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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Fri 24, 2020 6:00 am 
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Awwwwlrighteee,

The previous posts gave me some inspiration and better search terms and I found good reading on Class E drivers and eventually what bb.odin explained began to sink in, well some of it anyway. Using a single capacitor for DC block as well as to resonate the coil makes a lot of sense. My calculations are not predicting my results but there has been progress, or at least better understanding. I altered my circuit to remove everything after the mosfet and replaced it with Binh's first diagram, keeping my IRF620 mosfet into a 50 ohm load. The IRF620 has similar on resistance and output C values to Binh's simulation so I think there is no need to change for this testing. I figured I'd start with Bihn's other values, wound a 105uH coil (since I didn't have that value) and verified that it resonated with a 270pF cap at 950kHz, as the LC formula says it should. Wired up the circuit (ignore the printed values, I first wire up with L2 105uH and C2 270pF). With those values the waveform at the mosfet drain looked very, very close to Binh's "E_WF6" jpg, but audio 2nd harmonic distortion was pretty high particularly lower frequencies. Tweaking C2 values found 210pf worked pretty well for audio distortion, and L2 was 105uH.
Attachment:
test into 50ohm load.png
test into 50ohm load.png [ 92.92 KiB | Viewed 294 times ]

For fun I changed 105uH to a manufactured 82uH, and then 270pF for C2 works nicely. A 10pf change of C2, one way or the other, pushes 2nd harmonic distortion up by several percentage points. Here's a distortion sweep with L2=82uH. It isn't noted on the graph's scale but 2nd and 3rd distortions remain below 1% until the sharp upturn at higher audio frequencies. It is worth mentioning that the optimum LC tuning point (lowest distortion) is extremely narrow.
Attachment:
File comment: top line is audio fundamental and shows demodulated frequency response. 10kHz is down about 3.5db, which is about what I was seeing with the original circuit.
Distortion 82uH into 50 ohms.jpg
Distortion 82uH into 50 ohms.jpg [ 99.04 KiB | Viewed 294 times ]

The waveform at the mosfet is a mess though, not as pretty as Binh's simulations. To me, this suggests that the switch is closing much earlier than what is considered "ideal" although it results in lower audio distortion. It also results in lower overall output level to the 50 ohm load.
Attachment:
Vds waveform into 50 ohm load.jpg
Vds waveform into 50 ohm load.jpg [ 91.69 KiB | Viewed 294 times ]

Again, these are into a 50 ohm load. An antenna is not going to be as easy to tune even with the head start provided by Binh. But here's where I need to evaluate whether to continue down this path. Distortion isn't much better, just a wee bit flatter, than what I could tweak my original circuit to give, and both of them need to be tweaked away from peak amplitude to obtain best distortion. Both circuits have an extremely narrow tuning point where distortion of audio below 1kHz remains below 1%. Compare the above distortion graph to one I snapped last week with the original circuit. Distortions are below 1% until about 1kHz. Tuning is also very narrow, as I noted in my original post.
Attachment:
File comment: scale matches what is used above.
Distortion original with 395u airCoil tuned low dist.jpg
Distortion original with 395u airCoil tuned low dist.jpg [ 132.34 KiB | Viewed 294 times ]

More work is needed.

edit: cleaned up some grammar.

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Last edited by richfair on Jan Fri 24, 2020 4:06 pm, edited 2 times in total.

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Fri 24, 2020 3:56 pm 
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I would like to raise a couple more points. First, it should be noted that Binh (bb.odin) shows distortion plots for carrier distortion. I am focused on demodulated audio distortion. So far, though, it appears that reducing carrier harmonic distortion also decreases audio harmonic distortion.

In the texts I've read, the LC timing is set so the switch closes at or near the point of 0 volts and 0 current, which seems to require fixing the LC components to naturally resonate a little bit faster than the desired operating frequency (due to current phase being shifted later in time). Yet, bb.odin's two simulations use LC values that resonate slower, below the 1 MHz operating frequency (~960kHz). After I adjust values in my test to give lowest AUDIO distortion of the demodulated rf, the LC values resonate a bit above 1mHz. I obviously need to work the Sokal equations for myself, as painful for me as that will be, rather than just poke in values on a breadboard. That will be my next challenge. There is an article by George Slade that is filled with equations and good explanations.

Finally, class E modulation is obviously sensitive to circuit and antenna changes. Tuning is very delicate since any change to the antenna gets reflected backwards and throws off that perfectly tuned switch. Just a few pF here or there makes for big changes of audible distortion. Tuning by measurement should hopefully only need be done during design. That suggests tuning for a fixed frequency if one wants the best results. Tuning by ear is possible during user setup but only if you know what to listen for, again not very practical for most. Even if I give up the possibility of having variable frequencies my desire would be to have something a bit more tolerant of imperfect tuning of the antenna. A buffer is needed after the switch to do that, which complicates this project further. When is enough enough?

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Sat 25, 2020 3:43 am 
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Many papers I've read on the subject indicate that the series LC resonates lower than the switching frequency. For analysis, it's usually modeled as an ideal LC tuned to the switching frequency in series with a reactance jX. A formula for jX is given in the textbook by Albulet as X = pi*(pi^2 - 4)*R/16s = 1.1525*R. Since it's positive, it's an inductance. That makes the resonance frequency lower.

The class E amplifier is a current-fed circuit. The choke transforms the supply voltage into a current source. When there is a change in the supply voltage (such as in the case of high side envelope modulation), it takes a number of switching cycles for the average DC current through the choke to reach its steady-state value. This settling time depends on the choke value and the load network. With a 220uH choke and a load network tuned for 50-Ohm load, my simulation shows a 15 usec settling time which is good for modulation frequency up to 66.667 kHz.

The supply choke L1 and the shunt capacitor C1 across the switch form a lowpass filter for the envelope. At the operating frequency this filter is loaded by the load resistor. At the audio frequencies, it is loaded by a high impedance which makes it underdamped. This is shown below in the envelope step response of the supply voltage. That is likely the cause of high audio distortion that Richard has been seeing.
Attachment:
Class_E_Amp_Underdamped_Env_Step_Resp.jpg
Class_E_Amp_Underdamped_Env_Step_Resp.jpg [ 171.65 KiB | Viewed 266 times ]

Adding a resistor R3 in parallel with the shunt capacitor will solve this problem. The lowpass filter is critically damped or overdamped when R3 <= 0.5*sqrt(L1/C1). For L1 = 220uH and C1 = 720pF, a 270 ohm resistor will make the step response slightly overdamped. Note that the leading and trailing edges rise monotonically to their final value. I hope this will reduce the audio distortion. (Click on the figure to enlarge).
Attachment:
Class_E_Amp_w_Step_Resp_Damping.jpg
Class_E_Amp_w_Step_Resp_Damping.jpg [ 322.93 KiB | Viewed 266 times ]

Attachment:
Class_E_Amp_Damped_Env_Step_Resp.jpg
Class_E_Amp_Damped_Env_Step_Resp.jpg [ 166.49 KiB | Viewed 264 times ]

The series inductor L2 is chosen for a desired loaded Q factor. For Q = 13 and R = 50 Ohm, L2 is about 100uH. The value of the choke L1 is not important. It's chosen for a desired ripple current while not making the envelope settling time too long. The value of the shunt capacitor depends on the load resistor. A formula given by Albulet is C1 = 0.1836/(w*R), where w = 2*pi*f and f is the switching frequency. To facilitate tuning, the only component that needs to be variable is C2. C2 could be a high-quality fixed capacitor that resonates with L2 in parallel with a small trimmer Ctrim. The value for Ctrim is 1/(w*(w*L2 - X)) - C2, where X = 1.1525*R.

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Sun 26, 2020 12:38 am 
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Binh, I sincerely appreciate the time you are taking to help me and others who may be lurking. Before installing a resistor across the shunt capacitor I thought it would be a good learning tool for me to work through calculations to try to arrive at numbers similar to yours. I have encountered difficulty.

I will get a chance to breadboard tomorrow and will first try your suggestion of a resistor parallel to the switch, as well as consider all that you posted above.

I have only just today realized I need to clarify a most important point that my mosfet is operating with 2.4 volts on the supply side of L1, NOT 9v. The transmitter's supply is 9v but the mosfet's dc supply is taken from the opamp+npn audio driver. The driver stage was adjusted to provide 2.4vdc during audio silence, which results in an average 42ma current draw from the npn's emitter and thus mosfet input power of 100mw to stay within part-15 guidelines.

This afternoon I spent time with a document by George Slade and then used it as a source of calculations: https://www.researchgate.net/publication/320623200_Notes_on_designing_Class-E_RF_power_amplifiers
It is quite difficult for me to follow some equations presented during his initial analysis but once into his practical example, the math is easier to follow. I adapted the author's equations 30 through 38 to an Excel spreadsheet. My spreadsheet returns the same results he cites during his practical example, which I hope proves the equations were entered to Excel as intended. BTW, he shows a 1.153 adjustment to the LC resonance but used in such a way as to return an "excess" inductance value that is later added to the L, to make L a bit higher and thus the LC resonance a bit lower, just as you said

I realized the supply voltage discrepancy when I plugged in my values to the spreadsheet. Once defining Vcc as 2.4 volts, the results are as follows. One of the design values to be inputted is Q. I'm not sure how to choose this value up front. I used Q=10, which may explain why the L value is so low. Slade's equations calculate a C value in parallel to the output load resistance. It is to help match the switch-LC network to the output resistance, I think. The switch "sees" a 32.2 ohm load according to my spreadsheet, which must be matched to 50 ohms for output. I will follow up tomorrow with physical results after review of your recent post.
Rload = 50ohms
Cshunt = 780pF (mosfet's spec sheet lists output as 100pF, so Cshunt's value is really 880pf)
Cres = 479pF
Ltotl = 63uH
Cmtch = 2.2nF


Attachments:
ClassE mod diagram for excel sml.png
ClassE mod diagram for excel sml.png [ 13.83 KiB | Viewed 216 times ]

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Sun 26, 2020 7:12 am 
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Richard,

I've been studying this amplifier class for quite a while and I'm still not able to understand all of its aspects. The circuit looks deceptively simple but the math used in the analysis is extremely complex and tedious. It was the subject of a couple of PhD theses.

I used 9V for the transmitter supply in a couple of simulations to check whether the transmitter is still somewhat tuned when it is 100% or over modulated. The actual supply voltage is around 3V to give about 100 mW output across a 50-Ohm load.

The specifications I use in the design are:
- Operating frequency
- Output power
- Load resistance
- Switch transistor saturation voltage

The Q factor is simply chosen to give a reasonable harmonic rejection. It the combination of load resistance and Q factor results in too small a value for the series capacitance, the desired Q is kept and the load resistance is reduced.

The equations used in the design are adapted from various papers but mostly from the PhD thesis by David Paul Kimber, "Class E Amplifiers and their Modulation Behaviour". The order of calculations is summarized in the figures below.
Attachment:
Class_E_Amp_Design_Equations_1.png
Class_E_Amp_Design_Equations_1.png [ 64.04 KiB | Viewed 203 times ]

Attachment:
Class_E_Amp_Design_Equations_2.png
Class_E_Amp_Design_Equations_2.png [ 75.36 KiB | Viewed 200 times ]

Note on the damping resistor in parallel with the shunt capacitor: Using this resistor will significantly increase the input power. We can afford to use this since the output power is small and efficiency is not a great concern. It cannot be used in high-power transmitters. The remedies for modulation distortion take the form of forward pre-distortion or adaptive distortion cancellation.

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Sun 26, 2020 3:15 pm 
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Revised calculation steps for computer programming (click on attachments to enlarge):
Attachment:
Class_E_Amp_Design_Equations_v2a.png
Class_E_Amp_Design_Equations_v2a.png [ 298.33 KiB | Viewed 185 times ]

Attachment:
Class_E_Amp_Design_Equations_v2b.png
Class_E_Amp_Design_Equations_v2b.png [ 318.04 KiB | Viewed 185 times ]

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Mon 27, 2020 5:13 am 
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Gettiing this thing to modulate well into a 50 ohm load is hard enough, but then matching into a short wire antenna brings more problems, more parts, and more losses. I have decided to return to my earlier circuit where only 2 parts are required to resonate and match to the antenna. I have practically equal audio distortion and amplitude performance either way but can get more radiated power using the earlier circuit approach. Also, it is very easily tuned to match a range of short antennas and, with change of inductor, can cover the entire mw band. At my intended 1mHz I can easily tune a short whip or a 3 meter wire with a couple degrees twist of the variable cap, and get equally low audio distortion at the receiver. The longer wire radiates more power of course but the sound quality is quite good either way. When tuned carefully at 98% modulation, 3rd harmonic distortion stays below 1% while 2nd harmonic is at or below 1% up to 1kHz, and then curves up to 9-10% at 10kHz. Since most audio power is below 1kHz, I can live with it for now.

Regarding a 270 ohm resistor paralleled with the shunt cap, there was only a fraction of a db drop of audio distortion above 7kHz. No change at all below 7kHz. 100 ohms had a slighty stronger effect and also reduced carrier output a little more. There was still an obvious increase of 2nd harmonic distortion above 2kHz so the benefit may not outweigh the downside, but I'll play around with it more.

There is a bit more work to do on this project after which I will update last year's thread with the changes.

Binh, I cannot thank you enough for your time and graciousness. You have taught me a lot.

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Tue 28, 2020 3:44 am 
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Richard, I'm a learner too.

Since the output series LC is tuned below the carrier frequency, the sidebands have different amplitudes and phases. The USB has lower amplitude and longer delay than the LSB. The imbalance increases for higher modulation frequencies and so the distortion, exactly what you observed. As Bob pointed out early, this is not a problem for receivers with synchronous detector. If you receive the USB only, there will be less distortion than the simple envelope detector even though you have to boost the high audio frequencies. For the LSB, you have to do the reverse by boosting the low audio frequencies.

The imbalance is larger for lower Q factors because of the larger difference between the tuned and carrier frequencies. A paper I read suggests a minimum Q of 15 and an upper modulation frequency at least 50 times smaller than the carrier (e.g. 20 kHz at 1 MHz carrier) for 1% distortion. It also suggests a minimum choke of 3.5 times the load resistance divided by the carrier frequency (3.5 R/fc) so that it doesn't make distortion worse. A higher Q makes sense because the excess reactance X for the series inductor L2 is proportional to the load and a power series of Q inverse. By making Q larger, the ratio L2/X gets larger and thus the lower frequency separation.

Another source of distortion is the nonlinear drain-source capacitance. This could be mitigated by making the shunt capacitor relatively larger that goes with a lower load resistance. The drain-gate capacitance is also nonlinear but you already partially compensate for it by the inverse carrier injection via a matching capacitor.

Due to its high efficiency, there have been big interests in using class E PA for mobile digital communications. It's particularly suitable for polar modulation. Instead of a quadrature carrier oscillator, two mixers and one adder for QAM, polar modulation only requires a class E PA drain modulated by the magnitude of the baseband signal. The baseband signal phase is used to modulate the carrier phase which can be easily done with a DDS chip. If interested, do a google search on "envelope elimination and restoration".

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 Post subject: Re: AM trans, audio distortion related to antenna matching?
PostPosted: Jan Wed 29, 2020 8:11 pm 
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Yeah, so I typed in those search terms and ended up in a paper titled "High Efficiency Power Amp Based on Envelope Elimination and Restoration Technique" available on researchgate. All I can say is, "eyes wide open". Eyes wide open like what happens just before one is hit by a bus! There is so much, so much to these subjects, who was Ignorant I to think this simple am transmitter would be a suitable project?

Ah well, we shall press ahead anyway despite its faults and our own.

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