Middlebrooks Extra Element theoremDennis Feucht - September 15, 2013

Following historically from the earlier work of Blackman, Gray and Searle, Cochrun and Grabel (andRosenstark) are some newer circuit analysis methods developed by R. D. Middlebrook of Caltech.This article presents the Extra Element Theorem (EET), a powerful and simple method of problemreduction that reduces circuits so that they can be analyzed with one reactance at a time. Somevariations are also presented (the impedance EET) along with the original, non-obsolete theoremfrom Blackman for feedback loops.

Extra Element Theorem (EET)

The extra element theorem (EET) was developed by R. D. Middlebrook as a refinement of a longhistory of related methods. As the genealogy chart of dynamics methods from the first article ofthis series, Circuit Dynamics for Design, shows, the EET combines ideas that are found in a lessrefined form in Gray and Searles MIT textbook on active circuits (Electronic Principles: Physics,Models, and Circuits, Wiley, 1969) and in Blackmans Impedance Theorem (BZT). BZT shows thepower of port-oriented methods of circuit analysis.

More can be inferred from port analysis than is at first apparent by subjecting the ports to differentconditions, and this is in part a consequence of the properties of linear systems. (All these methodsare based on linearized circuit variables that vary incrementally around a static operating-point.)The EET is the culmination of a century of development of port-oriented analytic techniques and isthe featured method to master, though simple methods such as the OCTC and quadratic Cochrun-Grabel-Rosenstark methods are quite useful to know and apply.

The EET is based on the following diagram of a circuit (box) with input and output ports, xi and xoand an additional port somewhere in the circuit with external impedance Z across it having portvoltage v and current i.

The Z port is that of a circuit element - the extra element. With Z attached to the circuit, the portv-i relationship is

The negative sign signifies that Z is external to the port. By port convention, the port driving-pointimpedance, v/i, is that looking into the circuit port. The current direction must be reversed (itspolarity changed) to refer to Z. The port equations are

The four port parameters can be found:

Aoc is the gain from xi to xo with the Z port open-circuited. ZD is the Z-port driving-point impedance,the impedance of the circuit from the port without the external Z and with the condition on ZD thatthe input be set to zero.

From the port equations, substituting for v and solving for i,

Substituting for i in the port equation for xo,

Eliminate Ti and Tv

While this gain expression is in port parameters, it can be made more useful by eliminating Ti and Tv.To do this, another condition is imposed on the amplifier: xo is nulled, or adjusted to be zero. Nullingdoes not mean forcing xo to be zero by shorting its node or opening its loop. A dependent variablesuch as xo can be nulled only by adjusting circuit conditions so that xo = 0. To do this, assume xo = 0and solve for the impedance at the Z-port that makes it so. This might sound difficult but it is usuallyeasier than finding ZD. Often the assumption of zero output ripples backwards through the circuit,reducing analysis significantly. The first port equation becomes

xo = 0 = Aocxi + Tii

Solving for i with the output nulled,

The Z-port impedance is then

where ZN is the Z-port output-nulled impedance. We can find both ZD and ZN by imposing conditionson the circuit. Solving the ZN equation for the superfluous port parameters,

They are eliminated by substituting them into

This is the final form of the EET. The factor in parentheses is called the correction factor because itmodifies the otherwise unmodified gain Aoc by the effect of Z on the circuit.

What the EET enables us to do is to find the gain of the circuit when it is affected by the addition ofZ. With Z removed (port open), the gain is Aoc. When Z is included, the modified gain can be found byfinding the correction factor by finding ZD and ZN. The process is

1. Find Aoc with Z removed.

2. Find ZD by setting the input to zero and deriving the port impedance at Z.

3. Find ZN by nulling the output (with xi applied) and find the port impedance.

4. Substitute ZD and ZN into the correction factor and solve for the modified gain.

The EET has a dual theorem that is expressed by exchanging v and i. It is derived from the dual ofthe above port equations and applies when an internal port is normally shorted and is opened toinsert Z. For it,

The open- and short-circuit EET formulas differ only by the condition on A and by the Z ratios in thecorrection factor. To remember whether Z is in the numerators or denominators of the Z ratios,when Z is shorted, the correction factor reduces to one, leaving Asc, the short-circuit gain. Thus Z forthe short-circuit formula must be in the numerators. The open-circuit formula has Z in thedenominators, and it must be infinite (open port) to cause the correction factor to be one.Additionally, although the Z subscripts D and N stand for driving-point and nulled, they canequally stand for denominator and numerator, where they are found in both (dual) formulas.

Single CB Stage Cc

Single CB Stage Cc

An example of the use of the EET is to find the gain of the CB stage with and without Cc. Ce = 0 pF,the ideally fast transistor without hf (s) effects. Z becomes 1/sCc and the b-c port is the port acrosswhich Cc is placed to modify the circuit, shown with an external voltage source, v, applied to the b-cZ-port.

The quasistatic voltage gain is found with the Cc port open using the transresistance method frominspection;

Next, ZD = RD is found by opening the b-c port and finding the open-circuit resistance. We havealready done this for the general single-stage circuit; RD = Rbc. To find RN, assume vo = 0 V. Then thecurrent in RL must be zero and by KCL at the collector node,

i = 0ib

The base voltage is

Then

Combining these resistances along with the extra element, Z = 1/sCc, into the open-circuit formula,

The EET method provides a complete transfer function by including the zero at z = 1/(RB/0)Cc inaddition to the open-circuit pole that the OCTC and Cochrun-Grabel methods produce. Thus the EETis a more powerful and complete circuit theorem that in this case was only more work to use thanthe previous pole-only methods in finding ZN which gave the zero.

Single CE Stage Cc

The EET can also be used to find the RHP zero of the single CE stage.

Given the quasistatic Av = Avoc and the extra element, Z = 1/sCc, then ZD = Rbc, the OCTC resistanceacross the b-c port. (This resistance was derived in a previous article in this series, Single-StageBJT Dynamics, OCT12.) ZN is derived as follows.

Then

The gain with Cc is thus

The RHP zero time-constant resistance is the resistance of the emitter circuit whereas for the CB, itwas of the base circuit.

Impedance EET (ZEET)

Impedance EET (ZEET)

A form of the EET for finding port impedances is the impedance EET (or ZEET) and is derived fromthe following diagram showing two ports. The Zi port is an adaptation of the xi and xo variables of theEET, applied as port voltage and current. The correspondences are

The goal is to determine how a port of the circuit (on the right) affects the input impedance, Zi, ofthe left-side input port. By opening and shorting the input port, the effect of Z can be determined onZi. The circuit Z-port, for our interest, is chosen to be across a circuit capacitance such as Cc.

The first of three port equations is at the Z port:

The other two equations are port parameter equations:

These equations are port functions of the form vi(ii, i) and v(ii, i). The circuit is assumed linear and bysuperposition the effects of sources at the ports add as in the port equations. The coefficients mustbe impedances and are so designated, though they have yet to take on meaning. If the Z port isopened, i = 0 A and

Zioc is the open-circuit Zi. For the other parameters,

ZD is the circuit-port driving impedance with the input port open. The circuit-port impedance can befound by setting vi = 0 V in the port equation;

Solving for ii in the first equation and substituting for ii in the second,

Then the Z-port impedance with the input port shorted (to satisfy the vi = 0 V condition) is

Now solve the two port equations for input impedance, vi/ii without constraints applied to the ports.First, substitute for v from the first equation into the third and solve for i;

Then substitute for i into the first equation:

The input impedance is

This result can be put into a better form for use as a formula by writing it as

This simplifies to its working normalized transfer-function form as

If the circuit port is opened, then Z and Zi = Zioc, the original open-circuit Zi. The effect of Z on Ziis found by finding the three parameters. ZN is the circuit-port Z when the input port is shorted andZD is its impedance when opened. It might be easier to remember the theorem in the following form:

The rational factor after Zioc is the correction factor because it changes the original impedance, Zioc,to account for Z.

The dual of the open-circuit ZEET has a circuit port that is a conductor opened for modification. Theformula is

To remember whether Z is in the numerators or denominators of the ratios, for the open-circuit port,as Z , Zi Zioc. For the short-circuit port, setting Z = 0 results in the original circuit condition,that Zi = Zisc. In both cases, the correction factor becomes one when the effect of Z is removed.

Blackmans Impedance Theorem (BZT)

Blackmans Impedance Theorem (BZT)

Receding backward in time (but not relevance) from the ZEET, the power of port-oriented analysiswas demonstrated earlier by Blackmans Impedance Theorem (BZT) which is presented here asBlackmans Resistance Theorem (BZT). BZT is equivalent to the ZEET with a change ininterpretation of the parameters. BZT is developed here in the context of the ZEET with mostly thesame nomenclature.

BZT is used to find the resistance of a port within a feedback loop. It can be anywhere in the loop

and need not be at the error input or fed-back quantity output. The port configuration is like that ofthe ZEET though BZT interprets the ZEET circuit port as a feedback loop. ZEET is converted to BZTby setting v xo and i xi so that the BZT loop gain is from xi to xo. The BZT port equations are

(Keep in mind that xi is of the other port than vi and ii.) The feedback loop is broken at some circuitbranch to form a port (corresponding to the ZEET Z port), and the two ends are labeled xi and xo,based on the direction of waveform propagation. Then the open-circuit loop gain is xo/xi = Toc whenthe resistance port (corresponding to the Zi port of the ZEET) is open so that ii = 0 A. The firstparameter extracted is the loop gain under the condition that the resistance port is open;

The R port is short-circuited when vi = 0 V, and the loop gain is then

Both Toc and Tsc are measured parameters. To find Tsc, solve for ii in the first port equation.

Substitute ii into the second port equation;

From this, set vi = 0 V and solve for

The closed-loop condition is that xo = xi. Applied to the second port equation,

Substitute xi into the first port equation and solve for the closed-loop resistance;

Then substituting for (TvTi/Rol),

Blackmans resistance formula results:

where the open-loop input resistance, Rol corresponds to Zioc in ZEET, Tsc ZN and Toc ZD. Measureloop gain with the R port open (Toc) and shorted (Tsc), measure the open-loop port resistance, Rol, andthe closed-loop resistance is given by Blackmans formula. Although the formula is given forresistance, it can be generalized to impedance.

Closure

With the advent of Middlebrooks theorems, design is made simpler through design-orientedanalysis. The EET and its variant interpretations all simplify the extraction from circuits of theparameters of interest for design. These theorems have been around for several decades, yet (like hftransistor theory) have not diffused widely within the electronics engineering world.

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