## General Network Theorem

These pages are under construction.

In general N injections. No interpretation of decomposition results.

Morphs into the General Feedback Theorem, Extra Element Theorem and Chain Theorem depending on test signal injection configuration.

## General Feedback Theorem

These pages are under construction.

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.

## Extra Element Theorem

These pages are under construction.

The N-Extra Element Theorem (NEET) derives from the NGNT using N injections. A transfer function is decomposed in terms of its value when $N$ extra elements (EEs) $W_i$ are absent, and a correction factor expressing the modifications due the EEs. An element is absent if it is assigned either a zero or infinite value, select by appropriate injection.

The NEET can be written as, a.o.:

\begin{align} H &= \Href D \Dn \end{align}

#### 1EET

For one extra element $W$ and a single injection, the 1GNT can be written as the 1EET:

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

Depending on whether current or voltage injection is used, $T$ and $\Tn$ can be interpreted as: \begin{align} T &= \frac{W}{W_d} \\ \Tn &= \frac{W}{W_n} \end{align} or \begin{align} T &= \frac{W_d}{W} \\ \Tn &= \frac{W_n}{W} \end{align} with $W_d$ and $W_n$ the driving point immitance and null driving point immitance, respectively.

#### 2EET and NEET

See references.

References:

1. R D Middlebrook. Null double injection and the extra element theorem. IEEE Transactions on Education 32(3):167–180, 1989. URL, DOI

2. R D Middlebrook, V Vorperian and J Lindal. The N Extra Element Theorem. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 45(9):919–935, 1998. URL, DOI

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

4. R D Middlebrook. Low-entropy expressions: The key to design-oriented analysis. łdots in Education Conference, 1991. Twenty-First łdots, pages 399–403, 1991. URL

## Chain Theorem

These pages are under construction.

References:

Chain Theorem explained

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