If you have a problem with this example please EXPLAIN your reasoning.
The problem is that you don’t know the difference between closed and open loop properties and haven’t bothered to try and see if the math backs up your assumptions.
For a positive feedback loop (LTI or within a range that can be approximated as LTI) the response to a change in input is:
Ochange = Ichange * [A/(1-A*B)]
where A is the open loop gain and B is the feedback factor
This is a well documented formula that can be found in any introductory textbook on system theory.
In your example you erroneously base the effect of CO2 on the system input rather then the system output. It’s a feedback element which means it samples the system output and adds/subtracts to the system input based on it. This is what it should look like drawn out properly.
- Insolation (a) goes up from 100 W to 101 W which creates a 1W energy imbalance
- The forward gain is 1deg/W giving a 1 deg increase in T
- The increase in T releases enough CO2 to reduce outgoing energy by 0.5W creating it’s own 0.5W imbalance
- this causes a further 0.5 deg of warming, which releases more CO2, which causes another 0.25W imbalance.
- etc etc
Each iteration of the loop adds a term to the series that represents the final change in T
T = Tstart + 1 + 0.5 + 0.25 + 0.125 + …
You can work it out by hand or computer to find out that the series converges to an increase of 2 deg. Alternatively you can use the formula I gave above.
Now reverse it
- Insolation (a) goes up from 101 W to 100 W which creates a -1W energy imbalance
- The forward gain is 1deg/W giving a 1 deg decrease in T
- The decrease in T causes the absorption of enough CO2 to increase outgoing energy by 0.5W creating it’s own -0.5W imbalance
- this causes a further 0.5 deg of cooling, which causes the absorption of more CO2, which causes another -0.25W imbalance.
- etc etc
T = Tstart – 1 – 0.5 – 0.25 – 0.125 -…
Again the series converges to Tstart – 2 deg.