Guest post by Nick Stokes,
People outside climate science seem drawn to feedback analogies for climate behaviour. Climate scientists sometimes make use of them too, although they are not part of GCMs. But it gets tangled. In fact, all that the feedback talk is usually doing is describing the behaviour of variables that satisfy a few linear equations. Feedback talk adds a way of thinking about this, but does not change the mathematics of linear equations.
A couple of articles I’ll refer to are a survey article by Roe, and a frequently cited 2006 article by Soden and Held.
The basic calculus behind feedback and linear signal analysis goes like this. You have a device or system with a number of state variables, which I’ll bundle into a vector x. And the physics requires that they satisfy a set of equations that I’ll write just as
f(x)=0
There is a particular set of values x0 which satisfy those equations that for an amplifier, say, would be called the operating point. Generally it is a state existing prior to perturbation by an amount dx (a vector of state changes). After perturbation it still has to satisfy the equations, so
f(x0)=0 and f(x0+dx)=0
For linear amplifiers, the perturbed state can be well approximated by the derivative expression
f(x0+dx) = f(x0)+f'(x0) dx = 0
and since f(x0) = 0, that leaves the set of linear equations in the perturbation
f'(x0) dx = 0
We don’t have to worry too much about the form of f'(x0), or indeed f(x0). The point is that it is linear, so all terms are proportional to perturbation. We can just take it that f'(x0) is a matrix operating on the vector of perturbations dx. Roe (p 99) has a section headed “Feedbacks Are Just Taylor Series in Disguise”. Actually “Taylor Series” overstates it, since only first order terms are used. But it is getting close to the correct treatment as linear equations of perturbations.
Usually we think of one of the components of dx as the input, or forcing, and another as the output. Then the equations can be shaken down to make output proportional to input, or gain. This is just a property of a linear system of n equations in n+1 variables, and the feedback algebra just expresses it. But you don’t have to think of it that way. I’ll give some examples leading up to climate.
One thing that is important is that you keep the sets of variables separate. The components of x0 satisfy a state equation. The perturbation components satisfy equations, but are proportional to the perturbation. You can’t mix them. This is the basic flaw in Lord Monckton’s recent paper.
Example 1 – the abstract feedback system
The Wiki description is as good as any. It’s labelled negative feedback, but applies generally. The diagram is:
with the accompanying text
Note that it starts with two equations in three unknown voltages. Two are overall input and output, and the third, V’, is the voltage at the input to the amplifier (triangle). V’ is eliminated, leading to an equation relating input and output (red star). This is then manipulated to a gain ratio. But all these steps are just standard high-school manipulations; they don’t add anything. A computer (or a student?) could have solved them at any stage.
Example 2 – a junction transistor
Here is a very simplified AC circuit, with bias arrangements and capacitors omitted. The voltages are the perturbations (AC). Simplified transistor properties are assumed – zero input impedance, infinite output, and a current amplification β=100. So the AC voltage at the base (V’) is held to zero. There are 3 unmarked currents, denoted by the suffices of the resistors I0, I1, If. Directions are I0 right, I1 down, If right. 5 variables in all.
So we write down linear relations. There are 3 Ohm’s Law
| Vin=I0*R0 | Vout=-I1*R1 | Vout= -If*Rf |
and one current gain relation:
β*(I0 – If) = I1 + If
Again, anything further done with these equations is just high school manipulation. But it can be shaken down to a voltage gain by eliminating currents, written in gain/feedback style:
V1 = -β (R1/R0) V0 / ( 1 + f) where f=(β+1)R1/Rf
Note that it is an inverting amplifier, and the feedback is negative.
Example 3 – Climate feedbacks
Again, it’s just a matter of writing down linear equations, resulting here from equilibrium flux balance. I’ll follow this 2006 article of Soden and Held. Unfortunately, they don’t actually quite write the flux equations, but I’ll do it for them. They write:
ΔR is the change in flux at TOA, which is the GHG forcing. ΔT is the surface temperature response. The feedback factors are T for temperature,w water vapor, C clouds and α (=a) for albedo. What they are actually doing (multiply by λ) is writing a flux balance
ΔR = λTΔT + λwΔT + λCΔT + λaΔT
Each term on the right represents a flux due to that factor. They do a bit extra, which I won’t go into, to deal with the fact that flux is at TOA and response is at surface. Their T flux is what people often call the Planck feedback; they roll into it other kinds of temperature dependent cooling, but it is mainly radiation (Stefan-Boltzmann etc).
This hopefully demystifies all the stuff about positive, negative feedback and runaway. The first is a big term that determines what is thought of as feedback-free (open-loop) gain. It is the 3.2 W/m^2/K figure that is often quoted, and turns into the 1.05K/doubling which forms the basis for Lord Monckton’s ECS. That comes from this paper. The other terms are mostly negative, so they diminish the coefficient of ΔT and so increase the amount ΔT must respond to stay in balance. That is interpreted as positive feedback.
It actually gives a perhaps less scary picture of thermal runaway. If these negative fluxes increase, there will come a point where the coefficient of ΔT is zero. That doesn’t mean instant flames. It just means there is nothing to counter heat accumulation from the forcing flux. So the temperature will indeed rise without limit (until some nonlinearity intervenes), but only as forced by the few W/m2 of ΔR. Not good, but not perhaps as dramatic as imagined. If the coefficient became negative, then there could be exponential rise, which might get more dramatic.
So did climatology make a startling error in omitting “reference temperature”.
I may have given away the answer, but anyway, it is, no! Soden and Held is a typical exposition. They correctly gather the perturbation terms – that is, the forcing, in terms of GHG heat flux, and the proportional responses. It is wrong to include variables from the original state equation. One reason is that the have been accounted for already in the balance of the state before perturbation. They don’t need to be balanced again. The other is that they aren’t proportional to the perturbation, so the results would make no sense. In the limit of small perturbation, you still have a big reference temperature term that won’t go away. No balance could be achieved.
So are all sets of linear equations to be regarded as amplification/feedback?
Well, nothing really hangs on it except the way you talk about them; the algebra is the same. But what characterises amplification is that one of the coefficients is large relative to the others. That means that changing that variable induces a large response in others (hence amplified). What is characterised as feedback is where this variable appears in at least one other term, and is also multiplied by the big coefficient. That makes a big proportional change in the output variables. That modifies the apparent performance in ways described as feedback.
So what is the outcome here? Mainly that you can talk about feedback, signals, Bode etc if you find it helps. But the underlying maths is just linear algebra, and the key thing is to write down correct perturbation equations, and manipulate them algebraically if you really want to. Or just solve them as they are.
I see Nick’s point in a way. CO2 has gone up, and temp has gone up. Now just explain the time lags and short term inverse relationships and…done (not really but you can see the temptation).
But other things in the atmosphere have gone up too. For every gallon of motor fuel burned, 6 gallons of water are created and sent out the tail pipe, for instance. Does the earth have more water (and water vapour) than in 1850 ? Water vapour is a real effective GHG. Or is it a drop in the bucket ? It is water vapour, and does rise up in the atmosphere, so already in the correct form to effect albedo, etc.
For CO2 being the variable to isolate in an equation with many others, then perturb it, then look at the temperature increase, for the last 160 years, appears to be the main effort of the models. But CO2 is also the weakest of all the things that have an effect on the surface temperature (assuming constant sunlight). And there are significant periods of time when the effect is negative as well as positive.
Until that is understood, why spend trillions on it when a paper boy would have a better understanding of maintaining his ins and outs and predict the weekly profits to save for the Schwinn ?
Where is this model actually documented ? How is version control and testing and CM performed ? Each change in source code that implements each functional change in the document is kept where ?
Charlie June 7, 2019 at 6:13 pm
Nobody knows about since 1850 … but here is some recent data. Short answer is that total preciptiable water has gone up about 5% since 1988.
w.
Willis, 5% seems surprisingly large, since I would only expect about 7% more TPW per +1°C, and temperatures since 1988 have risen only somewhere between 0.3 and 0.5 °C since 1988.
But then I looked at your graph, and it appears to me that half of that 5% increase was really just the 2015-16 El Nino spike at the end.
Do you agree?
I’m not a climate scientist or a system theorist but I am an empirical scientist and I believe that equating the very complex system that is earth’s climate to a linear approximation of a transistor (or any other such readily deterministic system) is rather unrealistic to say the least. Clearly, there are inputs to the climate system that we don’t understand and that are therefore not accounted for by such models. Take the obvious causal relationship between the sun’s magnetic field strength and climate. There is no consensus mechanism to explain it so it is ignored. This is the only reasonable approach for a theorist to take – of course you have to ask about what else is being left out – but it also makes the reliability of these climate models highly suspect IMHO.
Nick, the “feedback diagram” you show immediately after “Example 1 — the abstract feedback system” and the words you employ in the “blue box” aren’t internally consistent. [Note: I assume “A” as depicted in the feedback loop diagram is the same thing as the “AOL” term in equations in the blue box.] To illustrate the inherent inconsistency between the “diagram” and the “words,” I choose to set AOL=1 and B=1. I know, it’s not a very good amplifier, but there’s nothing preventing me from using a unity-gain amplifier. You restricted B to be a fraction; but since 1 is a fraction, I’m free to set B=1.
Let Vin be the input voltage applied to the circuit–i.e., the input to the “+” side of the subtractor in your feedback diagram. As you wrote: “Without feedback, the input voltage V’in” {which at this point isn’t defined, but can’t be anything other than Vin} “is applied directly to the amplifier input. The according output voltage is Vout = AOL * V’in.” In the case of a unity gain amplifier (AOL=1), the output voltage, Vout=Vin.
You then continue: “Suppose now that an attenuating feedback loop applies a fraction B*Vout of the output to one of the subtractor inputs so that it subtracts from the circuit voltage Vin applied to the other subtractor input.” Okay, let’s do that. For B=1, the voltage B*Vout=B*Vin=Vin. For this case, both inputs to the “subtractor” are Vin, which means the subtractor output will be 0. The subtractor output is the input to the amplifier so the amplifier’s input is 0, which for a linear amplifier means the amplifier’s output will also be 0.
All this sounds fairly straight forward. All I’ve done is subtract the output of a unity gain amplifier from the amplifier’s input and fed the difference back to the amplifier. But according to your formula for the closed-loop gain [AFB = AOL/(1 + B*AOL)], the value of AFB for the above circuit parameters is AFB=1/(1+1)=1/2.
In summary, I have a feedback-loop circuit with a closed-loop gain of 1/2 to which I input a non-zero voltage signal and get 0 output. Say what?
I don’t know how the “amplifier community” does things; but in the digital signal processing world, when you represent a feedback device (a device whose output is fed back as an input to the device) or for that matter a feed-forward device, a shift (time delay) is inherently assumed somewhere in the feedback/feed-forward diagram. Without the shift, it is impossible to decide the order in which numerical operations should be carried out. [Note: A similar thing happens in a spreadsheet when operations involving multiple cells loop back on themselves–the spreadsheet doesn’t know where to start or stop operations.] Then if you want to know how the device behaves to a constant input signal (i.e., to a direct current or DC input), you input a DC signal and observe the output as a function of time.
Reed,
“the words you employ in the “blue box” “
As said, the words in the blue box are the accompanying text – ie in Wiki. My point in quoting it was just to show how it was solely manipulating linear expressions. But I think the fault in your analysis is
“Okay, let’s do that. For B=1, the voltage B*Vout=B*Vin=Vin. For this case, both inputs to the “subtractor” are Vin”
You are subtracting the pre-feedback Vout. But it should be the post-feedback value, as yet unknown. That is why I urge people to stick with the linear equations, and solve them as they appear. In your case, with AOL=B=1, that gives
Vout=V’; V’=Vin-Vout and so
Vout=Vin/2
which is a perfectly straightforward gain=.5 amplifier.
Nick, I apologize. I may have entered this comment into the wrong location. I’m going to re-enter it here so that I know it’s in the right place.
Okay Nick try this.
For all time AOL=1 and B=1. For all time up to but not including time 0, no signal has been input to the circuit. At time 0 and for all time thereafter the input signal is 6. I argue that immediately prior to time 0, the only “acceptable” value of the circuit output is 0, because any non-zero value of the circuit output at time 0 would have to be justified.
I’m going to “loop” through the circuit a number of times. I start by computing the circuit output at the end of the first “loop.” Multiply the circuit output (0) at the start of the first “loop” by the factor B=1. That product is 0. Subtract that product from the input signal giving a difference of 6. Multiply that difference by AOL=1 to get the circuit output at the end of the first “loop.” Thus, the circuit output value at the end of the first “loop” is 6.
Now make a second “loop” through the circuit. At the start of the second “loop” the circuit output is 6 so that the product of B=1 and the circuit output is also 6. Subtract that product from the input signal (6) giving a difference value of 0. Multiply that difference by AOL=1 to get the circuit output at the end of the second “loop.” Thus, the circuit output value at the end of the second “loop” is 0.
Now make a third “loop” through the circuit. At the start of the third “loop” the circuit output is 0 so that the product of B=1 and the circuit output is also 0. Subtract that product from the input signal (6) giving a difference value of 6. Multiply that difference by AOL=1 to get the circuit output at the end of the third “loop.” Thus, the circuit output value at the end of the third “loop” is 6.
Repeat the above ad infinitum. The circuit output oscillates between 0 and 6, which has an average value of 3, but at no time is 3.
Now if you allow the circuit output to be 3 immediately prior to starting the first “loop,” then you get the following. Multiply the circuit output (3) at the start of the first “loop” by B=1. That product is 3. Subtract that product from the input signal (6) for a difference of 3. Multiply that difference by AOL=1 to get 3, which is the circuit output at the end of the first loop. Repeat ad infinitum. The output value at the end of each “loop” will always be 3, which is consistent with a gain of 0.5.
But if the input signal had been 8 instead of 6 and I wanted to ensure that the circuit output value was always 4 (i.e., corresponded to a closed-loop gain of 0.5), then I would have to have set the circuit output value at the start of the first “loop” to 4, not 3.
Thus to ensure a circuit gain of 0.5, the circuit would have to have a priori knowledge of the input signals strength. Where did the circuit get this knowledge?
If this doesn’t convince you there is a problem, try the following. Open a spreadsheet. Enter into Cell F1 the value of AOL—in our case, enter 1 into Cell F1. Enter into Cell F2 the value of B—in our case, enter 1 into Cell F2. Enter into Cell F3 the value of the input signal—in our case, enter 6 into Cell F3. Your equation for the output is “output = AOL*(input – B*output)”. Let Cell F4 store the output. Then the equation to be entered into Cell F4 is: “ = F1*(F3 – F2*F4)”. What you’re going to see when you hit the “return” button is a “circular reference warning” indicating that the spreadsheet doesn’t know what to do. In your post you wrote: “A computer (or a student?) could have solved them at any stage.” Microsoft Excel must be deficient , because it can’t solve “them.”
“Repeat the above ad infinitum. The circuit output oscillates between 0 and 6, which has an average value of 3, but at no time is 3.”
This again is emblematic of the tangles you can get into by not just using the linear equation. The equation is, dropping the out on Vout
V = V_in – V
Easy to just solve. But you insist on creating a sequence
V_n = V_in – V_(n-1)
Well, V_n = .5*Vin is a valid solution to that relation. But it turns out that you can’t converge to that solution by forward recurrence. OK, that just means that it is a bad way of solving a very simple linear relation. Don’t do it.
In fact, if the relation were V_n = V_in – p*V_(n-1), it would converge to the correct answer if p1. But there is no reason to do it that way. My point in this post is that you start with the linear relation, and then derive different ways of looking at it. If you derive something that doesn’t help, go back.
“p1”
Bah, HTML. p<1
Nick, to my comment (‘Repeat the above ad infinitum. The circuit output oscillates between 0 and 6, which has an average value of 3, but at no time is 3.’) you responded with “This again is emblematic of the tangles you can get into by not just using the linear equation.”
Including that response, here’s a summary of where we stand to date. You created a circuit and developed an equation that represents the behavior (input to output) of that circuit. Specifically, for the circuit in your guest post, after re-arranging terms the equation you came up with that relates the output to the input is
Vout = constant * Vin
Someone comes along and using (a) valid values for the circuit parameters (AOL and B) and (b) a valid input signal (in this case a DC or non-zero constant input) shows that when that signal is input to the circuit, the output is not a “constant times the input.” Your response is in essence You’re getting entangled in details. Just trust the equation. In math you can’t get entangled in details. The equation is either right or wrong. If by (a) inputting a valid signal to the circuit and (b) performing a valid analysis of the circuit output signal for that input signal the result contradicts the equation you purport represents the circuit, then the equation you purport represents the circuit is wrong. There is no wiggle room. In mathematics one counter example is sufficient to prove an equation is wrong.
Next, you select a different valid set of values for the circuit parameters (you required that P be less than 1) and argue for that circuit and the valid input signal used previously, the output signal converges to the correct answer. There is nothing in your derivation of the “input-signal/output-signal equation” that restricts its use to “converged signals only.” Your derivation applies to any output signal, not just a “converged output signal.” Arguing that for a specific input signal the converged output signal equals the input signal times a constant does not make your equation valid. One would still conclude your equation does not represent the circuit.
This completes my summary of where we are to date. Now to go forward. Another valid input signal is the sum of two components: a DC component and a sinusoidal component. For example the DC component might be “6” and the sinusoidal component might be “4*SIN(w*t+theta)”, where “4” is the amplitude of the sinusoid, “w” is the non-zero angular frequency of the sinusoid, and theta is the phase of the sinusoid at time t=0. If this signal is input to the circuit, then even when P is less than 1, not only does the output signal NOT equal “the input signal times a constant,” the output signal will not even converge to “the input signal times a constant.” The reason being that the “gain” of the circuit for a DC input is not equal to the “gain” of the circuit for a non-zero frequency sinusoidal input [See https://www.allaboutcircuits.com/textbook/semiconductors/chpt-1/amplifier-gain/%5D. For the input signal
6 + 4*SIN(w*t + theta)
the output signal converges to
C1*6 + C2*4*SIN(w*t + theta + phi)
where (a) C1 and C2, which are functions of the circuit parameters AOL and B, are NOT in general equal but may be equal for a specific input frequency w, and (b) the angle phi may not be zero. Since (a) C1 does not equal C2 and (b) phi may be non-zero, even after waiting an infinite amount of time (until convergence is reached), the output signal
C1*6 + C2*4*SIN(w*t + theta + phi)
will not be a constant times the input signal
6 + 4*SIN(w*t+theta).
This is another example (the third) of how your equation
Vout*(1 + B*AOL) = AOL*Vin
that purports to relate the output signal, Vout, to the input signal, Vin, does not represent the operation of your circuit diagram.
I’ve given three examples of where a valid signal is input to the circuit and the output signal is not a constant times the input signal. As far as proving that your equation does not represent circuit operations is wrong, a single example would have been sufficient. I used three examples only because you made objections (in my opinion, invalid objections but that’s for the reader to decide) to the first two examples. Your claims that I am somehow “entangled” in the details and that I “should just use the linear equation” to relate the output signal to the input signal may be acceptable to the AGW community, but they require a leap-of-faith I am not willing to take. The fact is your equation that relates the circuit output to the circuit input is just flat wrong. It may be correct of a few specific input signals, but it is not correct for a general valid input signal.
Reed,
“a valid input signal (in this case a DC or non-zero constant input) shows that when that signal is input to the circuit, the output is not a “constant times the input.””
No, you don’t show that. The system (which is not mine, but Wiki’s) says that the output will be half the input, with your parameters. And it will.
What you are doing is imagining a sequential way in which the system might respond. The amplification first, then the feedback etc. But that isn’t part of the system specification.
In fact, what you are doing is an established solution technique, called Hadamard iteration. To solve
x=f(x)
you take an x value, substitute on the right, get a new value on the left, and so forth. For an equation F(x)=0, you can write the sequence as
x_{n+1)=b*F(x_n)+x_n
for any nonzero b, you can see that if that converges, it will converge to a solution of x=0. But it may not converge. Then you have to try another value of b.
You have to distinguish between the existence of a solution and the ability of any particular method to find it.
Of course, all your argument is irrelevant to an application like climate, where there is just linear equation and no circuit diagram. You can create that if you think it helps understanding. And you can try to solve it by an iterative method, though there is no good reason for doing so. But if it doesn’t help, don’t do it.
“You’re getting entangled in details. Just trust the equation.”
The equation was the starting point. You created the details in trying to solve it. If tangled, try another way. The equation is the datum.
No, Nick, the starting point isn’t the equation, it’s the circuit diagram. If the starting point is the equation, what does it have to do with the feedback loop diagram? It’s every bit as logical to claim your equation represents the number of flapjacks it takes to cover the roof of a doghouse.
By the way, the way you depicted the circuit, if AOL = 4 and B=0.3, for a constant non-zero input signal the output of your circuit doesn’t even converge to a finite value–it grows without bound. Just try it Nick and see what happens. This non-convergence will occur whenever |AOL*B| is greater than 1. Your formula says the gain should be 4/(1 + 4*0.3) = 1.8181818—i.e., for a constant input signal of magnitude 1, the output signal should converge to 1.8181818. The output signal doesn’t converge to 1.8181818; it grows without bound. Your formula is valid only when the output signal converges; and for a DC input signal, that only occurs when the absolute value of AOL*B is less than 1.
You’re an interesting guy, Nick. You draw a circuit diagram with (a) an input, (b) two multipliers (AOL and B), (c) one summer (actually a differencer), (d) a feedback loop, and (e) an output. You apply reasoning to that circuit and derive a formula for the circuit gain—the relationship between the magnitude of the input signal and the magnitude of the output signal. Someone (and you can do this yourself) generates (a) a valid input signal (a unit DC step function—i.e., a signal that is 0 before some time and 1 after that time) and (b) valid values for the circuit parameters (AOL=4, B=0.3). He/she inputs that signal to the circuit and monitors the output signal. The output signal grows without bound—i.e., the output signal alternates sign with the absolute value of both the positive and negative terms increasing without bound. Faced with the discrepancy between your theoretical circuit gain and the observed signal output, you don’t question either your derived gain formula or the circuit input/output analysis. You simply say to that person “You’re getting entangled in details. Just trust the equation.” I sure hope that way of thinking is not representative of the AGW community.
PS. I don’t care what Wikipedia says. Just set your AOL and B circuit parameters to, respectively, 4 and 0.3, and input the unit DC step signal to your circuit. Execute the circuit and let me know what the output signal looks like.
Reed
“You’re an interesting guy, Nick. You draw a circuit diagram with…”
You seem determined to overlook that it is actually a Wiki article, not mine. But you might like to think about where this gets you. Are you saying that feedback amplification can’t work? That this perfectly orthodox feedback loop can’t work?
There is not much I can add to what I’ve said above.
Nick,
“You seem determined to overlook that it is actually a Wiki article, not mine. But you might like to think about where this gets you. Are you saying that feedback amplification can’t work? That this perfectly orthodox feedback loop can’t work?
There is not much I can add to what I’ve said above.”
It may be a Wiki article, but you used it to “Demystify Feedback.” Where it gets me is: “Wiki’s analysis is wrong.” I’m not saying that feedback amplification can’t work. I am saying that Wiki’s analysis of feedback amplification as represented by the diagram you presented in your Guest Post is wrong.
In a previous comment, I wrote: “I sure hope that way of thinking is not representative of the AGW community.” Upon reflection, that does seem to be a fairly common practice in the AGW community. Build a model for how you believe the earth behaves and will behave in the future. Run that model and compare its predicted outputs with measurements of the quantities predicted by the model. If the two disagree, say: Quit getting entangled in the measurements; the model is the datum. Trust the model.”
At the end of your most recent comment, you wrote: “ There is not much I can add to what I’ve said above.” I feel the same way – not that there isn’t anything you can add, but I’ve pretty much said all I feel is helpful. Given that, unless you specifically request I answer a question, this will be my last comment on this issue.
Sincerely,
Reed Coray
Nick writes “That this perfectly orthodox feedback loop can’t work?”
I’ll add that in the real world, of course amplifiers like that work because there is no discrete time step.
Only in the world of AGW’s CGMs where the calculations are necessarily time stepped is this a possible issue.
Re: Your reference
[…]Additional conditions being necessary (as a matter of fact, conditions of regularity), Paul Levy imposes such conditions either (1) at infinity—a difficult subject, which we shall leave aside—or (2) in the neighborhood of a determinate value, which the successive itératives of ƒ are assumed to approach.
Unless I’m misinterpreting this, sounds a lot like your recommendation
“Well, V_n = .5*Vin is a valid solution to that relation. But it turns out that you can’t converge to that solution by forward recurrence. OK, that just means that it is a bad way of solving a very simple linear relation. Don’t do it.”
…which is essentially just to use the “answer” which you must have got to by other means.
In my comment https://wattsupwiththat.com/2019/06/06/demystifying-feedback/#comment-2720155, I wrote: ‘Given that, unless you specifically request I answer a question, this will be my last comment on this issue.” I lied—or in the words of the Watergate politicians, “That statement no longer applies.” In a previous comment I had claimed: “a valid input signal (in this case a DC or non-zero constant input) shows that when that signal is input to the circuit, the output is not a “constant times the input.” Nick pointed out that for the values I chose for the Negative Feedback Circuit parameters (AOL=1, B=1), I had done no such thing. His words were: “No, you don’t show that. The system (which is not mine, but Wiki’s) says that the output will be half the input, with your parameters. And it will.”
So I’ll illustrate my point with better examples. In particular, I select three cases of valid Negative Feedback Circuit parameter values and valid (bounded) input signals. In none of the cases is the output signal a constant multiple of the input signal.
[Note: For any bounded input signal, when the absolute value of the product of AOL and B is less than 1, the output signal will always be bounded—i.e., the Negative Feedback Circuit is stable. For a bounded input signal, when the absolute value of the product of AOL and B is greater than 1, the output signal will become unbounded—i.e., the Negative Feedback Circuit is unstable. For a bounded input signal, when the absolute value of the product of AOL and B is equal 1, the behavior of the Negative Feedback Circuit is likely to be a function of the frequency content of the input signal.]
In the first case, the input signal is the unit step function (0 for time t<0, and 1 for time t≥0). For this case, the output signal converges to a constant multiple of the input signal, and that multiple is the purported gain of the Feedback Circuit. So the first case partially illustrates Nick’s contention that a negative Feedback Loop can/will attenuate an input signal.
In the second case, the input signal is the unit step function …plus… a sinusoid multiplied by the unit step function. For this case, the output signal has the same frequency components as the input signal (a DC term and a single sinusoid term); but the gain of the DC component is not equal to the gain of the AC component. The gain of the DC component is equal to the purported Closed-Loop gain; but the gain of the AC component is NOT equal to the purported Closed-Loop gain. Thus for the second case even after convergence, a single gain (constant multiplier) does not relate the output signal to the input signal.
In the third case, the input signal is again the unit step function, but the output signal is unbounded—i.e., does not converge to a finite value.
The contention of the Wiki portion of your Guest Post is that for a Negative Feedback Circuit with Feedback Parameters AOL and B, (a) the output signal will be a constant times the input signal, and (b) the Closed-Loop Gain, AFB, (which is defined to be the output signal divided by the input signal) is given by
AFB = AOL/(1 + B*AOL).
The three cases below illustrate that this is not the case.
Case 1:
Negative Feedback Circuit Parameters:
AOL = 4
B = 0.2
Product of AOL*B = 0.8—therefore the Negative Feedback Circuit is stable.
Purported gain, AFB = AOL/(1 + B*AOL) = 2.2222222222…
Input Signal:
Unit Step Function (0 for time t<0; 1 for time t≥0)
Output signal:
0 for time t<0; at time t=0 starts fluctuating (first 6 outputs: 4, 0.8, 3.36, 1.312, 2.9504, 1.63968), but converges fairly rapidly to a constant value of 2.2222222222.
Conclusion: (a) The output signal is NOT a constant times the input signal; (b) but the output signal converges to a constant value that IS the product of the input signal and a constant (the purported gain).
Case 2:
Negative Feedback Circuit Parameters:
AOL = 4
B = 0.2
Product of AOL*B = 0.8—therefore the Negative Feedback Circuit is stable.
Purported gain, AFB = AOL/(1 + B*AOL) = 2.2222222222…
Input Signal:
Unit Step Function …plus… Unit Step Function multiplied by a Sinusoid of Amplitude “A=0.5”, Frequency “F=0.25*FS”, and 0 Phase at time 0, where FS is the sampling rate. Thus, the Input Signal is 0 for time t<0 and 1 + 0.5*SIN(2*pi*n*0.25) for time t≥0.
Output signal:
0 for time t<0; at output index n=0, starts fluctuating (for indices n = 0, 1, 2, 3, 4, 5, 6, 7 the output signal values are, respectively, 4, 2.8, 1.76, 0.592, 3.5264, 3.17888, 1.456896, 0.8344832); but converges fairly rapidly to a four-point repeating sequence (for indices n = 99,992, 99,993, 99,994, 99,995, 99,996, 99,997, 99,998, 99,999 the output signal values are, respectively, 3.197831978353, 3.441734417317, 1.246612466137, 1.002710027090, 3.197831978327, 3.441734417339, 1.246612466140, 1.002710027088). Note: The converged four-point repeating output signal corresponds to a signal of the form
OUT(n) = 2.2222222222 + 1.561737619 * SIN(2*pi*n*0.25 + 0.674740942)
That is, the converged output signal is the sum of a DC term (magnitude 2.2222222222) and a sinusoid term [Amplitude of 1.561737619, Frequency of 0.25*FS, and Phase (radians) at time t=0 of 0.674740942]. Thus, both the input and output signals consist of a DC term and a common-frequency Sinusoid term. However, (a) the output DC term is 2.2222222222 times the input DC term, (b) the output sinusoid amplitude is 3.123475238 times the input sinusoid amplitude, and (c) at time t=0 the phase of the output sinusoid is not equal to the phase of the input sinusoid. Thus after convergence, (a) the actual DC gain, 2.2222222222, is equal to the purported Closed-Loop gain, but (b) the actual Sinusoid gain, 3.123475238, is NOT equal to the purported Closed-Loop gain. As such, the purported Negative Feedback Circuit gain of 2.2222222222 at best applies only to the input signal’s DC component and does NOT apply to the input signal’s AC component.
Conclusion: (a) The output signal is NOT a constant times the input signal; and (b) ignoring the phase shift in the AC term, even after the output signal converges to a repeating pattern, the output signal is NOT a constant times the input signal.
Case 3:
Negative Feedback Circuit Parameters:
AOL = 4
B = 0.3
Product of AOL*B = 1.2—therefore the Negative Feedback Circuit is unstable.
Purported gain, AFB = AOL/(1 + B*AOL) = 1.8181818…
Input Signal:
Unit Step Function
Output signal:
0 for time t<0; at time t=0 starts growing without bound (just try it).
Conclusion: (a) The output signal is NOT a constant times the input signal, and (b) the Negative Feedback Circuit gain is NOT the purported gain.
Now Nick claimed all of the above is irrelevant to climate. In particular, he wrote: “Of course, all your argument is irrelevant to an application like climate, where there is just linear equation and no circuit diagram.” If Nick’s perspective is true, then why did he bring Wiki’s Negative Feedback Circuit to his Demystifying feedback guest post? In fairness, I’m not sure if Nick or the blog moderator titled the guest post Demystifying feedback; but independent of the title, it appears that by bringing up Wiki’s Negative Feedback Circuit diagram he has confounded the discussion.
“That statement no longer applies”
I think the word used by Ziegler was “no longer operable”. Anyway, here is the fallacy in your reasoning. You take a relation between input Vin and output V
V = Vin – V
and assume that an input Vin will first affect only the first V, via the amplifier, and not the second, via the feedback. Then you bring in the feedback, and so on. But there is nothing in the specification to justify that. The feedback is probably just a resistor. There is no reason to believe, as you say, that the output from the amplifier (first V) and the termination of the feedback, which is the same point, will remain stationary. Or to put it another way, that no current would flow through the feedback resistor while the amplifier output was adjusting to the input.
The sequence that you describe is sometimes used, as in the math sequence I described. The reason is that if somehow that could be implemented, and it converged, then the converged result would be a satisfactory solution. But it actually can’t be implemented, and it doesn’t converge.
The title was mine. The argument is that in each of the cases listed (and in other cases too) the circuitry described simply specified linear equations, and all the feedback does is elementary manipulation of those equations. If you don’t believe the Wiki circuit, which is bog standard, try the transistor circuit. Do you really think that can’t work?
“operable”
Sorry, operative.
Nick,
I’ve been giving the matter of feedback loop representations of heat transfer between objects a lot of thought. I would like to continue this discussion with you; but I would like to do so using an exchange of emails instead of an exchange of comments on this thread. There are two reasons for changing the method of communication. First, Microsoft Word contains features that I struggle with when using HTML blog comments. For example, equations and figures can easily be included in Word documents, but are awkward (at least for me) to include in comments; and since I can attach Word documents to an email, email exchanges allow me to more easily express my thoughts.
Second, at some time in the future Anthony Watts will close comments to this thread, which will make communication on this issue harder to track. [Note, I have encountered the “termination of comments” communication problem when I was discussing a different issue using Joanne Nova’s blog as the communication vehicle.] If either you or I decide he would like to make the content of an email attachment a comment on this blog (or any other blog), he is free to do so provided he can convert the equations/figures in the email-attached Word file to equations/figures compatible with comments for that blog.
I’m sure we can get Anthony to (a) send you my email address and (b) send me your email address. Let me know if you’re interested in such an exchange by responding to this comment and so indicating. If you’re interested, will start the process of exchanging email addresses.
Sincerely, Reed Coray
Nick,
In lieu of comments on this thread, as best I can tell you have declined to take up my offer to exchange information via emails–https://wattsupwiththat.com/2019/06/06/demystifying-feedback/#comment-2724233. Given that, I’ll describe in this comment why I believe the representation of heat transfer between two objects cannot always be, and when heat transfer via conduction or convection is present should not be, represented by a feedback loop whose input is the internal rate energy enters one of the objects.
Define a two-object system in the vacuum of cold space (near 0 Kelvin) as follows. Object “A” is a solid sphere of radius “RS”>0 whose surface acts like a blackbody—i.e., absorbs all radiation incident on it and radiates energy in accordance with Planck’s cavity (blackbody) radiation law. Distributed symmetrically just below the surface of the sphere is a constant source of internal thermal energy at a rate “H” Watts.
Object “B” is an inert (no internal source of thermal energy) concentric spherical shell of inner radius “RI”>“RS”, outer radius “RO”>“RI”, thermal conductivity “k” and inner/outer surfaces act like a blackbody.
If the system is in energy-rate-equilibrium (ERE)—i.e., if the rate energy enters the system (or any part thereof) is equal to the rate energy leaves the system (or any part thereof)—then accounting for the sphere’s internal source of thermal energy and the radiation from all surfaces (sphere, shell inner, shell outer), the total rate, Hsphere_total, energy enters the sphere is given by:
Hsphere_total = H + 4*pi*sigma*(RS^2)*{(RO – RI)*H/(4*pi*k*RI*RO) + [H/(4*pi*sigma*RO^2)]^(1/4)}^4
where Sigma is the Stefan-Boltzmann constant = 5.670373212*10^(-8) Watts per meter^2 per K^4.
As a function of (a) geometrical constants, and (b) the internal power input to the sphere, H, the form of the equation that gives the total rate Hsphere_total energy enters the sphere is
Hsphere_total = H + C1 * [C2*H + C3*H^(1/4)]^4
Where C1, C2 and C3 are constants that depend on the sphere/shell geometry, but are independent of H. Note that if C2=0, the form of the equation for Hsphere_total is
Hsphere_total = H + C4*H = H*(1 + C4).
That is, when C2=0 the total rate, Hsphere_total, thermal energy enters the sphere is directly proportional to the internal rate, H, thermal energy enters the sphere. As such, it is possible to express the total rate thermal energy enters the sphere in terms of (a) the rate, H, internal energy enters the sphere and (b) a closed loop feedback system whose “per-loop” multiplier is “0<D<1”. Specifically, for a “per-loop” multiplier of “D”, the total rate energy enters the sphere (including the internal rate) is
H/(1 – D).
By finding the value of “D” such that
(1+D)*(1+C4)= 1,
a feedback loop can be constructed that will result in the correct total rate, Hsphere_total, of thermal energy entering the sphere. I’m not sure (a) how to determine what the feedback multiplier should be other than to use independent logic to determine the total rate energy enters the sphere and “back out” the feedback multiplier, or (b) what is gained by constructing such a feedback loop; but such a loop can be constructed.
However, if C2 is not equal to zero, Hsphere_total cannot be written as a linear function of H. If a feedback loop converges, the output is directly proportional to the input—i.e., is a linear function of the input. Thus, for the example in this comment, a feedback loop whose input is a constant, H, cannot be used to compute the total rate energy enters the sphere.
I believe that whenever thermal energy (heat) transfer exists via convection and/or conduction, the likelihood of representing heat transfer as a feedback loop is nil. For the Earth/Earth-atmosphere system to a high degree energy enters/leaves the system via radiation. It is for this reason that I believe that the AGW community overly focuses on radiative heat transfer relative to conductive/convective heat transfer. Using a feedback loop to characterize heat transfer between the atmosphere and the Earth’s surface is one such example.
Nick, the real world isn’t that simple
https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2015MS000493
Abstract : The effect of global climate model (GCM) time step—which also controls how frequently global and embedded cloud resolving scales are coupled—is examined in the Superparameterized Community Atmosphere Model ver 3.0. Systematic bias reductions of time‐mean shortwave cloud forcing (∼10 W/m2) and longwave cloud forcing (∼5 W/m2) occur as scale coupling frequency increases, but with systematically increasing rainfall variance and extremes throughout the tropics. An overarching change in the vertical structure of deep tropical convection, favoring more bottom‐heavy deep convection as a global model time step is reduced may help orchestrate these responses. The weak temperature gradient approximation is more faithfully satisfied when a high scale coupling frequency (a short global model time step) is used. These findings are distinct from the global model time step sensitivities of conventionally parameterized GCMs and have implications for understanding emergent behaviors of multiscale deep convective organization in superparameterized GCMs. The results may also be useful for helping to tune them.
When changing the time step size changes the way a GCM works, that confirms the feed backs depend on the calculations themselves.
“When changing the time step size changes the way a GCM works, that confirms the feed backs depend on the calculations themselves.”
GCM’s do not use feedbacks, but people sometimes calculate them from the solutions, as a way of interpreting. The solutions themselves are of differential equations, which are approximated using small differences. As the differences get larger (timestep), that approximation deteriorates, and the solution changes. That isn’t due to feedback, just to the behaviour of discretised pde’s.
Nick writes “GCM’s do not use feedbacks”
Everything a GCM calculates is a feedback against the sun’s energy simply radiating away like for the barren moon. Or do you want to play some sort of definition game?
Nick writes “As the differences get larger (timestep), that approximation deteriorates, and the solution changes. That isn’t due to feedback, just to the behaviour of discretised pde’s.”
And confirms they were only ever a fit in the first place.
Nick,
I thought some more about your reply to my comment. One way to analyze the behavior of a “loop” is to walk through the loop. For a feedback loop, this means walking through the loop an infinite number of times. Each time you walk through the loop, the various “loop” values change. In your writeup where you developed your set of linear equations, the value of V’in in your first equation (input to the triangular multiplier) represents V’in for the first pass through the loop. The value of V’in in your second equation represents V’in for the second pass through the loop. Thus, these two values are not the same, and setting them equal for the purpose of developing a set of linear equations misrepresents the operation of the loop.
For each pass through the loop you should assign to each loop variable (signal input, triangular multiplier input, output), an index (subscript) that corresponds to number of times you have gone through the loop. If you had done this, your first equation would have a subscript value of n=0 for V’in and your second equation would have a subscript value of n=1 for V’in. Equating these values, as you did when you substitute V’in0 equal to V’in1 is not valid; and any linear equations you generate from that substitution does not represent operation of the loop.
Actually, it’s only according to Lord Monckton that their “test rig” proves their point; we haven’t seen that test rig.
According to our Lord all specs and circuit design are in the paper they wanna publish. An independent laboratory also built such rig, according to the specs. Let’s wait for the paper – I’d like to believe sooner or later it will be published. But in fact we don’r need to even wait for that – I reckon for proof of concept that should be fairly easy rig. I think authors used actual temperature input/output but in fact any input, as voltage, can be used. Arduino board would do a charm.
Just as we haven’t seen his “eminent” co-authors entering into the rough and tumble of defending what Lord Monckton says is their belief.
Thanks God for that. If they engage in the blogospehere starting to fight with critics because ‘someone at the Internet is wrong’ that would be pretty worrying sign for me – amateurs in the game. The fact they keep the low profile is for me a very good sign that we’re actually dealing with respected specialists.
And I’m pretty sure the “test rig” merely proves that using average rather than local slope works if the system is linear—but also that local slope, whose use Lord Monckton tells us is the “grave error” that “climatology” makes, works, too.
I reckon the whole purpose of such rig is to demonstrate that a feed back unit acts upon entire input plus disturbances, not just disturbances. And that’s far better approach than analogies and ‘thought experiments’ we’re exercising here. Firstly, analogies cannot replace hard evidence, secondly they can be misleading. You’ve seen Nick’s analogy with hand pump – if that works as Nick imagines the tyre would be actually inflating our hand pump, and not vice versa.
Unfortunately, Mr. Watts stopped running my posts when I exhibited insufficient deference to Lord Monckton’s (exceedingly questionable) expertise, so you won’t see my test-circuit design.
You can stick it in the comments, as the ling to drawings.
PS – our Lord just posted another text! Looks like discussion is getting hotter.
Okay Nick try this.
For all time AOL=1 and B=1. For all time up to but not including time 0, no signal has been input to the circuit. At time 0 and for all time thereafter the input signal is 6. I argue that immediately prior to time 0, the only “acceptable” value of the circuit output is 0, because any non-zero value of the circuit output at time 0 would have to be justified.
I’m going to “loop” through the circuit a number of times. I start by computing the circuit output at the end of the first “loop.” Multiply the circuit output (0) at the start of the first “loop” by the factor B=1. That product is 0. Subtract that product from the input signal giving a difference of 6. Multiply that difference by AOL=1 to get the circuit output at the end of the first “loop.” Thus, the circuit output value at the end of the first “loop” is 6.
Now make a second “loop” through the circuit. At the start of the second “loop” the circuit output is 6 so that the product of B=1 and the circuit output is also 6. Subtract that product from the input signal (6) giving a difference value of 0. Multiply that difference by AOL=1 to get the circuit output at the end of the second “loop.” Thus, the circuit output value at the end of the second “loop” is 0.
Now make a third “loop” through the circuit. At the start of the third “loop” the circuit output is 0 so that the product of B=1 and the circuit output is also 0. Subtract that product from the input signal (6) giving a difference value of 6. Multiply that difference by AOL=1 to get the circuit output at the end of the third “loop.” Thus, the circuit output value at the end of the third “loop” is 6.
Repeat the above ad infinitum. The circuit output oscillates between 0 and 6, which has an average value of 3, but at no time is 3.
Now if you allow the circuit output to be 3 immediately prior to starting the first “loop,” then you get the following. Multiply the circuit output (3) at the start of the first “loop” by B=1. That product is 3. Subtract that product from the input signal (6) for a difference of 3. Multiply that difference by AOL=1 to get 3, which is the circuit output at the end of the first loop. Repeat ad infinitum. The output value at the end of each “loop” will always be 3, which is consistent with a gain of 0.5.
But if the input signal had been 8 instead of 6 and I wanted to ensure that the circuit output value was always 4 (i.e., corresponded to a closed-loop gain of 0.5), then I would have to have set the circuit output value at the start of the first “loop” to 4, not 3.
Thus to ensure a circuit gain of 0.5, the circuit would have to have a priori knowledge of the input signals strength. Where did the circuit get this knowledge?
If this doesn’t convince you there is a problem, try the following. Open a spreadsheet. Enter into Cell F1 the value of AOL—in our case, enter 1 into Cell F1. Enter into Cell F2 the value of B—in our case, enter 1 into Cell F2. Enter into Cell F3 the value of the input signal—in our case, enter 6 into Cell F3. Your equation for the output is “output = AOL*(input – B*output)”. Let Cell F4 store the output. Then the equation to be entered into Cell F4 is: “ = F1*(F3 – F2*F4)”. What you’re going to see when you hit the “return” button is a “circular reference warning” indicating that the spreadsheet doesn’t know what to do. In your post you wrote: “A computer (or a student?) could have solved them at any stage.” Microsoft Excel must be deficient , because it can’t solve “them.”
A few thoughts come to mind, after reading Nick’s article, but especially thinking back to his earlier reply about how models forecast weather accurately for a couple of weeks, but increasingly fail beyond that. This behavior might be due to a couple of mental reservations I’ve had about the modelling idea in general, and go to explaining why I’m in the sceptic camp.
1. Accuracy of linearized models relies on deviations being small enough so that higher power terms are unimportant. The failure of the weather (and climate) models might be due to development of excursions requiring cubic (time-symmetric) and higher-order treatment.
2. Truncation artifacts of a trivial type might apply to time variation as well. Nick models the state function as f(x0), rather than f(x0,t). An audio analogue to the latter might be predicting output from an overheating amplifier, where the operating conditions shift enough to substantially change gain or frequency response.
Time dependence of the climate state function f could arise, for instance, by oceanic upwelling: this could change the surface temperature, which would not only slightly disrupt the thermal balance, but might modify the viscosity and turbulence of the mixing layer, possibly affecting thermal transfer rates between air and water and CO2 ocean solubility and sequestration. The time scale for this variation is 100-1000 years.
Changes in biomass might also have an affect. Why was C14 sequestered so rapidly (~8 months) after the 1960’s nuclear tests, while the Bern profile persists much longer? On the face of it, one might conclude that an initial sequestering of CO2 triggers de-sequestration processes that go on much longer (micro-biomass is a possible suspect). If so, the state function f is being altered in a way indescribable by a linear perturbation.
To account for this in modelling terms, the state function f(x0) needs to be extended to f(x0,t) = f(x0, t0) + df(x0)*t+…. Unlike (1), this won’t de-linearize the equations (at least for shorter times), but it does break the steady-state assumption. However, since underlying hydrodynamic processes may well be cyclic, corresponding cyclic or stochastic descriptions, more complex than simple linear “drift”, might be necessary. After all, climatic variations show little indications of a tendency to achieve a steady state, if past history is a guide.
Despite some complexity, the math itself is “tedious but straightforward”. The larger question is, have ALL the relevant factors been identified, or are some missing?
I do think that the models ought to work to a degree where they can explain the colonization of Greenland, or the cold spells of the Justinian plague and the little ice age. We have a couple of thousand years of civilizational (proxy) behavior available for testing. After validation there, we can discuss to what extent we should trust their predictions to set policy.
“It is wrong to include variables from the original state equation. One reason is that the have been accounted for already in the balance of the state before perturbation.”
I can’t follow the math from either side. Take a balance sheet. That’s 1850. An income statement from that point on, makes no reference to that balance sheet. (This is a simplification.) You need to track inputs and outputs to get an income statement. You don’t say, we lost money, but look at the beginning balance sheet from 1850. That’s a distraction. We aren’t interested in the beginning or ending balance sheets as much as we are the income statement. The income statements drive the various balance sheets from different dates. Not the other way around. (This is a simplification.)
Balance sheets are still important. If you have a lot of money, you can lose money for a long time. The thermal mass of all the oceans are like a huge amount of cash on a balance sheet. Which means you can add a lot of CO2 before those change a lot. So, we should give the correct amount of weight to each thing. The balance sheets and the income statements.
Yes, that is a reasonable analogy. If you thing of total growth of wealth, you might include capital appreciation. Value of asset after, compared with before. All these go into a rate of change of worth (income) statement.
What Lord M is doing, in effect, is adding total asset values in as income.
Mr. Stokes, Mr. Watts, I need your help.
I have been debating the climate change farce with a nephew of mine who is a professor of chemistry at Yale, his wife is an assistant professor in Physics at Yale. Me, I am a retired carpenter.
They told me to take two two liter plastic bottles, fill each one with water one quarter full, drop a couple of Alka Seltzer tablets in one bottle, seal up both bottles tight , let them sit in the sun for an hour and then measure the temp of the air. I did tell them that this is a variation of Al Gore/Bill Neys experiment that was shown to be absolutely fraudulent, by you Mr. Watts. I told them I would do the experiment. I told them that I would do it 3 times ; 1 tablet, 2 tablets, 3 tablets. Take initial, final temp of both the water and air, I know that the reaction is endothermic, I know that the C02 bottle will be pressurized, there by retard evaporation, which cools. Kind Sirs, what else do I need to know?
John D, I guess you have googled;
https://principia-scientific.org/the-deliberately-false-greenhouse-gas-co2-experiment/
TonyN, thank you very much. I clicked on and read that article. I have one last favor.
What would be the resulting ppm of C02 caused by reacting 1000mg of anhydrous citric acid with a healthy excess of sodium bicarbonate in a 2 liter plastic container that has 600 ml of water. Add 400 ppm of CO2 to that number. I am hoping, sir, that the resulting ppm of C02 will be sky high, thereby invalidating the experiment. As a tangible means of saying thank you for your response, I have put you on my daily prayer list, you never know, huh.
This is in reply to
Beeze June 8, 2019 at 9:24 am
Consider this problem:
A 32 kg steel casting at a temperature of 425°C is quenched in 140 kg of oil initially at 20°C. Assuming no heat losses and the steel casting and oil to have constant specific heats of 502.416 and 2512.1 J/kg-K respectively, determine the change in entropy for a system consisting of the oil and casting.
There are three systems: 1) the steel casting which acts as a closed system, 2) the oil which is also acting as a closed system, 3) the oil and casting together which are acting as an isolated system because there is no heat gained or lost.
In this case, heat lost from the casting is gained by the oil, so total heat is zero:
or
Here I observe the standard where heat lost by a system is negative heat, and heat gained by a system is positive heat.
Next we need the definition of heat capacity:
We can usually igore the limit definition and use the constant pressure value. Also, specific heat capacity, little
, is big 
divided by mass, or:
Solving for
, we have:
where
is final temperature 
minus initial temperature 
. Plugging these terms into the above heat equation and solving for 
we get:
Substituting the values from the problem, we get
The change in entropy for the system is:
The Clausius definition of entropy is:
Although this definition is for a reversible heat transfer, entropy is a state variable and that lets us usually ignore the reversible requirement. We also need the definition of heat capacity, but this time we will apply the limit:
Using the definition of specific heat capacity:
We solve for
:
Substituting and integrating we get:
Now we can calculate the entropies:
Oops, we have a negative entropy. However, the second law only applies to isolated systems. So it’s possible for a closed system to have an entropy less than zero and not violate the second law. Now we compute the system entropy:
There are several things this example shows:
1. Heat is a conserved quantity. We couldn’t even begin the calculation if it wasn’t for the conservation of energy and heat.
2. The conservation of heat did not prevent this system from reaching equilibrium.
3. The conservation of heat does not violate the second law of thermodynamics.
4. The second law only applies to isolated systems.
And there you go.
Jim
Operating points are quite interesting. This all assumes that there is room for change. (the quantity can be increased by a factor of 2 say. )
And if the “amplifier” is actually saturated? Increasing the input by 10% with a gain of 100 will have near zero effect.
Look at Water Vapor Absorption. It nearly matches CO2 absorption. And there is 100XS as much in the atmosphere. Saturating the bands it absorbs in.
If there is already 99.9% reflection from water vapor adding CO2 will do very little.
GHG theory destroys itself on its own terms.
The frequent misuse of the analytic concept of feedback is one of the more egregious features of “climate science” as it seeks to wrap its ill-founded conclusions in the terminology of well-established system science . What is not revealed here is that operational amplifiers in electronic systems are invariably INDEPENDENTLY powered–an important factor found nowhere in the planetary climate system, which has no appreciable power source other than solar irradiance. While highly fungible in its manifold earthly expressions, solar energy may be locally stored, but its flux CANNOT be multiplied system-wide. Moreover, the usual analytic treatment of control systems assumes that there’s no drain of output power by the feedback loop. In the geophysical case, by contrast, we have only partial recirculation of thermal energy.
A far more credible system analogy would be a complex RC or RLC system, with no signal feedback whatsover. Such an analogue would provide a more tractable means of treating changes in system response characteristics induced as a follow-on effect by changes or modulations of input power. That, after all, is the poorly articulated sense of what is often really meant by “feedback” in the climate context.
“but its flux CANNOT be multiplied system-wide”
It is indeed solar energy that provides the power supply, in the form of the up to 240 W/m2 of net IR that flows upward through the atmosphere. This is analogous to the power current flowing through a triode valve, say. GHG’s modulate that current, with gain, just as does the grid in the triode. The amplified signal can then be fed back.
But as said here, the enthusiastic use of circuit concepts comes mainly from folks here, not climate scientists. They mainly use feedback terms to describe the behaviour of coefficients in a linear relation, usually from a flux balance. If there is a sum of coefficients multiplying what is regarded as a driving variable, or input, then the negative ones are described as positive feedback, and vice versa, anchored by one dominant positive one (Planck) which keeps the sum positive. That is a descriptor; nothing much hangs on its use.
The gain factor for triodes applies to the voltage (i.e., energy level), not the power current, which involves the inexorable flow of time.
There’s simply no way that power can be amplified in a passive system; it needs to be continually produced. (Otherwise, power utilities would use mere looping of transmission lines to multiply their profits.) That’s what makes the arithmetic “climate science” take on ostensible surface power fluxes (in the 500+ watts/m^2 range) wholly fantastic. You will not find any such aphysical nonsense in presentations by bona fide scientists (e.g., Peixoto & Oort). “Feedback” is not the hand-waving meme of ‘folks here,” who generally tend to grasp the rigorous sense of the term far better.
Some climate feedbacks have nothing to do with power fluxes, or even temperatures.
Among the most important climate feedbacks are the strong negative (stabilizing) feedbacks which regulate the CO2 level in the atmosphere. The higher CO2 levels go, the faster CO2 is removed from the atmosphere, by terrestrial greening and the oceans:
higher CO2 level → faster uptake by biosphere & oceans → lower CO2 level
“It is like pumping air into a tyre with a puncture: the harder you pump the faster the air escapes.” –Clive Best (though he was actually discussing a different negative feedback)
“The gain factor for triodes applies to the voltage (i.e., energy level), not the power current, which involves the inexorable flow of time.”
The anode current is not inexorable. The grid voltage modulates it. The voltage gain is not the key to the device. A transformer can increase the voltage too. The point is that the triode delivers its voltage at the output with a lower output impedance than a transformer. More current can be drawn, hence more power. This comes as a deduction from the idle current. The equivalent of idle current in climate is that 240 W/m2 upward IR.
When global temperature rises because of CO2, more water vapor enters the air. It absorbs part of that 240 W/m2, and this gets reradiated. Some adds to the downwelling IR, and so warms the surface. That warming is due to the diversion from the 240 W/m2. Just as in the triode.
Paradoxically, a passive system can indeed “amplify” power, at least in the sense you’re referring to.
No, wait, hear me out.
Simplify the climate system radically enough to consider only steady-state radiative transfer. From the thousand-mile-high view we have, net of reflection, simply 240 W/m^2 flowing into the earth and 240 W/m^2 flowing back out. Now, the system’s passive. But that passive system has “amplified” the 240 W/m^2 coming in to cause 390 W/m^2 at the surface: a surface temperature of 288 K.
How does it do that without an internal power source? Multiple counting: part of the incoming power absorbed and re-emitted by the surface gets absorbed by the atmosphere and re-emitted back toward the surface, where it’s again absorbed and re-emitted so that we count it again. We need no internal power source, because we really haven’t added power. We’ve only counted it more than once, so incoming power is “amplified” to arrive at the surface’s emitted power.
The surface temperature thus depends on the degree to which we double count, i.e., on the degree to which the surface again absorbs and at steady state therefore emits again power it has already emitted previously. And the degree to which the surface re-emits depends on the atmosphere’s infrared-light opacity, which depends in part on atmospheric water-vapor concentration—which in turn depends on surface temperature.
So there’s positive feedback: a surface-temperature increase increases atmospheric opacity, which increases surface radiation, i.e., the surface temperature. That opacity is like a valve, and there’s no reason in principle why the valve can’t control far more power than it takes to operate it.
Now, I’m not saying the opacity change and thus the positive feedback actually is very great; I don’t think it is. (And, yet again, we’ve intentionally left out many other real-world mechanisms for the sake of discussion.) But there’s nothing about feedback theory generally that prevents power “amplification” of the type climatologists postulate.
In anticipation of the above responses, I’ve already pointed to:
There’s nothing at all new presented here that would justify the analytic treatment of climate as a feedback control system in any rigorous dynamical sense of the term. Nick’s would-be counter-example of a triode is that of partial recirculation of stored energy (not power), typical of the static gain seen in process engineering. By contrast, the transfer function of a genuine (closed-loop) feedback system is given by H/(1-HG), where H is the open-loop transfer function, G is that of the feedback loop, and both are functions of frequency. His conception of back-radiation is that of a mathematician, not a geophysicist who properly distinguishes between extensive heat fluxes and mere intensive expressions of state.
ΔR is the change in flux at TOA, which is the GHG forcing. ΔT is the surface temperature response. The feedback factors are T for temperature,w water vapor, C clouds and α (=a) for albedo. What they are actually doing (multiply by λ) is writing a flux balance
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Small question,
If CO2, the GHG in question, is itself increased by the surface temperature response. Increased water temp more CO2, Where is this factor taken into account?
Demystifying feedback. Guest post by Nick Stokes,
A hoot.
The trick is to distract the audience from the real game at hand.
Well done, sir.
The point being that the article is not about demystifying feedback at all.
The goal is to muddy the waters wherein CM is wading.
Not that it needs much as the eloquent but overladen word salad he uses hides understanding from mere mortals.
But,
the mere fact that Nick has felt the need to make this attack means that there must be some substance, some message, some understanding to glean from it.
This could come in 3 ways, a simpler precise explanation by CM [unlikely, he doesn’t get it], An explanation in reverse by Nick in a a Damascene conversion [unlikely, he doesn’t get it] or one of the regulars with a bit of inspiration
Feedback is a very difficult concept due to the interplay of numerous other factors than CO2 plus the unknown interactions of both CO2 itself and the temperature rise that should be associated with it on the other factors.
Ceteris paribus there should be a consistent temperature rise with CO2 rise.
A linear relationship one would be stoked to admit.
And yet there is none.
One can see the CO2 at Mauna Loa doing its little sawtooth movement upward in an impossibly regular manner, yet the Temperature runs to its own beat. True they both finish up from the start to the end but not one sign of synchronicity.
Now one could put this down to Natural Variation.
But Natural Variation of the magnitude to completely obliterate any vestige of relationship would imply that the signal is not there.
The interesting part of Nick’s summary is that there should be a positive feedback of 3 times the passive signal. Which would make even the hint of a relationship much more obvious [magnified].
But zip nil nada
I will give him a possible CO2 temp relationship, on science, but virtually no positive feedback yet detectable on practice.