Resonance with nonlinear impedances

My book uses nonlinear impedance for modelling loudspeakers and electrodes.  The loudspeaker models include an inductor-like component with and the electrode models include a capacitor-like component with . Standard linear components are special cases of the nonlinear impedance: inductors have , resistors have , and capacitors have .

This past week I was thinking about the resonance that I see in some loudspeakers around 4MHz, which can be modeled with a capacitor in parallel with the main nonlinear impedance.  How can I estimate the capacitance to get initial values for model fitting?  In general, when do I get resonance with two nonlinear impedances?

Although I initially looked at the capacitor case, I realized that the general case of two nonlinear impedances in series gives me simple math that I can easily generalize to special cases and to parallel connections, so let’s look at .  The first thing to do is to replace by the polar form , getting .  The we can apply Euler’s formula to get the real and imaginary parts:

.

.

We will get resonance whenever the imaginary part goes to zero:

, or

.

The special case of an inductor and a capacitor sets , , , and ,  yielding , which is the standard result.

We get a resonance whenever and have opposite signs.

We can deal with parallel rather than series impedances by looking at the sum of admittances instead of the sum of impedances.  To get the admittances, the exponents and get negated and the coefficients and inverted, giving us 

.

Note: this post is a much simpler analysis than last year’s in Resonance for non-linear impedance, because here I am just looking for where the phase goes to zero, rather than where the magnitude of impedance is minimized.

Update 2021 May 4: The two definitions of resonance I’ve used (minimum and ) are not the same—I tried doing a parametric plot of the magnitude vs. the phase for one asymmetric example ( and ) and saw that the minimum magnitude did not occur at 0°.  So I’ll need to think some more about what I want “resonance” to mean for nonlinear impedances.  

 

Forty-ninth weight progress report

This post is yet another weight progress report, continuing the previous one, part of a long series since I started in January 2015.

I have continued to put on weight and have been staying in the “overweight” range. This is most definitely not where I want to be, but I’ve been having a hard time motivating myself to do any exercise.

I had a couple of months when I was much heavier than I wanted to be, but holding steady, but now I’m off in a range I never expected to be.

My self-image was formed in my early years as a very skinny kid—always the smallest and skinniest boy in my classes. I weighed only 85 pounds going into high school, and only 115 when I graduated at 16. I got up to 135 pounds by the time I got my bachelor’s degree, and only put on a little weight in grad school. So up until I was 30 or so, I was extremely thin. I could eat large quantities of high-fat food (like premium ice cream) and not put on weight. My concern in my early years was being able to get enough calories to keep my inefficient metabolism fueled.

It has been very difficult to adjust both my view of myself and my behavior to match the new reality—now I have to watch how much I eat and make extreme effort to lose weight. Sitting around all day on my computer with snacks just a couple of steps away has not been a healthy lifestyle this past year.

I got a new bicycle computer 37 days ago, but I’ve only put 14 miles on the odometer: 0.38 miles/day.  I probably need to average 5 miles a day to stop gaining weight, and 10 miles a day to start losing weight (along with better control of my eating). I’ve been finding it very difficult to motivate myself to do anything physical (about the most strenuous thing I do is mow the lawn every 2–3 weeks with an electric mower).

I need to find something to motivate myself to exercise again.

Electric lawnmower repaired again and again

In Electric lawnmower repaired yet again I reported

After I reassembled the motor I tested the mower without replacing the cover—it seemed to work ok.  I replaced the cover, and lawn mower worked just fine.  I was too tired to mow the lawn (very little sleep last night, and the trip back from Berkeley had taken 6 hours, rather than 3 hours, because of BART delays, the BART train we were on going out of service, and missed connections), but I should be able to mow the lawn sometime in the next week.

I did mow the lawn the next week, but about halfway through I started having trouble with the mower.  The motor ran fine, but if I squeezed the handle all the way it turned off again, making the mower rather difficult to control.  I finished mowing the lawn, but decided to investigate the problem.  At first, I thought that there might be a mechanical problem, with the handle releasing the switch when pulled too far, but I saw no evidence of that when I disassembled the handle and looked at the action of the cam on the handle.

Using my multimeter, I determined that the switch operated normally when the button was pressed part way, but not when the button was pressed in all the way.  I suspected that this meant that the switch was damaged (perhaps by corrosion), so I ordered a new switch.  Unfortunately, the part is quite expensive (about $15 on eBay), because it is a custom-made switch for Black and Decker.

I ordered the switch, and ebay said it had been shipped. I thought that the Post Office had reported it delivered, but I never saw it.  I even asked the mailman about it two days after the claimed delivery, but he could not remember.  He thought he had stuffed a thick envelope in the mail slot, but wasn’t sure.  So either the post office did not deliver, or porch pirates had taken the package. [See update below.] Both outcomes seem to be likely here, based on reports on Nextdoor.

I ordered another switch which was shipped on 4/20 and arrived on 4/22.  Today, I installed the switch.

The old switch, showing how the wires were connected. I took this picture before disconnecting any wires, in case I needed a reference for where they went.

The new switch, showing the part number. I transferred one wire at a time from the old switch to the new, so I didn’t need the reference photo that I took.

The new switch in place in the handle.

I disassembled the old switch to see if the contact damage was visible. The problem was not corrosion but burned material on the contacts—probably from the short circuits from the earlier rectifier failures.

I’ll try out the mower again this afternoon, to see whether this latest fix does the job. So far I have bought two bridge rectifiers, one set of brushes, and two switches to lengthen the life of this mower. At some point (perhaps at the next failure) it may be worth getting a new mower.

Update (still 2021 April 24): I mowed the lawn with the repaired lawnmower, and it worked fine. While I was mowing, the mailman came, with the package that had the first switch I had ordered. I went back to the USPS site to find out what had happened.  It turned out that the ebay package that had been delivered was one for my wife, not the lawnmower switch.  The switch had not arrived at the post office for ten days after the company (Parts Sales) had said they had shipped it.  So the missing switch was neither the post office’s fault nor was it porch pirates—it was due to lies by the company claiming they had shipped when they hadn’t and confusion on my part about what package had been reported as delivered.

Electric lawnmower repaired yet again

In Vaccine vested! I reported

I also took apart the lawn mower to see it if is fixable.  As always, it took me a long time to clear out the grass packed into the recessed screw holes to get the cover off, and a long time to vacuum all the grass out of the interior of the mower.  When I did finally get access to the motor and electronics, I determined that the bridge rectifier had failed again—this time with a short circuit instead of an open circuit.  I’ll buy another GBPC5010-G‎ 50-Amp bridge rectifier, and see if this one lasts a little longer.

There is one mistake in the quoted section—the first failure of the lawnmower (back in 2017) was also a short circuit.  The new behavior was identical to the old.

When I got back from Berkeley today (where my wife and I were visiting our son), I found the rectifier and a couple of other packages that had been delivered on Friday sitting on our porch.  I took apart the lawnmower and replaced the bridge rectifier.  I verified that the old rectifier really had shorted out two of its diodes, so I was pretty confident that I had fixed the problem and I put the mower back together.

It still didn’t work.

There was no longer a short circuit that blew fuses, but the mower just ran for a few seconds and died, as if the blade were blocked.  With the mower unplugged, I could turn the blade by hand, but took the opportunity to chip some of the dried-on grass from the bottom of the mower anyway.

After removing the cover again, I inspected the motor more closely, and I decided to take off the top plate of the motor (which holds the brushes), to see if there was anything wrong with the rotor or commutator.  The commutator looked very dirty, and it looked like the carbon from the brushes (or perhaps some carbonized grass) had gotten stuck between adjacent plates of the commutator, so I scrubbed the commutator with an old toothbrush.

After I reassembled the motor I tested the mower without replacing the cover—it seemed to work ok.  I replaced the cover, and lawn mower worked just fine.  I was too tired to mow the lawn (very little sleep last night, and the trip back from Berkeley had taken 6 hours, rather than 3 hours, because of BART delays, the BART train we were on going out of service, and missed connections), but I should be able to mow the lawn sometime in the next week.

Incidentally there was another error in Vaccine vested!: the 500 is a “Rapid bus”, not light rail.  It does provide a pretty quick connection between Diridon station and the new end of the BART line at Berryessa. It was also free, because VTA is not charging for transit until April 1.  I could have ridden the Highway 17 Express for free also (because I’m old), but I just took their half-price offer for regular passengers.  I did use my new senior Clipper Card on BART for the first time, though only on one leg of the trip, as I had some money on my old regular Clipper Card to use up.  The whole round-trip to Berkeley cost only about $17, which is about what it would cost me with all the senior discounts.  I don’t plan to take senior discounts on the local buses (SCMTD really needs the money), but I will take them on BART and VTA, which have a much bigger and wealthier tax base.

Santa Cruz County hits 200 Covid deaths

A while back, I posted about the Santa Cruz County Health Department’s COVID-19 dashboard.  Yesterday, 2021 March 23, the county hit 200 COVID-19 deaths, so I thought I would update the post.  We have been doing a little better in the nursing homes—now only 51% of the deaths are there (102/200) instead of 66% (79/120). The four big outbreaks are the same, with a couple more deaths at Watsonville Post Acute, and on more at Hearts and Hands, but Sunshine Villa is close behind with 7 deaths and many of the other nursing homes now have 1–4 deaths.  Statewide, the fraction of deaths from nursing homes is about 23% (down from 33%), so we are still way overloaded with nursing-home deaths.  Or, more cheerfully, we’ve been better about keeping down deaths outside the nursing homes.

As before, most of our positive tests are in young adults, but most of the deaths are in older adults—the case fatality rate starts going up somewhere in the 60s and really soars in the 80s and 90s:

age range	cases	deaths	case fatality rate
0–9	1085	0	0%
10–19	1865	0	0%
20–29	3122	0	0%
30–39	2649	4	0.15%
40–49	2232	6	0.27%
50–59	1878	5	0.27%
60–69	1269	27	2.13%
70–79	611	42	6.87%
80–89	339	61	17.99%
90+	195	55	28.21%

The case fatality rate is very slightly higher in each age category, which may be due to the California strain that circulated in January and February being a bit more deadly.

I’ve been looking at the lagged case fatality rate (dividing the number of deaths by the number of cases 2, 3, or 4 weeks earlier) on http://91-divoc.com/pages/covid-visualization/.  The 4-week lag seems to give the most constant case rate, but even it shows a pretty big increase in California in the last month—the cases now seem to be more deadly.

Although the case rate still indicates that more Latinx people are getting infected than their share of the population (55.98% of the cases are Latinx vs. 33.49% of the population),  the death rates match the population statistics (35.5% of the deaths are of Latinx people).  Most of the old people in the county are white, and age is a much stronger predictor of who will die than race is.

I can’t check the number of hospital beds, because the state moved the page and the county has not updated the link.  The forwarding link provided on the state page goes to Alameda County, not Santa Cruz County, and I don’t see how to fix that.  From the information that is available, we seem to have hit peak usage around January 16, with about 85 hospitalized, and we are now down to around 5 hospitalized—so we’re pretty much out of this spike.  Let’s hope that a large enough fraction of the population is vaccinated before we get hit with another spike.