Crystal Qu and Other Doggerel

Hello!

A reader saw a photo of my old W7ZOI crystal characterization oscillator (with the suggested switch from Dr. Gordon-Smith, G3UUR) in the site archive. I had removed its crystal holders to yet another such oscillator and just soldered in a crystal as a PROP for the photo. This photo "suggested" it was OK to solder in the various crystals under test. 

The reader found serious temperature drift during crystal frequency measures and then emailed me with deserved concern. Yikes.  My bad — I apologize.  Securing a crystal under test in its holder must never involve heat. Wes, W7ZOI even puts a crystal into its holder while wearing gloves to minimize body temperature effects.

I prefer to characterize crystal motional L and C with the G3UUR formula and measure C0 with a LC meter. For Qu, I'm adopting Wes' series trap method shared in EMRFD Chapter 7. I'll blog my humbling Qu experiments after this introductory bit.

 

 

Above — 3 photos of an actual oscillator I've used to characterize crystal motional parameters. In the third photo I replaced the gimpy toggle with a slider switch. Studying the effects of switch types might require experiments, however, I've learned that switch lubricants may affect measures. Regardless, the C of the toggle switch plus the 33 pF fixed series capacitor get accommodated in the calculations.

The ultimate switch might include the latching microwave relays like those found in the K2 VCO, or perhaps small pin bridging connectors like those found on motherboards? Building on a single-sided Cu board seems prudent to avoid unwanted Q side effects.

My alligator clip crystal holders might not rank as rave stuff, but they've worked well for me from 2 - 12 MHz.  I'm not a builder known for handsome breadboard techniques — I'm more a framing carpenter: but hope to grow in my understanding of component-level engineering over time.

A new circuit with the the holders for the xtal + capacitor & shorting bar built above a copper ground plane is under development. This fixture uses a tracking generator plus spectrum analyzer to get the values needed to calculate Lm with the G3UUR formula. Further, an IL measure and 3 dB measurement will provide ESR and Qu. "One and done" it will be!
I need to characterize some xtals for the 30-40 MHz crystal ladder IF filters in my future NBFM receiver experiments. I'll share all these experiments in the future.

Above — My current oscillator schematic. The original version by Wes shown in EMRFD works great and I recommend it . The 2 Colpitts caps should be >=10 times the series 33 pF, so my 390 pF caps work OK. I added a green "power on" LED with a 1K current limiter resistor in my build.

Most of my lower current DC power supplies connect to my breadboards via RCA jacks. This helps prevent DC supply alligator probe shorts during experiments involving many components. Whatever works.

Above — Measuring the capacitance of the open switch plus the 33 pF fixed value cap wired in-situ. This will give you the total circuit capacitance of the open switch, the 33 pF fixed value capacitor, and any stray capacitance from your crystal holder, wires, etc. The switch itself plus stray wiring  will be a few pF so the total should be 36 to 40 pF or so.

Above — My open switch total circuit capacitance.

 Qu Method

Whenever I write about crystal characterization, I spark passionate emails. I've read it all — "gotta use a VNA" , "there's a better way to measure crystal holder capacitance", "...run a VXO signal generator", "you should use this program, or that fixture". All are good comments and appreciated. 

I think we're lucky to have so many crystal ladder filter experts in our midst.

Certainly, professionals measure crystal parameters with network analyzers, however, we amateurs could debate this forever. Gear and measurement techniques change over time. Digital boxes keep getting cheaper to make, write code for, or buy: even the new Ham down the street bought a VNA to match his antenna. I had to show him how to use it though.

The proof of the pudding lies in in your filter breadboards. Sticking measures/values such as Lm into a program should result in a build that matches the simulated transfer response. If you make and then confirm a good filter; all is well!

Sometimes standard value capacitors, or matching errors from stray L and C, or the lack of shielding may produce extra ripple, or reduce the stop-band response and/or boost insertion loss. Then, too, we sometimes have to make a narrow filter with low Q crystals and suffer high insertion loss because we can't find better xtals at the time. 

Instrument accuracy, measurement quality and arithmetic errors also factor. My gear gets calibrated regularly and it costs me dearly. I know a "expert" builder with old, crusty gear who never pays for professional calibration and then goes off on a new builder because he rounded too much during calculations.

Do your best — forget the folklore — relax, and enjoy your test bench. A desire to know, grow and learn brings its own rewards. I've mentioned this before: ours is often a difficult, frustrating hobby.

Whatever method you characterize your crystals with — it should match your budget, skill and needs.

Measuring Qu

Qu or Unloaded Quality Factor in RF tuned circuits and filters can help with simulations and breadboards alike. I'll show an early attempt to measure crystal Qu. We enjoy a few methods for Q and Qu — enough, at least, to make it a fun challenge.

To preface, I recently bought my first non-home brew signal generator. It's old, but accurate, low noise, and freshly calibrated for I'm on a mission to run a 3 GHz bandwidth lab. A signal generator plus tracking generator/spectrum analyzer will serve as the main signal power measurement instruments as I move well above HF.

In part, wishing to improve my measurement techniques with this gear informed my choice of  Qu measurement technique.

I examined a 4.0 MHz crystal from a batch of 10 "high quality, well matched crystals" purchased in 1996 from a reputable vendor from Germany. To my delight, I found 6 within ~10 Hz frequency and built a N=6,  500 Hertz wide CW filter with 0.1 dB ripple and an IL ~1 dB.

Here's the crystal motional L and C measures plus C0 of a remaining, unused crystal:

Above — The series trap set-up.  You need a flat amplitude versus frequency signal generator with 1 Hz tuning resolution, plus a detector that can measure power/amplitude and frequency in a narrow notch. I used a spectrum analyzer with a narrow span + RBW to best pick off the trap or notch frequency. 

The crystal was soldered in a fixture and allowed to cool for 6 hours. I recorded the power measure with the crystal in place, then substituted a through connector to learn the crystal's attenuation at its series resonant frequency,

I've learned that Z0 is the critical piece. I got better data by threading a SMA 10 - 15 dB attenuator pad on either side of the crystal to boost measurement fidelity. 

Above — I measured the attenuation of the notch as 9.46 dB and took equation 7.4 from EMRFD with Lm substituted for Lu. These are fabulous crystals and a Qu of ~275K explains why the IL was so low in my aforementioned narrow CW filter. I've got some glass encased crystals with a Qu of 720K in my parts collection. Now those are sweet.

Above — As a lark, I tried to measure Qu with a direct 3 dB method. Lots of attenuation made the measure difficult and I'm not sure I've truly got Qu. Transformers would work better, but I abandoned this little adventure since I'm happy with the series trap method. I'll also apply the trap technique to infer resonator Q at VHF for filter measures.

Big thanks to all those greats who pioneered crystal measurement and to Wes, W7ZOI for his support.

If you're building the Minima, I recommend viewing this page/site on the xtal filter and more by Steve, VK2SJA. He's got a link to a great crystal ladder filter summary by Nick Kennedy, WA5BDU and also applies the latest version of Dishal by Horst Steder, DJ6EV that embeds the work of Jack Hardcastle, G3JIR and he. Horst, DJ6EV link

Update:

Here's a quick Qu measurement of a resonator using the series trap method with a tracking generator and spectrum analyzer. Please see EMRFD Figure 7.66. A 96 nH heavy gauge Cu wire coil was soldered in series with a small air variable capacitor. I measured the resonator's attenuation at SRF after normalizing/zeroing the sweep system. I forgot to photograph the coil and cap in my fixture.

Above — The attenuation at the resonator's series resonant frequency.

Above — The Qu of this resonator at 112.3 MHz. I then measured and calculated Qu using the 3 dB method shown on November 11, 2014 and got 280 -- pretty close.

 

Above — A solid copper ground plane and connector holder eliminates any possible unwanted "capacitor" effects of FR4 board. The series resonator was soldered to the center of the copper wire at 1 end and ground at the other.

Best!

Measuring Resonator Q at VHF

Greetings!

I've long enjoyed broadcast FM DX chasing. This post covers the first chapter in my pursuit to design and make a homebrew broadcast (wideband) FM DX receiver.

Traditional FM DX superheterodyne receiver ran single conversion with a 10.7 MHz IF and usually a dual-gate MOSFET mixer behind 1-2 dual gate MOSFET preamplifers embedded in 2-4 doubled-tuned bandpass filters.  To cover 88-108 MHz with decent skirts + bandwidth, the band-pass filters were tuned with ganged, air-variable capacitors that simultaneously tuned the VFO.

Fast forward to this day in time. Following low-pass filters and T/R switch circuitry, some modern narrow band FM transceivers run varactor tuned band-pass filters, 1 or 2 dual gate MOSFET amplifier(s) and a MOSFET mixer in the receiver chain.

I explored varactor tuned band-pass filters for wideband FM and learned a few things along the way.

Above — My first FM band-pass filter that tuned all 20 MHz of the FM band without a double humped filter response. In order to get the BB639's capacitance down to its lowest value (~ 3 pF to tune 108 MHz), you need 28 volts applied reverse DC. I keep a homebrew bench DC-DC converter for just that purpose. The filter's 3 dB bandwidth ran from about 2.2 MHz at 88 MHz to 3.3 MHz wide at 108 MHz

In order to boost the filter low-pass response, I tapped the inductors for the input and output ports. Light loading improved the filter response, but reduced the input/output return loss. Tuning a filter over a 20 MHz span proved a lesson in compromise.

60 nH isn't a lot of L and my coils were 6 turns of bare copper 22 gauge wire wound on a small bolt and then stretched to allow room for tapping and to set the correct measured inductance. I further tweaked them in-situ. Later, I placed this filter after various preamplifiers including a dual-gate MOSFET and a common gate JFET amp.

Above — A snippet of 1 of my amplifier + band-pass filter circuits. I got a better filter response with a 96 nH inductor, although this should technically worsen it.

1 major problem arose with my filters: a horrible insertion loss of 8-9 dB!!  I expected about half that. Later I wrote some great friends for advice and after reading their wisdom, I came up with 2 sane theories: the insertion loss was due to lowered Q and input + output port mismatch.

Resonator Q Measurement

I then realized that I'd never measured resonator Q at VHF. If you own a Q meter such as the
HP4342a stop reading now, get a coffee and go to another blog-site.

As amateur experimenters, to derive resonator Q we may employ 2 techniques: calculate Q after measures with a parallel tuned L C tank, or calculate Q after measuring with the L and C set in a series tuned trap circuit.

I've not enjoyed much success with the latter, so will present the method where our parallel L and C are loosely coupled to a 50 Ω source and load. Please refer to EMRFD page 7.36 for more details.

Supplies:

• Signal generator with level output amplitude,  50 Ω output Z and enough output power to allow measurement with your particular detector.
• 50 Ω detector: spectrum analyzer,  50 Ω terminated 'scope, or measurement receiver etc.
• 50 Ω patch cables.
• A homebrew jig with RF connectors, coupling capacitors and ground plane.
• Frequency counter.
• Through connector.

Setup:

Above — The basic paralleled tuned resonator measurement set up including gimmick probes as the input and output coupling capacitors.

Above — My test jig with the inductor and air variable capacitor soldered in place. I copied this jig from Bob, K3NHI and received advice from Wes, W7ZOI. Bob made a circuit at his QTH, measured the resonator Q and sent me a photo and some measures by email. I re-created his circuit to compare results. The coil = ~ 300 nH or 10 turns on a 1/4 inch bolt. The capacitor to resonate it @ ~100 MHz on my bench = 7.27 pF.

 Method to Set Jig Insertion Loss at VHF

In order to measure a resonators unloaded Q, or Qu, the insertion loss of the jig must minimally be 30 dB. To clarify, join the input and output cables together with a barrel or through connector and measure power. Unsplice and then connect the cables to the jig and measure power once again — power should drop by at least 30 dB at the test frequency.

To create the >= 30 dB insertion loss we lightly couple the jigs input and output with low value capacitors. At HF, we may insert small series capacitors, but this is nearly impossible at VHF unless you own some special microwave parts. Instead of series capacitors, we couple with gimmick wires.  Experiment to find the correct wire distance from the resonator to create the needed insertion loss.

Here are my 2 jig measurements with a DSO plus the IL calculation:

Above — My insertion loss fulfills the require >= 30 dB needed for proper resonator coupling
I calculated the IL in dB with JavaScript Tool G. It's difficult to measure under 50 mV with a 'scope for some; your power meter or spectrum analyzer might work better.

To get the needed IL or measure for Qu calculations, set your signal generator to the desired frequency and then tweak the resonator variable capacitor to give the highest possible AC voltage/power. Then re-tweak your signal generator to ensure you've peaked the signal. You might have to re- tweak the variable cap again and so forth.

Optionally, If your resonator capacitor is fixed, adjust your signal generator frequency to peak the signal.

After properly setting the IL and peaking for the strongest signal, the final measures go quickly:

1. Record the frequency where you measured maximum power: that's FO, or center frequency.
   
   
2. While watching your detector, lower the signal generator frequency until FO power drops by 3 dB [easier to do with an SA or power meter]. Record that frequency.
   
   
3. Bring the signal generator back up to FO and then increase frequency until the power drops by 3 dB. Record that value.
   
4. Calculate Qu as Frequency/Bandwidth.

I'll show my measures performed with a homebrew signal generator with less than ideal tuning resolution; however, you'll get the idea.

Above — FO or center frequency = -25.84 dBm, therefore my 3 dB down target = 28.84 dBm when I change my signal generator below and above FO.

 

Above 2 images — The 3 dB measures below and above FO (getting as close as possible with a homebrew VCO).  My VHF VCO sports a >= 30 dB output return loss from 98 - 149 MHz.

Calculate B or the 3 dB bandwidth by subtracting the lower frequency from the higher.

My calculated resonator Qu = 289. Bob, K3NHI measured then calculated 300 on his. Pretty close.

Above — Bob's homebrew jig and resonator. Bob measures everything: including his breadboard length! 

Varactor Measurement

I removed the air-variable trimmer cap and inserted a small piece of copper clad break-out board to hold a tiny BB639 varactor (size SOD-323). I voltage tuned it to resonance and then repeated the whole resonator Q measurement routine. Q = 174: a drop of 115 which would boost my original FM band-pass filter insertion loss by at least 1.5 dB compared to an air variable capacitor.

Above — Measuring resonator Q with a varactor.

Through experiments with the the aforementioned parallel + series resonator measurement techniques, I learned that carved squares and traces in boards may also lower Q. Even a Manhattan or carved pad nearby may couple to the resonator and drop its Q during measurement. My worse case measure produced a drop in Q of 20 from nearby islands carved in the copper board. Clearly we need board traces, but they can affect resonator Q and thus add to filter insertion loss.

Further, good VHF filter designs stick each resonator in an RF tight compartment. Whatever filter I eventually keep, I'll mind my Qs.


Footnotes

Thanks to Bob, Bob, Wes, Ken, John and others who kept me on track  — I know just enough to act foolishly on the bench.

I made 1 varactor tuned band-pass filter on single-sided copper board and compared it to the double-sided board versions. The single-sided board suffered poor stopband shape and didn't tune as well.

Above — A board set up for a dual-gate MOSFET surface mount circuit where the FET source runs a shunt resistor and capacitor to ground, plus has DC voltage on both G1 and G2. Some via holes connect the top and bottom ground plane.

Above — I placed 22 gauge copper wire in the via holes and soldered them top and bottom.
Grounded parts are placed near a via wire. Sometimes, I'll add more via wires near grounded parts.

Above — A single frequency amplifier carefully matched to see how much gain the BF998 could deliver. 21.2 dB rocks my world.

Above — Some of my (mostly) 50 Ω homebrew bench modules for test and measurement.

Above — My favorite design project of 2014: a return loss bridge with directivity >= 30 dB from 5 MHz to 1.5 GHz.   You may read more about it in the old site pops.net archive: Topics 2012 - 2014 : Caitlyn 310 — UHF Beginnings : 3. Return Loss Bridge Experiments : Bridge #4

AF Power Amp Experiments

Although VHF focused, I spent some time studying AF power amp design this fall. Even SDRs need audio amplifiers for the ear interface.  If a speaker guy wants some serious wattage blaring in his cottage, then a split DC supply audio power amp with DC -15v / +15v or so garners an easy ~12-16 watts clean average power.

However, back in the land of radio heads, swinging the AC even remotely near rail to rail with a 12 volt single-supply proves arduous. Thus, designing and building AF power amps that cleanly swing as close to the supply rails as possible seems like a good idea.

Before these, my single-supply AF power amplifier experiments usually employed single or Darlington emitter followers arranged as a complimentary symmetry pair. 1 bleak fall day; with coffee and cats, I changed it up and moved away from typical, symmetrical, emitter follower push-pull finals to learn about other topologies.

I tested these amps with an ultra-low distortion, variable amplitude 1 KHz signal generator, an 8 Ω resistive dummy load, a DSO with FFT and 2 DMMs. I'll share 3 progressive amp experiments:

Amplifier 1

First up is quasi-complementary amplifier. Unusual to see in 2014, but great fodder for learning what goes on in a PA.  After making, debugging and analyzing these circuits,  I felt humbled about how little I consider AC signal phase in addition to the easier concepts of amplitude and frequency during my AC signal analysis.

Above in Figure 1 — My first quasi-complementary amplifier.

Q1 = a class A driver with negative feedback that lowers its input impedance. Typically, a small series resistor goes on the input to raise its input impedance, however, I left it off on this experimental amp. 

Q2, an emitter follower [with no phase inversion] forms a standard Darlington pair with Q4 Their asymmetrical compliment are Q3 and Q5: Q3 is a PNP common emitter amp that inverts the signal phase.

When a negative swinging signal enters Q4, it draws current that flows into the speaker because simultaneously, a positive swinging signal enters Q5 and holds it near cut off. When the AC signal flips polarity, Q4 cuts off and Q5 conducts. Thus the quasi-complementary amplifier gives single-ended push-pull output.

Let's discuss applying forward bias on Q4 and Q5 to eliminate crossover distortion.

The simplest method involves placing diode(s) across the Q2 and Q3 bases to forward bias them enough to in turn, forward bias Q4 and Q5 almost to conduction in their quiescent state.

Above — Cross over distortion measured across the 8 Ω load with 2 bias diodes.

Above — Cross over distortion with 3 diodes (quite near to eliminating the cross over distortion).

I could have tried 4 diodes, but just replaced the diodes with an adjustable level shifter or so-called amplified diode to allow precise control of the forward bias on Q4 and Q5. Further, 1 replaced the original DC coupled feedback from the speaker back to the Q1 with AC coupled negative feedback (a 100 pF capacitor):

Above — Figure 1B. My final quasi-complementary amplifier. 

The 22 µF capacitor in both designs provides essential positive feedback across the 1K resistor. Bootstrapping feedback compensates for the asymmetrical output stage allowing the positive peak signal swing to approach its negative counterpart. The 22 µF capacitor maintains a charge to keep the DC voltage across R1 constant and their time constants must consider the lowest frequency to be amplified.

Above — Figure 1B breadboard; although it still has diodes to set the quiescent current at this point.

 

Above — FFT  in pink showing strong harmonics. You can see heavy cross-over distortion in the yellow time domain tracing too. I then tweaked the 10K level shifter bias pot to eliminate the crossover distortion:

 Above — FFT (pink) with minimal cross-over distortion after adjusting the level shifter pot. 

I'm now routinely setting my level shifter to remove cross over distortion with the aid of FFT plus simultaneous viewing of the signal in time domain ('scope viewing). I usually set the drive so the amp is making about 1/2 its maximal clean signal power when tweaking the bias. Then I increase the drive to "full" clean signal power to confirm that cross over distortion doesn't re-emerge, Usually at this point, harmonic distortion begins to dominate.

While it's possible to set the PA forward bias by just viewing the sine wave and tweaking the 10K pot to find where the cross over distortion just disappears, to me, the FFT takes this to the next level. 

My Figure 1B bench assessment offered a good new-bad news paradox. In the 'scope trace above, the tones are down 63 dBc, so I managed my personal best, single DC supply PA in terms of distortion at 1/2 power. The bad news = it took ~ 200 mA of quiescent current to deliver this performance. 

It seems that the output is stage is starved for gain on the upper half of the AC cycle — which is why it takes so much current and it's peak-peak is limited. Point X, or the collector of Q5 should ideally be close to 1/2 VCC, but in my quest for low distortion and big signal swing, it ended up at 5.68 VDC.

I think back in the day when they used amps like this, builders tolerated a lot more crossover distortion, but I’m surprised that this amp when biased in AB (or maybe deeper) gives such good distortion performance with the mere addition of that 100 pF feedback cap. Further, I liked setting the PA bias with the aid of FFT instead of just looking at the old sine wave.  I'm learning, and for sure, making mistakes.

Above — FFT of Figure 1B's maximal clean signal power = 3.82 Vpp [228 mW]. The worst tone is -53 dBc. Clearly with ~ 200 mA of idle current and only 228 mW clean signal power, this power amplifier is not a keeper, but I enjoyed analyzing + working on it and felt a boost in confidence going forward.

Signal Squaring

A home brew square wave board might help assess your AF power stages in time domain. If your PA can accurately reproduce a square wave, you're on the right track!

A square wave provides a symmetrical waveform that alternates instantaneously between 2 levels and allows you to see rise times, overshoot and other phenomena. I used square wave analysis to help me choose the 100 pF AC feedback capacitor in Figure 1b.

Ensure you watch your DC current so you don't suffer final amplifier thermal runaway during square wave testing as your amp may consume significant power when driven hard. I blew 1 final in my experiments. With PA experiments, heat sinking the finals proves worthwhile. I did this in Amplifiers 2 and 3.

Above — A simple, signal squarer I keep in my lab. With appropriate RF bypass and series coupling caps  it also works great at radio frequency. For example, 0.1 µF capacitors @ HF.

Above — A ragtag squarer breadboard that's seen much use over the past few years.

Above — Square wave output of Figure 1b.

Amplifier 2

I converted Amplifier 1B into a full complimentary version:

Above — Figure 2 schematic. A Darlington — complimentary circuit lies on both halves of the push-pull scheme, Function is similar to Figure 1B, but the symmetry adds the advantage of a remarkably low quiescent current for proper AC/DC operation. I added emitter degeneration (series feedback) to the Q1 voltage amplifier. Nearly every resistor was adjusted or swapped to find the sweet spot in order to swing the largest AC voltage 'tween the rails.

Above — Breadboard of Figure 2 connected to an 8 Ω resistive load (2 parallel resistors).

Above — Bias setup: Even at half-power, the lowest distortion possible gives a 2nd harmonic of only 32 dB down. I tried many schemes to lower this distortion, but failed.

 

Above —Maximum clean signal power = a disappointment. Even with lower drive, this amp suffered from harmonic distortion. No point in continuing to work on it. Onto Amplifier 3:

Amplifier 3

In review; Figure 2 featured a complementary NPN—PNP driver + PNP—NPN output pair with a level-shifter that sets the bias differential on the drivers which in turn establish the bias for the finals since they're directly coupled. Sadly, Figure 2 suffers from harmonic distortion.

Amplifier 3 fixes these woes:

Above — Figure 3 schematic. Optimized for low distortion and quiescent current @ 1W; it's driven with a low noise op-amp and ranks as the best single-supply audio PA I've built to date. 

Above — A combined time and frequency plot at maximum clean power: 1 Watt. The 2nd harmonic is 63 dB down!  

Following FFT analysis, I performed the most important test of all — listening through a speaker. 

On my bench top sits an old cassette player with line-level output. The audio tape plays the clear — booming — voice of a loud, male Russian professor. In Russian language, only 1 syllable is ever accented and his taped voice peaks rip like thunder — well testing audio amps plus scaring cats. Wow, классный , superb --- for sure, a version is going in my next receiver.

When you connect an op-amp to an AF power stage expect oscillations. Surprisingly, mine were between ~1 and 2.7 MHz. With the output connected directly to the op-amps inverting input you might create a situation where the total phase shift at the feedback loop exceeds 360 degrees plus exhibits a gain > 1  — oscillations — 

Experimenting with all the parts connected to the op-amp's inverting input is the first place to start when debugging higher frequency oscillations. 

Decoupling and bypassing the DC supply with simple RC networks is warranted if motor boating [low frequency oscillations ] arise in any AF circuit.  On my JavaScript RF Tools page,  Section D. Calculate Cut off Frequency for an RC Pi Low-Pass Filter , you may assess combinations of caps and resistors.

Above — Figure 3 breadboard shown in audio test mode with an RCA female on the output and a "tacked on" 10K volume pot since the voltage gain is pretty high. This Cu board is the same 1 used for Figure 2, so by this time it's looking pretty worn. 

I think some radio enthusiasts feel more impressed by pretty looking circuits rather than well designed and properly tested stages... I thought I got a good exposure though. I have never used Photoshop and prefer to get it done on camera.

Also, we normally don't leave unused op-amp pins floating — I will have stripped parts and trashed this board by the time you read this.

Final Thoughts

I've got some other ideas, designs and advice to assess. Further, some rail-to-rail op-amps will arrive soon. In 2015, I'll see if I can make a linear PA without distortion at signal amplitudes even closer to the rails. Perhaps? Many thanks to my mentors + supporters and

Thanks to you for reading — catch you later!