An ammeter connected directly to an ideal source through ideal conductors measures all the time the same current whatever Rₛₕ?

I answered this SE EE question yesterday.

My initial answer

You have connected a real ammeter to an ideal voltage source.

The problem of this arrangement is that the voltmeter inside the ammeter actually measures the entire source voltage... and this voltage does not depend on the shunt resistance Rsh (here we ignore the fact that the current will be significant). Thus your ammeter will act as a voltmeter.

The problem with this 22nd century meter is that it is not a "true" ammeter such as a 19th century "coil ammeter" which directly measures current through the magnetic field it creates. The modern one measures current indirectly by voltage across a resistance and that is why it is "mislead" in this situation.

We can observe two cases:

1. Known resistance. If the meter "knows" what the resistance Rsh is and uses it to calculate the current (this is the situation when we switch ranges and Rsh changes stepwise), the reading is true (in the sense that this is the current flowing through the shunt). But actually this value is very wrong because it is entirely determined by Rsh (the ammeter). It turns out that by switching the ranges, we (the ammeter) set the current... and it should be exactly the opposite - the current should be set by the external circuit.

2. Unknown resistance. However, if the meter does not "know" what the resistance is (for example, we change it "invisibly" for it :-), the situation gets even worse. Now the ammeter not only changes the current in the circuit, but also reads it incorrectly.

Dual case

Above we considered the case where a real ammeter made by a voltmeter in parallel to a shunt resistor is connected in parallel to an ideal voltage source.

It would be interesting to consider the dual case where a real voltmeter made by an ammmeter in series with a ballast resistor is connected in series to an ideal current source.

Now the ballast resistor cannot change the current and the ammeter will show the current set by the ideal current source.