# General Feedback Theorem

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The 2-General Feedback Theorem (2-GFT) derives from the General Network Theorem with two injections and is used to analyze feedback systems. A transfer function is decomposed into the ideal transfer function, loop gain and null loop gain or direct forward transmission. In addition, the various components of the loop gain and null loop gain are calculated.

Analytically, this decomposition can be calculated easily using null injection techniques. Furthermore, typically a lot of the lower-level transfer functions can be ignored. This allows to quickly assemble an analytical model that is useful for design. This is explained in Middlebrook's course. This model can checked numerically against the simulation results.

A few of the equivalent 2-GFT representations are:

\begin{align} H &= \Hinf \frac{T}{1+T} + \frac{\Ho}{1+T} \\ &= \Hinf \frac{1+\frac{1}{\Tn}}{1+\frac{1}{T}} \\ &= \Hinf D \Dn \\ &= \Hinf D + \Ho \Do \end{align}

The second-level transfer functions can then be interpreted as:

 $$\Hinf$$ 'ideal, desired' transfer function $$T$$ principal loop gain $$\Tn$$ null loop gain $$\Ho$$ direct forward transmission through feedback path plus common-mode gain of the forward path

$D$ and $\Dn$ are the discrepancy factor and null discrepancy factor, respectively. They express the changes to the closed-loop gain ($H$) compared with the ideal transfer function ($\Hinf$) due the non-idealities in the system, including limited dc gain and bandwidth of the loop gain, reverse transmission through the loop and direct forward transmission.

If an ideal injection point exists in a circuit, the 2-GFT simplifies to the 1-GFT.

Consider a single-loop feedback system or the major loop of a multiple-loop feedback system (*).

### 2-GFT

Typically, the test signal configuration consists of both a current and voltage injection and must be chosen as follows to produce the GFT interpretation:

1. The test signal (current and/or voltage) must be injected such that $$i_y$$ and/or $$v_y$$ is the error signal.
2. The test signal must be injected inside the major loop, outside any minor loops.

The principal loop gain is composed of a forward and reverse part:

$$T = \frac{\Tfwd}{1+\Trev}$$

The forward (reverse) loop gain is the parallel combination of the open-circuit forward (reverse) voltage and the short-circuit current loop gains:

\begin{align} \Tfwd =\Tvfwd || \Tifwd \\ \Trev =\Tvrev || \Tirev \end{align}

The forward (reverse) null loop gain is the parallel combination of the open-circuit forward (reverse) voltage and the short-circuit current null loop gains:

\begin{align} \Tnfwd =\Tnvfwd ||\Tnifwd \\ \Tnrev =\Tnvrev ||\Tnirev \end{align}

A redundancy relation yields $$\Ho$$:

$$H_\infty T = H_0 T_n$$

Another redundancy relation is:

\begin{align} T_{v,fwd} T_{v,rev} &= T_{i,fwd} T_{i,rev} \\ T_{nv,fwd} T_{nv,rev} &= T_{ni,fwd} T_{ni,rev} \end{align}

(*): multiple-loop feedback systems are not trivial. This will be covered later.

### 1-GFT

Assume an ideal voltage injection point. Current injection can be ignored. The definition of $$H_\infty$$ simplifies and the definition and calculation of $$H_0$$ simplify as well. Reverse loop gain $$T_{rev}$$ is zero and the short-circuit current loop gain $$T_{i,fwd}$$ is infinite. The loop gain equals the open-circuit forward voltage loop gain:

$$T = T_{v,fwd}$$

The same is true for the null loop gain:

$$T_n = T_{nv,fwd}$$

In the dual case of an ideal current injection point, reverse loop gain $$T_{rev}$$ is also zero and the open-circuit voltage loop gain $$T_{v,fwd}$$ is infinite. The loop gain equals the short-circuit current loop gain:

$$T = T_{i,fwd}$$

$$T_n = T_{ni,fwd}$$

References:

1. R D Middlebrook. The general feedback theorem: a final solution for feedback systems. Microwave magazine, IEEE (April), 2006. URL

2. R D Middlebrook. Structured Analog Design. 2004.