# What is a single-loop feedback system?

There is both a theoretical and pragmatic side to this question.

### Theoretical definition

Bode defines single loop amplifiers [circuits] as follows:

A single loop amplifier [circuit] is one in which the return difference of any tube [active device] is equal to unity if the gain of any other tube in the circuit vanishes.

It follows that the transconductances of the active devices enter the circuit determinant only as a product $G_{m1}G_{m2}\ldots G_{mn}$. This implies that the active devices occur in cascaded fashion and that the return differences for all active devices are the same.

Bode proceeds to make two remarks, which are paraphrased next:

1. This definition excludes local feedback loops, such as [cathode, emitter, source]-followers. These can be replaced by equivalent devices given the original operating points. This last remark is important as different operating points result in a different linearized system, which might be unstable. A particular situation is during circuit start-up. Care should be exercised. Also see later.
2. This definition excludes an active device with multiple parallel paths, such as so-called multiple-feedback filters. These are actually single-loop amplifiers as there one active device (the opamp model).

Chen repeats this definition and remarks, but rewrites it as:

A single-loop feedback amplifier is one in which the return difference with respect to the controlling parameter of any active device is equal to unity if the controlling parameter of any other active devices in the network vanishes.

### Practical circuits

This is answered in Should I check all possible loops in my circuit.

References:

1. Wai-Kai Chen. Active network analysis. World Scientific Publishing Co., Inc., 1991. URL

2. HW Bode. Network analysis and feedback amplifier design. D. Van Nostrand Co., Inc., 1945. URL