...
BTW, I disagree that this circuit is impractical ...

One question which would help answer this is what is this circuit supposed to do, and what is the intended range of load impedance.

We already see that a high impedance load is impractical for two reasons. First, high impedance is likely to be marginally stable and prone to tolerance problems. Second, t_n_k has shown that a high impedance load results in a voltage gain equation consistent with a voltage source; and, we really don't want a high impedance voltage source.

If we consider the voltage gain equation with a finite (and relatively low) value of load impedance \( R_L \), under the condition that source impedance is set to infinity we get the following, assuming everything is perfectly balanced.

\( G={{R_3 R_5 R_L}\over{ R_2 R_4 R_6 }}\)

This is a voltage source with voltage proportional to load resistance. Or, in other words it's a good current source. Hence, at this point I'm inclined to agree with you that this is likely a practical circuit. I think to be sure, one needs to do a sensitivity analysis to see how resistor tolerance will affect the results. It seems that stability is not an issue, but the quality of the current source might be sensitive to tolerances.

For reference the full voltage gain formula works out to the following.

\( G={{R_1 R_3 R_5 R_L}\over{ R_2 (R_4 R_6 R_1 + R_4 R_6 R_L + R_4 R_1 R_L - R_5 R_3 R_L) }}\)

A sensitivity analysis can be done using this equation.

 

In the attached simulation circuit I was able to "achieve" a constant current of 50.00uA over the load range of zero (0.0) to approx 180kΩ. Which isn't too bad.

I had to tweak the feedback resistor to something slightly less than the expected 13k - presumably to allow for the amplifier output resistance.

The source was also 'tweaked' to an arbitrary (?) 260mV which is, I guess, what worked for the particular amplifier. As one would expect the circuit runs out of puff as the individual amplifier outputs swing closer to the supply rails with increasing load resistance.

No obvious instability issues.

 

No obvious instability issues.

This is as we expect based on the theoretical analysis. But, what about the tolerance issue? If you simulate with 1% tolerance on all resistors, how well does the current control work, and what is the useful range of load resistance? These are the critical questions for whether this circuit is practical.

 

In the attached simulation circuit I was able to "achieve" a constant current of 50.00uA over the load range of zero (0.0) to approx 180kΩ. Which isn't too bad.

I had to tweak the feedback resistor to something slightly less than the expected 13k - presumably to allow for the amplifier output resistance.

The source was also 'tweaked' to an arbitrary (?) 260mV which is, I guess, what worked for the particular amplifier. As one would expect the circuit runs out of puff as the individual amplifier outputs swing closer to the supply rails with increasing load resistance.

No obvious instability issues.

The circuit needed 260mV for 50uA out because the transconductance (Gm) is 1/R6, which can be seen from Steve's equation for G:

G=Vout/Vin. Since Iout=Vout/RL,
Gm=R3*R5/(R2*R4*R6)
In this case, R3*R5/(R2*R4)=1, so Gm=1/R6.
In your example,
Iout=GmVin=(1/5.2k)*260mV=50uA.

 

The circuit needed 260mV for 50uA out because the transconductance (Gm) is 1/R6, which can be seen from Steve's equation for G:

G=Vout/Vin. Since Iout=Vout/RL,
Gm=R3*R5/(R2*R4*R6)
In this case, R3*R5/(R2*R4)=1, so Gm=1/R6.
In your example,
Iout=GmVin=(1/5.2k)*260mV=50uA.

Yes thanks Ron - I was missing the bleeding obvious.

In response to steveb's question about sensitivity to component variations it was clear even very small changes in R1 (say) had a noticeable effect. I'll try to run some simulations with component tolerance variations to see what happens.

 

Had a go at doing a quick and 'rough' Monte Carlo simulation of previously loaded schematic behaviour with 2% resistor tolerances for two different (50uA and 1mA) nominal constant current outputs and varying load conditions. Table of results included here. Each set of results were based on averaged data from three different runs at nominal conditions in each case.

 

Had a go at doing a quick and 'rough' Monte Carlo simulation of previously loaded schematic behaviour with 2% resistor tolerances for two different (50uA and 1mA) nominal constant current outputs and varying load conditions. Table of results included here. Each set of results were based on averaged data from three different runs at nominal conditions in each case.

Nice work!

It looks like a practical circuit to me. Load resistance should be 10K or less for reasonable performance in my opinion.