Last weekend I posted a voltage-versus-current curve for a 1N5817 Schottky diode, to confirm the theoretical formula , where I_S is the saturation current of the diode, using the following setup

and measuring the results with an Arduino Leonardo.  I claimed that the resulting data fits the model well for over six decades (> 120dB):

Fitting over a wide range of currents is more robust than fitting over the narrower range that I can get with just one value for R2.
There is quantization error still on the voltages, but the overlapping current ranges give good data for most of the range. V_T is now 26.1mV and I_S is 0.91µA.
click to embiggen

I was a little dissatisfied at the low end, because of the low resolution of the Arduino analog-to-digital converter (only 10 bits).  This weekend I decided to repeat the measurements, but using a Freedom KL25Z board, which has a 16-bit ADC.  Of course, it doesn’t really get 16 bits of accuracy—the data sheet claims that when you use the hardware averaging of 32 samples in 16-bit differential mode (the most accurate) you get at least 12.8 equivalent bits and typically 14.5 equivalent bits. (For single-ended 16-bit, the effective number of bits is only guaranteed to be 12.2, and the typical is 13.9 bits.)  They claim a ±6.8LSB total unadjusted error.

My son helped me get the bare metal ARM system set up on my laptop, along with ADC and UART routines, so that I could write my own single-purpose data logger for this problem (he’s working on getting the KL25Z board integrated into the Arduino Data Logger, but it isn’t close to being ready yet).  My program used the longest sample times and hardware averaging of 32 samples, to get the most accurate conversions possible from the 16-bit ADC.  The first version of the program used differential inputs for the voltage across the diode (E20-E21), but single-ended readings (E21) for the voltage across the resistor.  I had to reduce the voltage for the test from 5v to 3.3v, because the KL25Z runs on 3.3v, not 5v. I got some rather weird results:

Two runs of measurement with R2=100Ω. The low-current measurements seem to be all noise.
click to embiggen

I could get decent measurement in the low-current range by using a larger resistor, so the problem was not noise in the measurement fixture or problems reading low differential voltages on the diode, but just with the small single-ended read for the current. It is pretty clear to me that the ADC does not work well when the input voltage results in less than about 50 counts.  (Note, that means that at the low end of the voltage range you only have about a 9.4-bit equivalent ADC.)

I modified the circuit to allow differential reading away from 0 for both the voltage across the diode and the voltage across the shunt resistor:

Adding an extra resistor ensured that the lowest voltage did not get too close to ground and I could use differential reads for both voltage (E20-E21) and current (E22-E23)/R2

This gave me a much cleaner reading, with problems only once the differential counts got below about 20:

Differential measurement with R2=100Ω. The low-current measurements have problems when the counts get small, but not nearly as severely as with the single-ended measurements.
click to embiggen

I replaced the 100Ω R2 resistor with a 15.56kΩ resistor (nominally 15kΩ), to extend to lower currents despite the noise in the ADC:

This plot extends the fit down to about 0.1µA, but only by adding an extra term—an offset to the voltage. I thought at first that overshoot to –11mV is an error in the analog-to-digital converter on the KL25Z, as I couldn’t see how my circuit could be back-biasing the diode.
Click to embiggen

I tried using larger resistors, but was unable to get any better data using them—I seem to be limited by the differential voltage measurement of the diode at the low end. I thought I might be able to improve the measurements by adding an instrumentation amp to increase the signal for low voltages.  But first I tried just hooking up a voltmeter, with no ADC or instrumentation amp connections.  When the voltage across R2 (100kΩ) is 0.31mV, the voltage across the diode plus R2 is only 0.05V, so there is -0.26mV across the diode.  The backwards voltage across the diode was not an artifact of the ADC!

I then tried looking at the voltage across the diode with my oscilloscope.  There is about 20mV of AC noise, independent of the DC voltage, until the diode has about 50mV across it (with the 15.5kΩ resistor for R2), by which time the noise has dropped to about 10mV (the Bitscope oscilloscope with the differential probe has a noise floor of about 3mV, if the two leads are connected together, so this is not just oscilloscope noise).  This noise seems to be white noise, not 60Hz hum pickup, so is probably coming from the diode.  This AC noise signal limits how accurately we can measure the DC current, and rectifying the noise could be the source of the mysterious “backwards” bias.

To reduce the noise, I put a 4.7µF ceramic capacitor in parallel with the diode, and redid all the measurements with 100Ω, 15.5kΩ, and 100kΩ resistors for R2.

Modified measurement circuit, adding a bypass capacitor to reduce AC noise on the diode and allow better DC measurement.

Now the signals are very clean down to nanoamp levels. I no longer need to add an offset to the voltage, as it is 0 to within the measurement repeatability. The noise from very small voltage differences for the 100Ω shunt resistor is still a bit of a problem, but that region is well covered by 15.5kΩ data. The curve was fit using just the 15.5kΩ and 100kΩ data, to avoid having to trim out the noise from the 100Ω data.
Click to embiggen

Lessons learned today:

• Higher-resolution ADCs do give smoother curves, with less digitization noise, but they aren’t a panacea for measurement problems. To get most of the resolution, I had to set the ADC to use long sample times and do a lot of averaging. I understand that Freescale Kinetis M series include 24-bit sigma-delta converters for higher precision at much lower speed (24 bits is 7 decimal digits), as well as the high-speed 16-bit successive-approximation converters. Unfortunately, they don’t have a low-cost development board for this series.
• Stay away from the bottom end of the ADC range on the KL25Z.  Scale single-ended inputs to have values at least 50, and differential inputs to have values at least 20.  There may be similar problems at the top end of the range, but I did not test for them.
  I wondered if the problem may be switching from the large value for the voltage across the diode to the small voltage across the shunt resistor that was the problem. I tried putting in a dummy read between the voltage and the current reads, but that didn’t help at all. At first I thought that the low-count readings were good with the large shunt resistors, but this is probably an illusion: errors in the current measurement for small currents aren’t visible on the plot, because the voltage across the diode is not changing, and so large horizontal errors in the plot are not visible there.
• Watch out for AC noise when trying to measure DC parameters.  If there are semiconductor junctions around, the noise may be rectified to produce an unwanted DC signal.
• The differential ADC settings have a range of ±VDDA, not ±VDDA/2. This means that the least-significant bit step size is twice as big for differential inputs as for single-ended inputs. For some reason the Freescale documentation never bothers to express what the differential range is.
• Serial USB connections are a bit flakey—the Arduino serial monitor missed a byte about every 200–300 lines.  I looked for anomalous points on the plot, then commented out the lines that produced them—they were almost all explainable by one character having been missed by the serial monitor; e.g., I commented out “662401069     86      19″ right after “660001069       865     17″,  because the last digit of the voltage (the second field) was missing.  The fields were a timestamp (in 24MHz ticks), voltage across the diode (in ADC units), and voltage across the shunt resistance (in ADC units).  [Actually, this was not a new lesson for me—I've had to do the same on almost all files collected from the Arduino serial monitor.  My son's data logger code is better at not losing data, but it is still worthwhile to check for anomalies.]
• The 3.3v supply from the Freedom board is much cleaner than the 5v USB supply that I get from the Arduino (unless I use an external power supply with the Arduino), but I can only take about 10mA from the 3.3v supply before it begins to droop.  If I want  more than that, I’d better provide my own power supply (or at least my own LDO regulator from the USB 5v supply).

Last weekend I posted a voltage-versus-current curve for a 1N5817 Schottky diode, to confirm the theoretical formula , where I_S is the saturation current of the diode, using the following setup

and measuring the results with an Arduino Leonardo.  I claimed that the resulting data fits the model well for over six decades (> 120dB):

Fitting over a wide range of currents is more robust than fitting over the narrower range that I can get with just one value for R2.
There is quantization error still on the voltages, but the overlapping current ranges give good data for most of the range. V_T is now 26.1mV and I_S is 0.91µA.
click to embiggen

I was a little dissatisfied at the low end, because of the low resolution of the Arduino analog-to-digital converter (only 10 bits).  This weekend I decided to repeat the measurements, but using a Freedom KL25Z board, which has a 16-bit ADC.  Of course, it doesn’t really get 16 bits of accuracy—the data sheet claims that when you use the hardware averaging of 32 samples in 16-bit differential mode (the most accurate) you get at least 12.8 equivalent bits and typically 14.5 equivalent bits. (For single-ended 16-bit, the effective number of bits is only guaranteed to be 12.2, and the typical is 13.9 bits.)  They claim a ±6.8LSB total unadjusted error.

My son helped me get the bare metal ARM system set up on my laptop, along with ADC and UART routines, so that I could write my own single-purpose data logger for this problem (he’s working on getting the KL25Z board integrated into the Arduino Data Logger, but it isn’t close to being ready yet).  My program used the longest sample times and hardware averaging of 32 samples, to get the most accurate conversions possible from the 16-bit ADC.  The first version of the program used differential inputs for the voltage across the diode (E20-E21), but single-ended readings (E21) for the voltage across the resistor.  I had to reduce the voltage for the test from 5v to 3.3v, because the KL25Z runs on 3.3v, not 5v. I got some rather weird results:

Two runs of measurement with R2=100Ω. The low-current measurements seem to be all noise.
click to embiggen

I could get decent measurement in the low-current range by using a larger resistor, so the problem was not noise in the measurement fixture or problems reading low differential voltages on the diode, but just with the small single-ended read for the current. It is pretty clear to me that the ADC does not work well when the input voltage results in less than about 50 counts.  (Note, that means that at the low end of the voltage range you only have about a 9.4-bit equivalent ADC.)

I modified the circuit to allow differential reading away from 0 for both the voltage across the diode and the voltage across the shunt resistor:

Adding an extra resistor ensured that the lowest voltage did not get too close to ground and I could use differential reads for both voltage (E20-E21) and current (E22-E23)/R2

This gave me a much cleaner reading, with problems only once the differential counts got below about 20:

Differential measurement with R2=100Ω. The low-current measurements have problems when the counts get small, but not nearly as severely as with the single-ended measurements.
click to embiggen

I replaced the 100Ω R2 resistor with a 15.56kΩ resistor (nominally 15kΩ), to extend to lower currents despite the noise in the ADC:

This plot extends the fit down to about 0.1µA, but only by adding an extra term—an offset to the voltage. I thought at first that overshoot to –11mV is an error in the analog-to-digital converter on the KL25Z, as I couldn’t see how my circuit could be back-biasing the diode.
Click to embiggen

I tried using larger resistors, but was unable to get any better data using them—I seem to be limited by the differential voltage measurement of the diode at the low end. I thought I might be able to improve the measurements by adding an instrumentation amp to increase the signal for low voltages.  But first I tried just hooking up a voltmeter, with no ADC or instrumentation amp connections.  When the voltage across R2 (100kΩ) is 0.31mV, the voltage across the diode plus R2 is only 0.05V, so there is -0.26mV across the diode.  The backwards voltage across the diode was not an artifact of the ADC!

I then tried looking at the voltage across the diode with my oscilloscope.  There is about 20mV of AC noise, independent of the DC voltage, until the diode has about 50mV across it (with the 15.5kΩ resistor for R2), by which time the noise has dropped to about 10mV (the Bitscope oscilloscope with the differential probe has a noise floor of about 3mV, if the two leads are connected together, so this is not just oscilloscope noise).  This noise seems to be white noise, not 60Hz hum pickup, so is probably coming from the diode.  This AC noise signal limits how accurately we can measure the DC current, and rectifying the noise could be the source of the mysterious “backwards” bias.

To reduce the noise, I put a 4.7µF ceramic capacitor in parallel with the diode, and redid all the measurements with 100Ω, 15.5kΩ, and 100kΩ resistors for R2.

Modified measurement circuit, adding a bypass capacitor to reduce AC noise on the diode and allow better DC measurement.

Now the signals are very clean down to nanoamp levels. I no longer need to add an offset to the voltage, as it is 0 to within the measurement repeatability. The noise from very small voltage differences for the 100Ω shunt resistor is still a bit of a problem, but that region is well covered by 15.5kΩ data. The curve was fit using just the 15.5kΩ and 100kΩ data, to avoid having to trim out the noise from the 100Ω data.
Click to embiggen

Lessons learned today:

• Higher-resolution ADCs do give smoother curves, with less digitization noise, but they aren’t a panacea for measurement problems. To get most of the resolution, I had to set the ADC to use long sample times and do a lot of averaging. I understand that Freescale Kinetis M series include 24-bit sigma-delta converters for higher precision at much lower speed (24 bits is 7 decimal digits), as well as the high-speed 16-bit successive-approximation converters. Unfortunately, they don’t have a low-cost development board for this series.
• Stay away from the bottom end of the ADC range on the KL25Z.  Scale single-ended inputs to have values at least 50, and differential inputs to have values at least 20.  There may be similar problems at the top end of the range, but I did not test for them.
  I wondered if the problem may be switching from the large value for the voltage across the diode to the small voltage across the shunt resistor that was the problem. I tried putting in a dummy read between the voltage and the current reads, but that didn’t help at all. At first I thought that the low-count readings were good with the large shunt resistors, but this is probably an illusion: errors in the current measurement for small currents aren’t visible on the plot, because the voltage across the diode is not changing, and so large horizontal errors in the plot are not visible there.
• Watch out for AC noise when trying to measure DC parameters.  If there are semiconductor junctions around, the noise may be rectified to produce an unwanted DC signal.
• The differential ADC settings have a range of ±VDDA, not ±VDDA/2. This means that the least-significant bit step size is twice as big for differential inputs as for single-ended inputs. For some reason the Freescale documentation never bothers to express what the differential range is.
• Serial USB connections are a bit flakey—the Arduino serial monitor missed a byte about every 200–300 lines.  I looked for anomalous points on the plot, then commented out the lines that produced them—they were almost all explainable by one character having been missed by the serial monitor; e.g., I commented out “662401069     86      19″ right after “660001069       865     17″,  because the last digit of the voltage (the second field) was missing.  The fields were a timestamp (in 24MHz ticks), voltage across the diode (in ADC units), and voltage across the shunt resistance (in ADC units).  [Actually, this was not a new lesson for me—I've had to do the same on almost all files collected from the Arduino serial monitor.  My son's data logger code is better at not losing data, but it is still worthwhile to check for anomalies.]
• The 3.3v supply from the Freedom board is much cleaner than the 5v USB supply that I get from the Arduino (unless I use an external power supply with the Arduino), but I can only take about 10mA from the 3.3v supply before it begins to droop.  If I want  more than that, I’d better provide my own power supply (or at least my own LDO regulator from the USB 5v supply).

Earlier today I posted a voltage-versus-current curve for a 1N5817 Schottky diode, to confirm the theoretical formula , where I_S is the saturation current of the diode:

Fitting over a wide range of currents is more robust than fitting over the narrower range that I can get with just one value for R2.
There is quantization error still on the voltages, but the overlapping current ranges give good data for most of the range. V_T is now 26.1mV and I_S is 0.91µA.

I also said that I should characterize the base-emitter junction of a PNP and an NPN transistor this way also, for setting the appropriate resistances for the log amplifier in the loudness circuit.  I did that this evening for the S9012 PNP and S9013 NPN transistors:

Base-emitter diode for the S9012 PNP transistor (collector and base connected together). V_T is 25.3mV and I_SO is 13.4fA.

Characteristics for the base-emitter diode of the S9013 NPN transistor (collector and base connected together). V_T is 25.5mV and I_SO is 8.85fA.

For both transistors, the region where the logarithmic fit is good is from about 0.5µA to about 50mA (maybe only 35mA for the NPN transistor). That gives about a 100dB working range for a log amplifier, if the largest current corresponds to 50mA. Of course, the op amps that are driving the input of the log amplifier don’t have that much drive capability, and we are probably limited to about 20mA—only a 90dB dynamic range on the log amplifier.

Earlier today I posted a voltage-versus-current curve for a 1N5817 Schottky diode, to confirm the theoretical formula , where I_S is the saturation current of the diode:

Fitting over a wide range of currents is more robust than fitting over the narrower range that I can get with just one value for R2.
There is quantization error still on the voltages, but the overlapping current ranges give good data for most of the range. V_T is now 26.1mV and I_S is 0.91µA.

I also said that I should characterize the base-emitter junction of a PNP and an NPN transistor this way also, for setting the appropriate resistances for the log amplifier in the loudness circuit.  I did that this evening for the S9012 PNP and S9013 NPN transistors:

Base-emitter diode for the S9012 PNP transistor (collector and base connected together). V_T is 25.3mV and I_SO is 13.4fA.

Characteristics for the base-emitter diode of the S9013 NPN transistor (collector and base connected together). V_T is 25.5mV and I_SO is 8.85fA.

For both transistors, the region where the logarithmic fit is good is from about 0.5µA to about 50mA (maybe only 35mA for the NPN transistor). That gives about a 100dB working range for a log amplifier, if the largest current corresponds to 50mA. Of course, the op amps that are driving the input of the log amplifier don’t have that much drive capability, and we are probably limited to about 20mA—only a 90dB dynamic range on the log amplifier.

Yesterday I posted a voltage-versus-current curve for a 1N5817 Schottky diode, to confirm the theoretical formula , where I_S is the saturation current of the diode, but I wasn’t really satisfied with the results, either in terms of dynamic range or the quality of the fit.

The voltage does fit nicely to the log of current, with I_S=1.32µA and V_T=27.1 mV. Given the quantization noise in the voltage measurement, these seem like a pretty good fit to theory. (click to embiggen)

One problem is that the serial variable resistor I used did not all really low currents.  I rewired it today so that I had a potentiometer providing the voltage, rather than a series variable resistor:

I also wrote a little Python program to merge different data files, so that I could combine files in which the resistance of R2 (for measuring the current) differed.

The resulting data fits the model well for over six decades (> 120dB):

Fitting over a wide range of currents is more robust than fitting over the narrower range that I can get with just one value for R2.
There is quantization error still on the voltages, but the overlapping current ranges give good data for most of the range. V_T is now 26.1mV and I_S is 0.91µA.

The measurements at the high-current end had to be redone with an external power supply for the Leonardo Arduino board (not just USB power), as the reference voltage for the A-to-D converter dipped as the load increased. There is a tiny effect still when using an external power supply, but only at the very highest current level, and it is buried in the noise.

At the low-current end, we can see the flattening of the curve from the “1+” term that is often omitted from the model.  The resolution in the voltage is poor there, but the current knee can be fairly accurately set by using a large value for R2.

I should probably characterize the base-emitter junction of a PNP and an NPN transistor this way also, for setting the appropriate resistances for the log amplifier in the loudness circuit.

Yesterday I posted a voltage-versus-current curve for a 1N5817 Schottky diode, to confirm the theoretical formula , where I_S is the saturation current of the diode, but I wasn’t really satisfied with the results, either in terms of dynamic range or the quality of the fit.

The voltage does fit nicely to the log of current, with I_S=1.32µA and V_T=27.1 mV. Given the quantization noise in the voltage measurement, these seem like a pretty good fit to theory. (click to embiggen)

One problem is that the serial variable resistor I used did not all really low currents.  I rewired it today so that I had a potentiometer providing the voltage, rather than a series variable resistor:

I also wrote a little Python program to merge different data files, so that I could combine files in which the resistance of R2 (for measuring the current) differed.

The resulting data fits the model well for over six decades (> 120dB):

Fitting over a wide range of currents is more robust than fitting over the narrower range that I can get with just one value for R2.
There is quantization error still on the voltages, but the overlapping current ranges give good data for most of the range. V_T is now 26.1mV and I_S is 0.91µA.

The measurements at the high-current end had to be redone with an external power supply for the Leonardo Arduino board (not just USB power), as the reference voltage for the A-to-D converter dipped as the load increased. There is a tiny effect still when using an external power supply, but only at the very highest current level, and it is buried in the noise.

At the low-current end, we can see the flattening of the curve from the “1+” term that is often omitted from the model.  The resolution in the voltage is poor there, but the current knee can be fairly accurately set by using a large value for R2.

I should probably characterize the base-emitter junction of a PNP and an NPN transistor this way also, for setting the appropriate resistances for the log amplifier in the loudness circuit.

Yesterday, I spent the day testing different transistors in the log amplifier, to see whether it made much difference which transistor I used.  I wanted to test all 11 transistors in the iteadstudio assortment, the 5 PNP transistors and the 6 NPN transistors, to see if it made much difference and whether one of the transistors would give me a larger mV/dB scaling than the others.

Here is the test circuit I made (essentially the same as for the tests in Logarithmic amplifier again):

(click image to embiggen) The same circuit can be used for either NPN or PNP transistors. The only difference is whether the 470µF capacitor starts at 0v (for PNP) or 5v (for NPN) before decaying to Vbias through the 16kΩ resistor R5. For many of the tests the voltage-to-current resistor R₂ was 1kΩ rather than 10kΩ.

The op amp at the top is a unity-gain buffer to provide a steady Vbias source that capable of providing some current (unlike the TL431ILP voltage reference shown here as an adjustable Zener diode). The nominal 2.5V reference was closer to 2.48V, which is within the ±2% spec for the part.

The unity-gain buffer on the left is to make the load on the RC circuit as high impedance as possible, so as not to disrupt the RC charging or discharging. The input current of the MCP6002 is typically ±1pA, which is far smaller than other sources of error (such as the parallel resistance of the capacitor itself). The parallel resistance of the capacitor makes the destination voltage of the RC circuit slightly lower than V_bias, which means that we will not get all the way to V_bias when testing PNP transistors, but will overshoot slightly when testing NPN transistors. (We should be able to reverse that by putting the capacitor between +5V and the switch, rather than ground and the switch.)

The right-hand op amp is just a non-inverting amplifier with 3× gain, to get better precision on measuring the output voltage with the Arduino ADC.  I made some measurements with V_out but most with the 3V_out signal. The scaling factor was 3.0 much more accurately than the repeatability of my fits to the data, so the gain in precision was not accompanied by a loss of accuracy. Ideally, I’d like to be able to use more than a 3× gain in the final stage, to get better precision, but I’m limited by the offset voltage of the log amplifier, which is determined by the transistor and by the voltage-to-current converter R₂.

The middle op amp is the log amplifier itself, which relies on the exponential relationship of the emitter current to the voltage across the base-emitter junction. More precisely, we can use the Ebers-Moll model of a bipolar transistor to get

The collector is held at Vbias by the feedback loop of the op amp, so V_BC is zero, simplifying the equation to

Assuming that β_R and I_S are constant, and that V_BE is very large compared to V_T (about 23 times bigger in my measurements), we have the desired exponential relationship: for some constant x.  Note that if we leave the –1 in, then we’ll have a small DC offset to i_C.

(I believe that all this analysis is correct for NPN transistors, where V_{BE is positive, but that some signs may need to be negated for PNP transistors.)}

The log-amplifier output is temperature-dependent, since , which is about 26mV at 300˚K. The scale factor can be multiplied by ln(10) to get 60mV/decade or 3mv/dB.

Note that this scaling is not affected by the current gain β of the transistor—that only affects the offset of the output voltage. This analysis agrees with the statement in Wikipedia, “At room temperature, an increase in V_BE by approximately 60 mV increases the emitter current by a factor of 10.” There is probably an even bigger temperature dependence for the offset of the output than for the scaling, because of changes in β_R and I_S, but that is not included in the Ebers-Moll model.

The theoretical result indicates that I should get about 3mV/dB for any transistor, but that the offset voltage will vary depending on the characteristics of the transistor.  I might also run into effects not included in the Ebers-Moll model, especially at very large or very small collector currents.

Adjusting R₂ to change the current can move the output offset around. If I make R₂ large and the current small, then V_BE will be small, and the approximation will be poor. If I make R₂ small, the current may exceed what the transistor is designed for and there may be saturation effects.

Looking at typical collector currents on the I-vs-V plots for the transistors, I’m seeing values like 8–20mA at the high end, so with a 2.5V drop across R₂, I want R₂ to be at least 300Ω. Initially, I picked 1kΩ, as providing a large current at 2.5V, hoping that this would give a large dynamic range.  Here are a couple of good examples of measurements using the 3× output and R₂=1kΩ.

(click image to embiggen) Typical 3× Vout vs Vin curve for a PNP transistor, here the A1015, using R₂=1kΩ. This uses Arduino-measured Vin, Vout, and Vbias, so is limited to about 2 decades (40dB).

(click image to embiggen) Typical 3× Vout vs. Vin curve for an NPN transistor, using R₂=1kΩ. Again we’re limited by the Arduino ADC to about 40dB of dynamic range.

The exponential decay of the capacitor to about V_bias allows us to extend the range of the fit well past the resolution of the ADC to measure V_in or V_bias. Combining the formulas for the log amplifier and the RC discharge gives us the general formula
.
Fitting the constants for this turns out to be difficult, because v₃ is close to zero. If it were exactly zero, the formula would be , and we could make arbitrary tradeoffs between v₂ and RC.

We can get a nice plot of V_out vs. time as the capacitor decays toward V_bias, particularly for the NPN transistor S9013:

(click picture to embiggen) The S9013 log amplifier shows what looks like a good fit over about 80dB, when using R₂=1kΩ, and good linearity for about 65dB. The upward tail shows where the collector-base junction begins to be forward biased, and the current is no longer controlled by the base-emitter voltage.

Note that this curve shows that the problems we had with direct measurement of RC discharge curves in the physics lab was due to limitations of the Arduino ADC, not to the underlying RC circuit. The tails of the discharge continue to follow the exponential well beyond the resolution of the 10-bit ADC in the Arduino.

Of course, I picked out the S9013 plot to show, because it was the nicest one. Some of the others were weird. For example, consider the 2N5551:

(click image to embiggen) Not that the 2N5551 Vout vs Vin curve is not a simple logarithm—there is some sort of saturation happening for large input voltages, so the straight-line fit is awful.

The same 2N5551 transistor with R₂ at 10kΩ behaves much better:

(click image to embiggen) With a 10kΩ resistor, the currents through the 2N5551 transistor are smaller, and no clipping occurs.

Even weirder was the behavior of the S9018 NPN transistor (the only transistor in the set not matched by a corresponding PNP transistor):

(click image to embiggen) With R₂=1kΩ, the S9018 transistor shows a flat spot in the response for  277mV < V_in-V_bias < 330mV (that is, 277uA < I_C < 330uA) . I have no idea what causes this flat spot.

I’m still mystified by the flat spot in the S9018 response—anyone have any ideas??

By changing R₂ to 10kΩ, we can push the flat spot to 10× higher voltage, just outside the input range that the log amplifier uses:

(click image to embiggen) With R₂=10kΩ, the response of the amplifier with a S9018 transistor is nicely logarithmic.

The time response to the RC discharge also looks good:

(click to embiggen) With R₂=10kΩ, we get a good fit for about 76dB on the S9018 transistor. The larger resistor gives a somewhat softer turn on for the transistor if we go past Vbias.

All of my transistors gave mV/dB scaling that was about the same (just under 9 mV/dB after the 3× gain, so just under 3mV/dB for the unamplified output), as is predicted by the Ebers-Moll model.  I got better dynamic ranges for the NPN transistors than for the PNP transistors, but this may have been due to artifacts of the test setup.  In any case, it looks like 60–70dB ranges are fairly easily achieved.

[UPDATE 2013 July 15: I know I said I was done with the log amplifiers, but I had to do just one more test.  In addition to the 16kΩ resistor to Vbias, I added a 5.7MΩ resistor to +5V, giving me a slightly higher target voltage so that I had some overshoot when testing PNP transistors.  With this setup I could test the S9012 PNP transistor for 82dB with R₂=1kΩ, and 63dB with R₂=10kΩ.  So the better dynamic range of the NPN was just an artifact of my test setup, as I thought.]

If one were to try to make a measuring instrument with a log amplifier, there would have to be some temperature compensation as the log-amplifier offset and scaling are both temperature sensitive.  Having a temperature-independent voltage source for calibration would be a good idea.

I think I’m about burned out on log amplifiers now.  Perhaps later his week I’ll try doing some precision rectifier circuits.

Yesterday, I spent the day testing different transistors in the log amplifier, to see whether it made much difference which transistor I used.  I wanted to test all 11 transistors in the iteadstudio assortment, the 5 PNP transistors and the 6 NPN transistors, to see if it made much difference and whether one of the transistors would give me a larger mV/dB scaling than the others.

Here is the test circuit I made (essentially the same as for the tests in Logarithmic amplifier again):

(click image to embiggen) The same circuit can be used for either NPN or PNP transistors. The only difference is whether the 470µF capacitor starts at 0v (for PNP) or 5v (for NPN) before decaying to Vbias through the 16kΩ resistor R5. For many of the tests the voltage-to-current resistor R₂ was 1kΩ rather than 10kΩ.

The op amp at the top is a unity-gain buffer to provide a steady Vbias source that capable of providing some current (unlike the TL431ILP voltage reference shown here as an adjustable Zener diode). The nominal 2.5V reference was closer to 2.48V, which is within the ±2% spec for the part.

The unity-gain buffer on the left is to make the load on the RC circuit as high impedance as possible, so as not to disrupt the RC charging or discharging. The input current of the MCP6002 is typically ±1pA, which is far smaller than other sources of error (such as the parallel resistance of the capacitor itself). The parallel resistance of the capacitor makes the destination voltage of the RC circuit slightly lower than V_bias, which means that we will not get all the way to V_bias when testing PNP transistors, but will overshoot slightly when testing NPN transistors. (We should be able to reverse that by putting the capacitor between +5V and the switch, rather than ground and the switch.)

The right-hand op amp is just a non-inverting amplifier with 3× gain, to get better precision on measuring the output voltage with the Arduino ADC.  I made some measurements with V_out but most with the 3V_out signal. The scaling factor was 3.0 much more accurately than the repeatability of my fits to the data, so the gain in precision was not accompanied by a loss of accuracy. Ideally, I’d like to be able to use more than a 3× gain in the final stage, to get better precision, but I’m limited by the offset voltage of the log amplifier, which is determined by the transistor and by the voltage-to-current converter R₂.

The middle op amp is the log amplifier itself, which relies on the exponential relationship of the emitter current to the voltage across the base-emitter junction. More precisely, we can use the Ebers-Moll model of a bipolar transistor to get

The collector is held at Vbias by the feedback loop of the op amp, so V_BC is zero, simplifying the equation to

Assuming that β_R and I_S are constant, and that V_BE is very large compared to V_T (about 23 times bigger in my measurements), we have the desired exponential relationship: for some constant x.  Note that if we leave the –1 in, then we’ll have a small DC offset to i_C.

(I believe that all this analysis is correct for NPN transistors, where V_{BE is positive, but that some signs may need to be negated for PNP transistors.)}

The log-amplifier output is temperature-dependent, since , which is about 26mV at 300˚K. The scale factor can be multiplied by ln(10) to get 60mV/decade or 3mv/dB.

Note that this scaling is not affected by the current gain β of the transistor—that only affects the offset of the output voltage. This analysis agrees with the statement in Wikipedia, “At room temperature, an increase in V_BE by approximately 60 mV increases the emitter current by a factor of 10.” There is probably an even bigger temperature dependence for the offset of the output than for the scaling, because of changes in β_R and I_S, but that is not included in the Ebers-Moll model.

The theoretical result indicates that I should get about 3mV/dB for any transistor, but that the offset voltage will vary depending on the characteristics of the transistor.  I might also run into effects not included in the Ebers-Moll model, especially at very large or very small collector currents.

Adjusting R₂ to change the current can move the output offset around. If I make R₂ large and the current small, then V_BE will be small, and the approximation will be poor. If I make R₂ small, the current may exceed what the transistor is designed for and there may be saturation effects.

Looking at typical collector currents on the I-vs-V plots for the transistors, I’m seeing values like 8–20mA at the high end, so with a 2.5V drop across R₂, I want R₂ to be at least 300Ω. Initially, I picked 1kΩ, as providing a large current at 2.5V, hoping that this would give a large dynamic range.  Here are a couple of good examples of measurements using the 3× output and R₂=1kΩ.

(click image to embiggen) Typical 3× Vout vs Vin curve for a PNP transistor, here the A1015, using R₂=1kΩ. This uses Arduino-measured Vin, Vout, and Vbias, so is limited to about 2 decades (40dB).

(click image to embiggen) Typical 3× Vout vs. Vin curve for an NPN transistor, using R₂=1kΩ. Again we’re limited by the Arduino ADC to about 40dB of dynamic range.

The exponential decay of the capacitor to about V_bias allows us to extend the range of the fit well past the resolution of the ADC to measure V_in or V_bias. Combining the formulas for the log amplifier and the RC discharge gives us the general formula
.
Fitting the constants for this turns out to be difficult, because v₃ is close to zero. If it were exactly zero, the formula would be , and we could make arbitrary tradeoffs between v₂ and RC.

We can get a nice plot of V_out vs. time as the capacitor decays toward V_bias, particularly for the NPN transistor S9013:

(click picture to embiggen) The S9013 log amplifier shows what looks like a good fit over about 80dB, when using R₂=1kΩ, and good linearity for about 65dB. The upward tail shows where the collector-base junction begins to be forward biased, and the current is no longer controlled by the base-emitter voltage.

Note that this curve shows that the problems we had with direct measurement of RC discharge curves in the physics lab was due to limitations of the Arduino ADC, not to the underlying RC circuit. The tails of the discharge continue to follow the exponential well beyond the resolution of the 10-bit ADC in the Arduino.

Of course, I picked out the S9013 plot to show, because it was the nicest one. Some of the others were weird. For example, consider the 2N5551:

(click image to embiggen) Not that the 2N5551 Vout vs Vin curve is not a simple logarithm—there is some sort of saturation happening for large input voltages, so the straight-line fit is awful.

The same 2N5551 transistor with R₂ at 10kΩ behaves much better:

(click image to embiggen) With a 10kΩ resistor, the currents through the 2N5551 transistor are smaller, and no clipping occurs.

Even weirder was the behavior of the S9018 NPN transistor (the only transistor in the set not matched by a corresponding PNP transistor):

(click image to embiggen) With R₂=1kΩ, the S9018 transistor shows a flat spot in the response for  277mV < V_in-V_bias < 330mV (that is, 277uA < I_C < 330uA) . I have no idea what causes this flat spot.

I’m still mystified by the flat spot in the S9018 response—anyone have any ideas??

By changing R₂ to 10kΩ, we can push the flat spot to 10× higher voltage, just outside the input range that the log amplifier uses:

(click image to embiggen) With R₂=10kΩ, the response of the amplifier with a S9018 transistor is nicely logarithmic.

The time response to the RC discharge also looks good:

(click to embiggen) With R₂=10kΩ, we get a good fit for about 76dB on the S9018 transistor. The larger resistor gives a somewhat softer turn on for the transistor if we go past Vbias.

All of my transistors gave mV/dB scaling that was about the same (just under 9 mV/dB after the 3× gain, so just under 3mV/dB for the unamplified output), as is predicted by the Ebers-Moll model.  I got better dynamic ranges for the NPN transistors than for the PNP transistors, but this may have been due to artifacts of the test setup.  In any case, it looks like 60–70dB ranges are fairly easily achieved.

[UPDATE 2013 July 15: I know I said I was done with the log amplifiers, but I had to do just one more test.  In addition to the 16kΩ resistor to Vbias, I added a 5.7MΩ resistor to +5V, giving me a slightly higher target voltage so that I had some overshoot when testing PNP transistors.  With this setup I could test the S9012 PNP transistor for 82dB with R₂=1kΩ, and 63dB with R₂=10kΩ.  So the better dynamic range of the NPN was just an artifact of my test setup, as I thought.]

If one were to try to make a measuring instrument with a log amplifier, there would have to be some temperature compensation as the log-amplifier offset and scaling are both temperature sensitive.  Having a temperature-independent voltage source for calibration would be a good idea.

I think I’m about burned out on log amplifiers now.  Perhaps later his week I’ll try doing some precision rectifier circuits.

Yesterday, in Logarithmic amplifier, I ended with the following plot:

(click plot to see larger version) The cloud of points is broad enough to be consistent with slightly different parameter values.

I was bothered by the broad cloud of points, and wanted to come up with a better test circuit—one that would give me more confidence in the parameters.  It was also quite difficult to get close to Vbias—the closest this could measure was one least-significant-bit of the DAC away (about 5mV).  A factor of 512 from the largest to the smallest signal is 54dB, but only about the upper 40dB of that was good enough data for fitting (and very little time was spent at near the Vbias value).

I think that part of the problem with the cloud was that the input signal was changing fairly quickly, and the Arduino serializes its ADC, so that the input and output are measured about 120µsec apart.  I decided to use a very simple slow-changing signal: a capacitor charging toward Vbias through a large resistor.  My first attempt used a 1MΩ resistor and a 10µF capacitor, for a 10-second time constant:

Output voltage from log amplifier (with 3x gain in second stage) as capacitor charges. (click picture for larger version) One fit uses the Arduino-measured Vin and Vbias voltages, the other attempts to model the RC charging as well. What are the weird glitches?

The capacitor charging should be a smooth curve exponential decay to Vbias, so the log amplifier output should be a straight line with time. There were two obvious problems with this first data—the output was not a straight line and there were weird glitches about every 15–20 seconds.

The non-straight curve comes from the capacitor not charging to Vbias. Even when the capacitor was given lots of time to charge, it remained stubbornly below the desired voltage. In think that the problem is leakage current: resistance in parallel with the capacitor. The voltage was about 1% lower than expected, which would be equivalent to having a 100MΩ resistor in parallel with the capacitor.  I can well believe that I have sneak paths with that sort of resistance on the breadboard as well as in the capacitor.

According to Cornell Dublier, a capacitor manufacturer, a typical parallel resistance for a 10µF aluminum electrolytic capacitor would be about 10MΩ [http://www.cde.com/catalogs/AEappGUIDE.pdf‎]:

Typical values are on the order of 100/C MΩ with C in μF, e.g. a 100 μF capacitor would have an Rp of about 1 MΩ.

So I may be lucky that I got as close to Vbias as I did.

The glitches had a different explanation: they were not glitches in the log amplifier circuit, but in the 5V power supply being used as a reference for the ADC on the Arduino board—I had forgotten how bad the USB power is coming out of my laptop, though I had certainly observed the 5V supply dropping for a second about every 20 seconds on previous projects.  The drop in the reference for the ADC results in a bogus increase in the measured voltages.  That problem was easy to fix: I plugged in the power supply for the Arduino rather than running off the USB power, so I had a very steady voltage source using the Arduino’s on-board regulator.

(click picture for larger version) With a proper power supply, I get a clean charge and the output is initially a straight line, but I’m still not getting close to Vbias. Again the blue fit uses the measured Vin and Vbias voltages, while the green curve tries to fit an RC decay model. Note the digitization noise on the measured inputs towards the end of the charging time.

To solve the problem of the leakage currents, I tried going to a larger capacitor and smaller resistor to get a similar RC time constant. At that point I had not found and read the Cornell Dublier application note, though I suspected that the parallel resistance might scale inversely with the capacitor size, in which case I would be facing the same problem no matter how I chose the R-vs-C tradeoff. Only reducing the RC time constant would work for getting me closer to Vbias.

Using a 47kΩ resistor and a 470µF capacitor worked a bit better, but the time constant was so long that I got impatient:

(click image for larger version) The blue fit is again using the measured Vin and Vbias, and has a pretty good fit. The green fit using an RC charge model does not seem quite as good a fit.

The calibration of 9.7mV/dB seems pretty good, so the 409mV range of the recording corresponds to a 42dB range. The line is straighter, but I’m still not getting as close to Vbias as I’d like.

I then tried a smaller RC time constant (hoping that the larger current with the same capacitor would result in getting closer to Vbias, and so testing a larger dynamic range on the log amplifier). I tried 16kΩ with the 470µF capacitor:

(click image to embiggen) I’m now getting a clear signal from the log amplifier even after the input voltage has gotten less than one least-significant-bit away from Vbias (the blue fit). I found it difficult to fit parameters for modeling the RC charge (the green fit).

The two models I fit to the data give me somewhat different mV/dB scales, though both fit the data fairly well. The blue curve fits better up to about 65 seconds, then has quantization problems. Using that estimate of 9.8mV/dB and the 560mV range of the output, we have a dynamic range here of 57dB. There is still some flattening of the curve—we aren’t quite getting to the Vbias value, but it is pretty straight for the first 50 seconds.

Note: the parallel resistance of the capacitors would not explain the not-quite-exponential behavior we saw in the RC time constant lab, since those measurements were discharging the capacitor to zero. A parallel resistance would just change the time constant, not the final voltage.

I was using the Duemilanove board for the log-amplifier tests. I retried with the Uno board, to see if differences in the ADC linearity make a difference in the fit:

(click to embiggen) Using the Uno Arduino board I still had trouble with the fit, and the Uno ADC seems to be noisier than the Duemilanove ADC. The missing parts of the blue curve are where the Uno board read the input as having passed Vbias.

The 625 mV range over 250 seconds corresponds to about 69dB, assuming that the 9.1 mV/dB calibration is reasonably accurate (and 64dB if the earlier 9.8mV/dB calibration is better).
My measurements of the log amplifier do not seem to yield a very consistent mV/dB parameter, with values from 9.1mV/dB to 9.8mV/dB using just the Arduino measurements (and even less consistency when a model of RC charging is used).  I’m not sure how I can do more consistent measurements with the equipment I have.  Anyone have any ideas?
Incidentally, my son has decided not to include a microphone in his project.  The silicon MEMS mic was small enough, but the op amp chip for the analog processing was too big for the small board area he had left in his layout, and he decided that the loudness detector was not valuable enough for the board area and parts cost. I believe that his available board area shrunk a little today, because he discovered that the keep-away check had not been turned on in the Eagle design-rule checker.  Turning it on indicated that he had packed the capacitors too close in places, and he had to spread them out. (At least, I think that’s what he told me—I’ve not been following his PC board layout very closely.)
I’m still interested in learning about log amplifiers and precision rectifiers, so I’m still going to breadboard the components of the design and test them out.  I’m not sure when I’ll ever use the knowledge, since the Applied Circuits course does not cover the nonlinear behavior of pn junctions nor the forward-voltage drop of diodes (we don’t use diodes in the course).

Yesterday, in Logarithmic amplifier, I ended with the following plot:

(click plot to see larger version) The cloud of points is broad enough to be consistent with slightly different parameter values.

I was bothered by the broad cloud of points, and wanted to come up with a better test circuit—one that would give me more confidence in the parameters.  It was also quite difficult to get close to Vbias—the closest this could measure was one least-significant-bit of the DAC away (about 5mV).  A factor of 512 from the largest to the smallest signal is 54dB, but only about the upper 40dB of that was good enough data for fitting (and very little time was spent at near the Vbias value).

I think that part of the problem with the cloud was that the input signal was changing fairly quickly, and the Arduino serializes its ADC, so that the input and output are measured about 120µsec apart.  I decided to use a very simple slow-changing signal: a capacitor charging toward Vbias through a large resistor.  My first attempt used a 1MΩ resistor and a 10µF capacitor, for a 10-second time constant:

Output voltage from log amplifier (with 3x gain in second stage) as capacitor charges. (click picture for larger version) One fit uses the Arduino-measured Vin and Vbias voltages, the other attempts to model the RC charging as well. What are the weird glitches?

The capacitor charging should be a smooth curve exponential decay to Vbias, so the log amplifier output should be a straight line with time. There were two obvious problems with this first data—the output was not a straight line and there were weird glitches about every 15–20 seconds.

The non-straight curve comes from the capacitor not charging to Vbias. Even when the capacitor was given lots of time to charge, it remained stubbornly below the desired voltage. In think that the problem is leakage current: resistance in parallel with the capacitor. The voltage was about 1% lower than expected, which would be equivalent to having a 100MΩ resistor in parallel with the capacitor.  I can well believe that I have sneak paths with that sort of resistance on the breadboard as well as in the capacitor.

According to Cornell Dublier, a capacitor manufacturer, a typical parallel resistance for a 10µF aluminum electrolytic capacitor would be about 10MΩ [http://www.cde.com/catalogs/AEappGUIDE.pdf‎]:

Typical values are on the order of 100/C MΩ with C in μF, e.g. a 100 μF capacitor would have an Rp of about 1 MΩ.

So I may be lucky that I got as close to Vbias as I did.

The glitches had a different explanation: they were not glitches in the log amplifier circuit, but in the 5V power supply being used as a reference for the ADC on the Arduino board—I had forgotten how bad the USB power is coming out of my laptop, though I had certainly observed the 5V supply dropping for a second about every 20 seconds on previous projects.  The drop in the reference for the ADC results in a bogus increase in the measured voltages.  That problem was easy to fix: I plugged in the power supply for the Arduino rather than running off the USB power, so I had a very steady voltage source using the Arduino’s on-board regulator.

(click picture for larger version) With a proper power supply, I get a clean charge and the output is initially a straight line, but I’m still not getting close to Vbias. Again the blue fit uses the measured Vin and Vbias voltages, while the green curve tries to fit an RC decay model. Note the digitization noise on the measured inputs towards the end of the charging time.

To solve the problem of the leakage currents, I tried going to a larger capacitor and smaller resistor to get a similar RC time constant. At that point I had not found and read the Cornell Dublier application note, though I suspected that the parallel resistance might scale inversely with the capacitor size, in which case I would be facing the same problem no matter how I chose the R-vs-C tradeoff. Only reducing the RC time constant would work for getting me closer to Vbias.

Using a 47kΩ resistor and a 470µF capacitor worked a bit better, but the time constant was so long that I got impatient:

(click image for larger version) The blue fit is again using the measured Vin and Vbias, and has a pretty good fit. The green fit using an RC charge model does not seem quite as good a fit.

The calibration of 9.7mV/dB seems pretty good, so the 409mV range of the recording corresponds to a 42dB range. The line is straighter, but I’m still not getting as close to Vbias as I’d like.

I then tried a smaller RC time constant (hoping that the larger current with the same capacitor would result in getting closer to Vbias, and so testing a larger dynamic range on the log amplifier). I tried 16kΩ with the 470µF capacitor:

(click image to embiggen) I’m now getting a clear signal from the log amplifier even after the input voltage has gotten less than one least-significant-bit away from Vbias (the blue fit). I found it difficult to fit parameters for modeling the RC charge (the green fit).

The two models I fit to the data give me somewhat different mV/dB scales, though both fit the data fairly well. The blue curve fits better up to about 65 seconds, then has quantization problems. Using that estimate of 9.8mV/dB and the 560mV range of the output, we have a dynamic range here of 57dB. There is still some flattening of the curve—we aren’t quite getting to the Vbias value, but it is pretty straight for the first 50 seconds.

Note: the parallel resistance of the capacitors would not explain the not-quite-exponential behavior we saw in the RC time constant lab, since those measurements were discharging the capacitor to zero. A parallel resistance would just change the time constant, not the final voltage.

I was using the Duemilanove board for the log-amplifier tests. I retried with the Uno board, to see if differences in the ADC linearity make a difference in the fit:

(click to embiggen) Using the Uno Arduino board I still had trouble with the fit, and the Uno ADC seems to be noisier than the Duemilanove ADC. The missing parts of the blue curve are where the Uno board read the input as having passed Vbias.

The 625 mV range over 250 seconds corresponds to about 69dB, assuming that the 9.1 mV/dB calibration is reasonably accurate (and 64dB if the earlier 9.8mV/dB calibration is better).
My measurements of the log amplifier do not seem to yield a very consistent mV/dB parameter, with values from 9.1mV/dB to 9.8mV/dB using just the Arduino measurements (and even less consistency when a model of RC charging is used).  I’m not sure how I can do more consistent measurements with the equipment I have.  Anyone have any ideas?
Incidentally, my son has decided not to include a microphone in his project.  The silicon MEMS mic was small enough, but the op amp chip for the analog processing was too big for the small board area he had left in his layout, and he decided that the loudness detector was not valuable enough for the board area and parts cost. I believe that his available board area shrunk a little today, because he discovered that the keep-away check had not been turned on in the Eagle design-rule checker.  Turning it on indicated that he had packed the capacitors too close in places, and he had to spread them out. (At least, I think that’s what he told me—I’ve not been following his PC board layout very closely.)
I’m still interested in learning about log amplifiers and precision rectifiers, so I’m still going to breadboard the components of the design and test them out.  I’m not sure when I’ll ever use the knowledge, since the Applied Circuits course does not cover the nonlinear behavior of pn junctions nor the forward-voltage drop of diodes (we don’t use diodes in the course).