EMRFD Message Archive 4058

Message Date From Subject
4058 2010-01-20 16:51:51 Glen Leinweber Leeson equation: Q
Am struggling with the Leeson equation that
models oscillator noise. Particularly what is
meant by "Q". I take it that this is noise-equivalent
bandwidth Q, not unloaded resonator Q.
My brain hurts when I think of oscillators...
Many oscillators unload the resonator for
much of the oscillatory cycle, then drive a short
burst of "regenerative" current that exactly
tops up circulating energy. So for much of the
cycle, Q is mostly "unloaded Q". Then during
the burst, Q is quite different - i'd guess much
lower. I realize that this is a minefield of non-
linearity beyond my ken.

Some regen users claim that a carefully tuned
receiver can have a nearly infinite Q. Have even
managed to build a crystal oscillator that took
tens of seconds to build amplitude. If these
examples had stable oscillator amplitude,
would their Leeson Q really be very high?
What limits noise-equivalent bandwidth from
approaching zero in a regenerative stage? Could
a truly linear regenerator approach zero NEqBW?
Can a non-linear regenerator do the same?
If so, what would their "Leeson Q" be?
4060 2010-01-20 19:26:41 w7zoi Re: Leeson equation: Q
Hi Glen and gang,

Hey, what's the expression that the "jocks" use? "No pain, no gain." So if the brain hurts, you are gaining. Clearly, you are looking at the oscillators in depth, and it is indeed fun, albeit slightly painful at times.

The Q in the Leeson equation is the loaded value of the resonator. Like any filtering situation, the loaded Q can be no better than the unloaded value. An interesting thing to ponder is the implication of Equation 3.7 in EMRFD. This says that if you build a filter with a single resonator, you will have an insertion loss through that filter the depends upon the ratio of loaded to the unloaded Q. As the loaded value approaches Qu, the insertion loss becomes infinite. So what happens if you do this in an oscillator? What happens is that you may have a good Q, but the active device must be operating at high gain to compensate for the loss. The net power gain is unity; it can be nothing else, for it it was more, the power would just keep growing. That conservation of energy thing again. You can model this as a high gain amplifier cascaded with a large attenuator. The noise figure of this topology is much higher than the NF of the amplifier alone. So if you have a resonator that operates close to Qu, the NF will be high. Note the presence of NF in the Leeson equation.

At the other extreme is an oscillator with a heavily loaded resonator. The noise figure can then be low, approaching the raw value of the active device. But the wider bandwidth means that there is still a lot of noise in the loop.

The hot spot is one where the loaded Q is close to half unloaded value.

We are indeed dealing with nonlinear behavior. However, Q is defined as something that is averaged over a full cycle. (See, for example, Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, p8, Wiley, 1965.) If you take a look at something that happens in just part of a cycle of the resonator natural oscillation, a wider bandwidth is implied.

Like all of us, I've built the Q multipliers, as well as the crystal oscillator that takes several minutes to build up into a stable oscillation. This is completely in line with a Q definition. The active device in the Q multiplier has gain that compensates for the energy loss per cycle. A really dramatic illustration of this is an AM regenerative receiver for the classic broadcast band where the bandwidth becomes so narrow that one sideband or another can be tuned, but is not detected for the lack of a carrier. I've also heard the claims of a regen receiver operating in a ham band and in the CW mode where quite narrow bandwidth is claimed, even in the absence of any other filtering. This seems to be inconsistent with a simple model of a regen detector as an oscillator where the device non linearity also serves to heterodyne the carrier down to baseband. I had a letter a while back from a fellow who was claiming a 1 kHz bandwidth at 3.5 MHz, which means a loaded Q of 3500 while listening to CW.

The key to this may be in the way the circuit is adjusted when this selectivity is observed. If you grab the regen control and advance it a lot to produce a robust oscillation, a narrow bandwidth is not seen. The high Q seems to happen in CW just after oscillation has started. The "softest" limiting is enough to keep the oscillation amplitude stable, but the gain can still be high in order to realize Q multiplication.

I've slipped into a sloppy mode here, sliding away from careful analysis into hand waving speculation. Some careful analysis would certainly be in order. Anyway, sorry for the hand waving.

I'm not sure what the term "noise equivalent bandwidth" means in this context. The term, or "noise bandwidth" is used in some situations to describe a filter. The filter will have a bandwidth, which is usually defined by the 3 dB or half power value. But if the shape is sloppy, the filter will let a lot of noise to pour through. One can look at the filter transfer function and calculate the total power that would emerge from the far end when a noise (flat) is applied. The noise bandwidth is then the bandwidth of a square "brickwall" filter that will allow the same noise power to emerge. See IRFD, equation 6.1-11
4063 2010-01-22 06:38:19 joop_l Re: Leeson equation: Q
I also wondered about loaded Q in oscillators. There are articles that examine loaded Q and may mention e.g. that an emitter coupled oscillator the crystal sees a low impedance load and therefore does not have a lot of Q reduction.
However, I played a bit with SPICE and it usually is quite predictable to see when conditions are met for oscillation. Namely when the negative impedance is more in magnitude than the resonator impedance (skipping the phase aspect for now). It does not matter if this is a "low impedance" emitter circuit, or a typical Colpitts or otherwise.

What I am looking for is how much the additional gain (or more negative resistance if you like) affects the oscillator performance. Like you mention, Wes, in the sustained state the gain will be limited in some manner anyway. And where loop gain reduces to one. Any present noise will have the circuit meander around this operating point. I can imagine (hand waving mode?) that it may be important to know how a circuit behaves around or slightly off this operating point.
1) Is the gain changing rapidly of more softly? (compare a super high negative resistance Franklin oscillator versus a variable gain limiting dual gate Hartley)
2) Is there a certain phase shift or DC operating bias shift involved in limiting?
3) Is the noise energy level small or big compared to the resonator energy level?
The above factors I can see will be different for the various oscillator topologies. And perhaps more of influence than "loaded Q" in the startup/linear phase of the oscillator. My gut feeling tells me that loaded-Q in a *clipping* oscillator is hard to define as the oscillator provides a negative resistance of the same magnitude as the resonator R. This confuses my perception of the "noise equivalent bandwidth" as well. I try to picture an oscillator with zero Ohms at the operating frequency, but increasing resistance moving away from this frequency, affected in different ways by both resonator and oscillator/limiter. Arrggg, Glenn what have you done to us... ;-)

Now I must admit I need to read up first on the Leeson stuff but any "easily digestible" articles are always welcome.

Joop - pe1cqp
4066 2010-01-22 10:53:19 w7zoi Re: Leeson equation: Q
Hi Joop, and gang,

Yea, you hit it on the head--it's all Glenn's fault for bringing this up.

Seriously, it is wonderful to see folks looking at things with regard to the fundamentals, so we are applauding you Glenn, even when we are pulling your leg a little!

One interesting place to look at some of this oscillator stuff is in the Clarke and Hess book, "Communication Circuits: Analysis and Design," Addison-Wesley Publishing, 1971. They do a wonderful job of looking at the change in bias in a bipolar oscillator as a function of operating level. This is really easy to measure too, although you want to put a large R in series with the DVM so you don't foul up the oscillator when you probe it to measure DC levels. The bottom line is easy enough -- when oscillating, the RF will supply some of the energy needed to bias the BE.

Incidentally, I think that looking at the bias conditions are a way of seeing if an amplifier with a microwave transistor is oscillating up high, way beyond the frequency of our static electricity frequency spectrum analyzers. I did this just yesterday when working with a 5 GHz F-t NEC part.

Some of the oscillator circuits that use a crystal with "low impedance terminations" are not what they are intended to be. For example, folks will sometimes extract energy from the crystal through a common base amplifier, "grounding" the crystal through the low impedance emitter. However, the common base stage must be bias to high enough current that it remains conducting for the entire operating cycle. If it opens up for part of the cycle, the impedance as averaged over a cycle, grows. Glenn hit
4067 2010-01-22 17:53:40 Glen Re: Leeson equation: Q
Have learned a great deal from the oscillator chapters in
"Introduction to Radio Frequency Design". Wes, when given a chance
to plug your own book, you should jump at it ;-) Joop, you might
gain some insights about how the active transistor changes with
bias, and how it chases a stable operating point.
A grounded-base Colpitts oscillator is tightly characterized
while adroitly avoiding the non-linear minefield. Amazing.

Have also discovered VHF or UHF oscillations in circuits meant for
much lower frequency, with a DVM. For example, the simple capacitor-
multiplier circuit used to smooth a DC supply having hum or noise
can easily oscillate. Its simplest form includes one resistor from
collector to base, and one capacitor from base to ground. Output at
emitter. A DVM probe at the base shows a slightly higher initial
voltage, that falls a few tens of millivolts after a second. In this
case, probing chokes off the oscillation. Or voltage falls when a
VHF bypass capacitor is added to the emitter of a more exuberant
oscillator.

Audio amps can oscillate at low-level high-frequencies. A 'scope can
often see these, but I'd bet VHF or UHF frequencies lurk there too.

Requiring loaded Q for Leeson's equation is discomforting. As Joop
points out, it is so very difficult to characterize impedance of the
active device. I gather from Lees
4068 2010-01-22 19:39:05 w7zoi Re: Leeson equation: Q
Hi Glen, and gang,

I did a bunch of experiments with the common base bipolar oscillator that is presented in IRFD. I wanted to see how well I could predict starting. This was back before most of us had out hands on simulation tools. The assumptions were extreme, but they worked amazingly well. I found that I could build 10 MHz oscillators and vary the standing current and the Colpitts capacitors and degeneration in the feedback. In all cases, at the low frequencies I could predict starting within one dB. That is, setting up a circuit that would have a starting gain of just 1 dB according to that analysis would always produce an oscillator that started.

All of that work was later extended with simulators. There were no surprises.

But none of this has a lot to do with oscillator noise. I had great plans to study the subject in greater detail. I ran into a good deal on a single run of coax cable and bought it. But alas, it's still in the garage. I have been tempted to use it with some antennas, but even that has been set aside. I was going to build a delay line discriminator that would work at HF; it was a long piece of coax.

I did build a UHF version of this when I was still working. At that time I was trying to build some fully integrated oscillators, including the resonator. The big thing that we fought with that project was the high 1/f noise in GaAs.

Some simple experiments were done by using just a receiver for the noise measurements. I never came up with any one topology that was better than others. But it was easy to observe that increasing the transistor bias current to increase the power, and hence the energy stored in the resonator did indeed produce a quieter oscillator. I never did find anything significant with regard to device noise figure. The ubiquitous 2N3904 is a moderately large area switching part and has a reasonable NF. I suspect that the reason JFETs are so good in oscillators is that the 1/f noise is low. I was able to resistively load coils to kill the Q, readjust the current or degeneration, to again obtain starting, and observe degraded noise, especially far from the carrier. But all of these experiments are difficult, for the measurements are difficult.

There are many sources of noise and they can all get you in trouble. Bias resistors can be sources. Varactor diodes can be especially bad. Resistors that bias varactor diodes can be especially bad -- watch out for those circuits that we have all seen where the varactor is biased with a 1 Meg.

If you want to do an interesting experiment (still on my to do list) build a VCO and then build a high gain audio amplifier driven with a resistor such that there is a high noise output. Then apply the noise source to the VCO and observe the noise. That's got to be interesting.

OK on the C-multiplier experience. I have used this circuit for years, but encountered an oscillation just recently. (Perhaps there were others that I missed?) This was the source of a spur in a receiver that a friend had built. I fixed it with the addition of a 100 Ohm R in the collector, but it took a borrowed microwave spectrum analyzer to find the problem.

Anyway, I'm sure that there is more than enough to do in all of these areas to keep us all busy. So many mountains and so little time, hi.

73, Wes
w7zoi


>
4069 2010-01-22 21:05:58 w4zcb Re: Leeson equation: Q
Hi Glen, and gang,

Some snipped

But none of this has a lot to do with oscillator noise. I had great plans to study the subject in greater detail. I ran into a good deal on a single run of coax cable and bought it. But alas, it's still in the garage. I have been tempted to use it with some antennas, but even that has been set aside. I was going to build a delay line discriminator that would work at HF; it was a long piece of coax.

I wonder if you still have that paper I sent to you back in the late 70's Wes, about the coax delay line self test circuit for oscillator noise done by the AFCEA frequency group? If you'd talk sweetly to your cohort Bob Larkin, betcha he would make you a variable delay line out of DSP! small, compact, light weight and low fuel consumption. (It's only software, or so I'm told!)

I did build a UHF version of this when I was still working. At that time I was trying to build some fully integrated oscillators, including the resonator. The big thing that we fought with that project was the high 1/f noise in GaAs.

Some simple experiments were done by using just a receiver for the noise measurements. I never came up with any one topology that was better than others. But it was easy to observe that increasing the transistor bias current to increase the power, and hence the energy stored in the resonator did indeed produce a quieter oscillator. I never did find anything significant with regard to device noise figure. The ubiquitous 2N3904 is a moderately large area switching part and has a reasonable NF. I suspect that the reason JFETs are so good in oscillators is that the 1/f noise is low. I was able to resistively load coils to kill the Q, readjust the current or degeneration, to again obtain starting, and observe degraded noise, especially far from the carrier. But all of these experiments are difficult, for the measurements are difficult.

There are many sources of noise and they can all get you in trouble. Bias resistors can be sources. Varactor diodes can be especially bad. Resistors that bias varactor diodes can be especially bad -- watch out for those circuits that we have all seen where the varactor is biased with a 1 Meg.

Anyway, I'm sure that there is more than enough to do in all of these areas to keep us all busy. So many mountains and so little time, hi.

73, Wes
w7zoi

Speaking of oscillators, Leesons equations, phase noise and such matters, you all might be interested in looking at Colin Horrabins (G3SBI and the designer of the "H"-mode mixer) two tank oscillator he designed for use in the CDG 2000 homebrew transceiver back in 2002. It is also used in John Thorpes AOR 7030 and the new (not yet available) AOR receiver. Using two tanks with a grounded base J-310, the LO noise falls off at twice the rate of a normal one tank oscillator.

I don't find it at the moment, and am searching for my copy, but a Hungarian chap published the mathematical proof for the characteristic in I believe one of Gary Breeds publications several years back. He maintained that the circuit is not defined by Leeson due to the topology.

W4ZCB

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4070 2010-01-22 21:13:00 w4zcb Re: Leeson equation: Q
Oh my, sorry for the formatting. I had Wes's remarks in one font and mine in another not realizing that Yahoo was going to make them all the same. You'll have to refer to Wes's original to see where I interrupted him and stuck in my own comments.

I'll get the hang of this computer stuff someday.

W4ZCB
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