupled.

 When the pen is on the tablet, we see the oscillating LC signal. 

We can take a close look at the frequency of the oscillation,

Zooming in at the peak, we can see the difference in the rising envelop when the tablet is driving and the falling envelop when the stylus is free oscillating.

At pressure level 0, the frequency is about 506KHz, and at the max pressure of 8192, the frequency is about 530KHz.  So the circuitry has a resolution of 3Hz.  When the upper button is pressed, the frequency is 484KHz, and the lower button 464KHz.  And the frequency changes from there if pressure is applied while the button is pressed.

We can make a circuit model and run simulation to compare with the measurements.

We will design a frequency to voltage conversion circuit to extract the pressure level in a future blog.

The stylus is detected with the tip about 1cm above the tablet; there is some hysteresis on that distance.  Placing the stylus flat lengthwise on the tablet  does not work, but vertically flat widthwise works.  This is a little curious; perhaps the excitation is only generate by traces in one direction.  It may be a deliberate design choice, because the stylus can be left on the tablet without affecting other pointing devices.

When we place a ferrite toroid around the stylus, the tablet can register button clicks with buttons being click.  This implies that the presence of the ferrite toroid lowers the resonant frequency to be in the range of that of button click.

NVMe on Raspberry Pi 5

Raspberry Pi 5 introduced a PCIe connector (16-pin 0.5mm FFC) in the place of the under-utilized display connector.  Now it is possible to use an NVMe SSD.  It would require a PCIe to NVMe adapter, which can be purchased for about $7 and comes with a board to mount an M.2 M-key NVMe SSD for size 2230/2242/2280 and a flexible flat cable to connect to RPi 5.  

Download the OS image.  xzcat | dd to the disk.  Two partitions are created and take up about 14GB.

The default EEPROM `BOOT_ORDER=0xf416`, the boot order should be NVMe, SD Card, USB.  But it still boots the SD card first.  With the SD card out, the RPi 5 boots from the NVMe.  The root partition is near full.  Use parted and resize2fs to grow the root file system.

A sustained read/write speed is 390MB/s read and 440MB/s write, which is better than the USB NVMe adapter 260MB/s read and 380MB/s write, but not dramatically.

The RPi 5 PCIe is rated for Gen 2 one-lane; the maximum throughput is 500MB/s.  

In config.txt, add `dtparam=pciex1_gen=3` to enables PCIe Gen 3 (1GB/s).  The reliability of the Gen 3 operation is not guaranteed, but it appears to work well.  The read speed is increased to 850 MB/s; the write speed is largely unchanged.  The disk is rated at up to 1700MB/s for sequential read and up to 1000MB/s for sequential write.  So the read speed is close to the bus speed, but the write speed is under performing.

The board averages about 1.2A/5V during writing.  The average idle current is 0.7A (without SSD, 0.6A).  With the NVMe SSD, the Raspberry Pi 5 does not shut down completely, still drawing 170mA despite of setting POWER_OF_ON_HALT=1 and WAKE_ON_GPIO=0 (without SSD, 4mA).

The PCIe interface appears to have some interference on the WiFi signal.  This FFC has no ground plane, which probably radiates more. The on-board WiFi has trouble picking up relatively weak signals.  Other people have reported similar issue.  The WiFi does work when it is close to the router. 

Nano Ammeter

One of "self-explanatory" circuits in AoE2 is a nano-ammeter circuit; no explanation is given.  I doubt it is really self-explanatory to a beginner.

This circuit appears to originate from PMI's OP-41 datasheet.  (PMI is Precision Monolithics, Inc which later became a part of Analog Devices.)  OP-41 is a JFET input opamp with very low input bias (5pA @ 25C) in the inverting configuration as a logarithmic amplifier.  (OP-41 also tauts the excellent CMRR of over 100dB for a FET input opamp.)  Q1 and Q2 are matching transistors.  Their current ratio is the exponential of the difference in Vbe. If the difference in Vbe is proportional to the temperature, the current ratio is constant.  Since the Q1 base is grounded, the difference in Vbe is just the Q2 base voltage.  The resistor divider provides the Q2 base voltage that is proportionally to the temperature, as the bandgap reference subtracts the diode connected Q3 to produce a voltage is approximately proportional to Vt (although there is more complex 2nd order temperature dependency).   How good is the approximation?   We can run a Spice simulation stepping the temperature.  The Q3 current is about 65uA at -55C and 115uA at 125C and approximately linear  I (uA) ~ 0.28 * T (K) . 

The trimmer adjusts the full scale value of the current meter. 

Also the compensation network requires some explanation, which is deferred to a separate discussion.

Quotes

"No one believes an analysis – except the person who did it.

Everyone believes a test – except the person who did it."

- ???

‘‘The best way to predict the future is to invent it.’’

 - Alan Kay

"... true knowledge can only be acquired piecemeal, by the patient interrogation of nature."

- Edmund Whittaker, A History of the Theories of Aether and Electricity

“Successful engineering is all about understanding how things break or fail.”

- Henry Petroski

"All science is either physics or stamp collecting."

- Ernest Rutherford

"The first time you go through the subject, you do not understand it at all. The second time, you think you understand it, except for one or two small points. The third time, you know that you do not understand it, but you are so used to the subject that it does not bother you anymore."

- Arnold Sommerfeld, about thermodynamics

"those who can, do; those who can't, teach."

George Bernard Shaw, Man and Superman

"engineering ... is the art of doing that well with one dollar, which any bungler can do with two after a fashion."

- Arthur M. Wellingtony, The Economic Theory of the Location of Railways

"Engineer - the man who can do for a reasonable cost what another would expend a fortune on"

- Rutherford Aris, Vector, Tensors and the Basic Equations of Fluid Mechanics 

“In God we trust. All others bring data.”

- Bo Lojek, History of Semiconductor Engineering

"Research is when you don't know what you're doing."

- ??? 

The cheapest, fastest, and most reliable components are those that aren’t there.

- C. Gordon Bell

"When you test you find something is wrong."

- Donald Trump (May 14, 2020)

I would rather have a general who was lucky than one who was good.

- Napoleon Bonaparte

"Perfect is the Enemy of Good Enough"

- Eric Johns (October 1988), U.S. Naval Institute Proceedings: 37

“If an expert says something can be done he is probably correct, but if he says it is impossible then consider getting another opinion.”

- Richard Hamming, The Art of Doing Science and Engineering 

People who are really serious about software should make their own hardware.

- Alan Kay

The management question, therefore, is not whether to build a pilot system and throw it away. You will do that. […] Hence plan to throw one away; you will, anyhow.

- Brooks, The Mythical Man Month (Page 116)

Brooks's Law: 

Adding manpower to a late software project makes it later.

 - Brooks, The Mythical Man Month

... this was going to be one of these Onion Syndrome deals - you peel off a couple layers, and cry; then you peel off a couple more layers, and cry some more.

- Bob Pease, What's All This Ground Noise Stuff, Anyhow?

Thinking is recommended. Heck, thinking is required.

- Bob Pease, Troubleshooting Analog Circuits

... a little known tenet of precision op amp circuits: Williams's Rule.  Williams's Rule is simple: always invert (except when you can't).

- Jim Williams, Analog Circuit Design: Art, Science, and Personalities

"Any idiot can count to ONE ..."

- Quoted by Samuel Wilensky in Analog Circuit Design: Art, Science, and Personalities

(maybe attributed to Bob Wildar)

Anyone can build a bridge that stands up, but only an engineer can build a bridge that just barely stands up.

- ????

In real estate, it is location, location, location.  In mathematics, it is notation, notation, notation.

- ????

In software, debugging is harder than writing code.  If you write the code as clever as you can, then you are not smart enough to debug it.

- ????

... where there is no confusion there is no prestige.

-Linderholm, Mathematics Made Difficult

All problems in computer science can be solved by another level of indirection, except for the problem of too many layers of indirection.

-????

... practicing what was called "the mushroom theory of management." ... defined it as follows: "Put 'em in the dark, feed 'em shit, and watch 'em grow."

- Tracy Kidder, The Soul of A New Machine

Hardware eventually fails. Software eventually works.

-Michael Hartung

Common Emitter Frequency Response using EET

Even the simple common emitter circuit can result in some pretty complicated expression for the frequency response. The inclusion of a single element, the base-collector capacitance, introduces much complexity.  This circuit is analyzed in detail by most of the textbooks, including Gray & Meyer's.  None of them makes use the EET.  We give a derivation using Middlebrook's EET to see if the analysis is simplified.

 We work out the factor involving Cf,

This is a little bit less work than the more direct method,

The two methods produce the same expression, so using EET does not give more insight.  

The base-collector capacitor results in an additional pole and a right-hand side zero and shifts the dominant pole.  The dominant pole can be approximated from the input capacitance and the Miller capacitance.