This paper describes a tool for the Cadence Virtuoso software that implements the Dissection Theorem (DT) or General Network Theorem (GNT) and its applications: the Extra Element Theorem (EET), Chain Theorem (CT) and General Feedback Theorem (GFT). The tool allows a circuit designer to gain additional circuit insight by providing all second- and third-level transfer functions of the DT. In particular, feedback networks are factored into their exact components, enabling a deeper insight into the structure of the loop gain, direct forward transmission and hence closed-loop behaviour

Bauwelinck, Johan

Moeneclaey, Bart

Verbrugghe, Jochen

English

Jochen Verbrugghe

Bart Moeneclaey

Johan Bauwelinck

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Implementation of the dissection theorem in cadence virtuoso

Ghent University Academic Bibliography

Implementation of the Dissection Theorem inCadence VirtuosoJochen Verbrugghe, Bart Moeneclaey and Jan VandewegeGhent University, Department of Information TechnologyEmail: see http://www.intec.ugent.be/design/group.phpAbstract—This paper describes a tool for the Cadence Virtuososoftware that implements the Dissection Theorem (DT) or Gen-eral Network Theorem (GNT) and its applications: the ExtraElement Theorem (EET), Chain Theorem (CT) and GeneralFeedback Theorem (GFT). The tool allows a designer to gainadditional circuit insight by providing all second- and third-leveltransfer functions of the DT. In particular, feedback networks arefactored into their exact components, enabling a deeper insightinto the structure of the loop gain, direct forward transmissionand hence closed-loop behaviour.I. INTRODUCTIONDuring circuit design, simulators such as Spectre are widelyused to verify circuit behaviour. Rather than examining trans-fer functions from input to output, additional insight can begained using several simpler transfer functions, each portray-ing a subset of the circuit properties. The tool for CadenceVirtuoso described in this paper incorporates the DT to providethese simpler transfer functions.The structure of this paper is as follows. Section II gives anoverview of the DT and its resulting applications. In SectionIII the tool as implemented in Cadence Virtuoso is described.Examples of the application of the tool are presented in SectionIV. Finally, in Section V, conclusions are presented.II. OVERVIEW OF THE GENERAL NETWORK THEOREMThe General Network Theorem or Dissection Theorem [1]states that any transfer function H (the first-level transferfunction) of a linear network, whether feedback is involved ornot, can be factored into three second-level transfer functions.This is done by using a single extra test signal (either voltageor current) injection:H = H∞1 + 1Tn1 + 1T(1)By injecting two extra test signals (voltage and current), T andTn can further be factored into third-level transfer functions:T =Tfwd1 + Trev(2)Tfwd = Tv,fwd||Ti,fwd (3)Trev = Tv,rev||Ti,rev (4)Similar expressions hold for Tn. Another—equivalent—formof (1) is given by:H = H∞T1 + T+H01 + T(5)with the redundancy relation:H∞H0=TnT(6)Various other equivalent forms exist [1], all of which aresupported. For feedback systems, (5) is formally equal tothe asymptotic gain model [2]. Remark that although thesymbols used in (1)-(6) indicate the use of the DT in feedbacksystems, their position in the factorization merely reflectsthe method used to calculate them. Most of the second-and third-level transfer functions are calculated using nulldouble injection (ndi) or double null triple injection (dnti)techniques [3]. These techniques use two (dni) or three (dnti)test signal injection sources, which are mutually adjusted tonull a specific signal. By propagation of this null throughthe circuit, these calculations are always easier than normalsingle injection calculations. This comes in handy for symbolichand analysis during the initial design phase. Furthermore, theresults automatically come out in low-entropy form [4].Depending on the test signal injection configuration, the DTmorphs into the Chain Theorem [5], Extra Element Theorem[6] or General Feedback Theorem [1], in which the second-and/or third-level transfer functions have a particular meaningand can be interpreted in relation to the circuit elements.III. DESCRIPTION OF THE TOOLThe tool consists of three parts: a probe to be inserted in thecircuit, a SKILL script to post-process the simulation resultsand a presentation layer.Fig. 1 depicts the probe and its properties. The probe hastwo terminals and a reference terminal. Terminals xx and yyinject and measure signals. The user a) inserts the probe in thecircuit such that the test signal injection configuration can nullthe correct signals—corresponding to the desired interpretationof the DT, b) indicates input and output signal and c) selectsthe correct type in the probe properties. The reader is referredto Section IV for examples. The probe utilizes a customnetlist procedure, such that during netlisting a number of ACanalyses are inserted in the netlist from which the final resultsare gathered. It is possible to specify parameters for theseanalyses. In addition the probe can be disabled.The post-processing script calculates the second- and third-level transfer functions using combinations of signals obtainedfrom the aforementioned AC analyses, extending the methodproposed in [7].Fig. 1. DT Probe and its properties.Fig. 2. DT Plot Form.The resulting transfer functions can be accessed via adedicated form, mimicking Cadence’s own Direct Plot Form,as depicted in Fig. 2. Second- and third level transfer functionsare logically split across two separate tab sheets. Additionally,stability margins are indicated for the GFT interpretation.IV. APPLICATION EXAMPLESIn order to illustrate the usage of the tool, an example isgiven for each theorem. Although no symbolic expressionsare given in the examples, they can easily be calculatedand compared to the numerical results. Conversely, initialsimulation results can be used to find out which lower leveltransfer functions are dominant, as a starting point for sym-bolic analysis.Fig. 3. CS amplifier loaded by an emitter follower to illustrate the CT.Fig. 4. Second-level transfer functions illustrating the DT as the CT.A. Chain TheoremThe Chain Theorem is deduced from the DT with singletest current or voltage injection configuration, such that theentire signal from the output of a first stage flows to the inputof a second stage [5] . For voltage injection, it can be shownthat H∞ represents the buffered voltage gain between the twostages. Tn is infinite and T equals Zi2Zo1 , the ratio of the inputimpedance of the second stage to the output impedance ofthe first stage, such that (1 + 1T )−1 is Zi2Zi2+Zo1 . The latterexpression is a voltage division and represents the loading ofthe first stage by the second one. A dual form exists for currentinjection and current gain.The common-source (CS) amplifier loaded by an emitterfollower, shown in Fig. 3, illustrates the CT. The current mirrorbiases the CS stage. The input voltage is injected betweenthe two gates, hence its The´venin output impedance is theoutput impedance of the diode-connected FET. Fig. 4 shows aWaveScan screenshot of the magnitude of the resulting second-level transfer functions. The flat part of H∞ is determinedby the FET transconductance, drain load resistance (firststage) and emitter follower gain (second stage), while theeffect of gate-source and Miller capacitances starts to showat higher frequencies. The loading due to the emitter followeris expressed in the (1+ 1T )−1 factor. Note in particular the low-pass effect due to the bootstrapped base-emitter capacitance ofthe follower.Fig. 5. Degenerated common-emitter amplifier to illustrate the Miller effectby EET.B. Extra Element TheoremThis theorem states that any transfer function of a linearsystem can be expressed in terms of its value when a givenextra element (EE) Z is absent, and a correction factorinvolving the EE and two driving point impedances seen bythe element [3]. Two dual versions exist, with either zero orinfinite reference impedance [8]. It is deduced from the DTwith single test voltage injection configuration. For the firstversion, the voltage injection should be placed such that thevoltage across the EE, and hence the current flowing throughit, can be nulled. It follows that H∞ then equals H|Z=∞,T represents the ratio of the EE impedance to the drivingpoint impedance at the EE terminals ZZd and Tn is the ratioof the EE impedance to the null driving point impedanceat the EE terminals ZZn . A similar argument holds for thethe version with zero reference impedance, resulting in H∞equaling H|Z=0 and inverted forms of T and Tn. Versions ofthe theorem for two [6] or any number [9] of extra elementsalso exist.Consider the degenerated common-emitter amplifier withexplicit collector-base capacitance in Fig. 5. The EET will beused to show the Miller effect. The probe has been placedsuch as to obtain the EET with infinite reference impedance(shunt capacitance absent). Fig. 6 shows the second-leveltransfer functions. H∞ is flat and is mainly determined bythe collector load resistance, transistor transconductance andemitter resistance. The factor (1 + 1T )−1 in (1) forms the low-pass filter created by the ohmic base resistance and the Millercapacitance. The well-known right half plane zero, createdby both a direct forward path and an inverting path frominput to output, appears in the factor 1 + 1Tn . It is mainlydetermined by the shunt capacitance and effective transistortransconductance. It is easily seen on the Bode diagram thatthe total transfer H is constructed from H∞ and these factors:it starts and ends out flat, with a pole and zero in between.C. General Feedback TheoremThe desired interpretation of the DT for feedback systemsis as follows [1]: H∞ represents the ‘ideal’ transfer functionFig. 6. Second-level transfer functions illustrating the DT as the EET.Fig. 7. Emitter follower with probe for feedback analysis.with infinite loop gain. T is the loop gain and Tn the null loopgain. H0 exhibits the direct forward transmission, importantwhen the loop gain becomes small. Their true values areobtained, without any approximations or assumptions, via theDT approach, by injecting both a voltage and current such thatthe total feedback error signal—both voltage and current—canbe nulled. Hence, the loop gain and null loop gain consist ofa forward and reverse part (see (2)), both of which consist ofthe parallel combination of a part due to voltage injection anda part due to current injection ((3)–(4)).As an example of local negative feedback, consider theemitter follower in Fig. 7. It is biased and driven by a1 kΩ The´venin source. The output voltage is taken from theemitter. The probe is inserted between the base and emitter,to null the error signal. A screenshot of the magnitude ofthe simulated second-level transfer functions is depicted inFig. 8. As expected for a voltage buffer, H∞ equals 0 dB forall frequencies. The null loop gain Tn is much bigger than theloop gain T , hence it can be safely ignored in this case. Statedotherwise, the direct forward transmission H0 is very low. Itfollows that the closed-loop transfer function H is determinedonly by T and H∞. As seen in the figure, T starts out flataround 25 dB and assumes a second-order slope asymptote athigh frequencies. A first pole of T occurs at relatively lowfrequency, while the second one lies close to the unity-gainFig. 8. Second-level transfer functions of the emitter follower illustring theDT as the GFT.frequency. This leads to peaking in H , which is apparent fromthe plot.Let’s now conduct a more detailed analysis of the loop gain.Consider Fig. 9. As the transistor is very much unilateral,the total reverse loop gain Trev is very low and can beignored. It then follows from (2) that the loop gain equalsthe forward loop gain Tfwd, which is the parallel combinationof a voltage injection part Tv,fwd and a current injection partTi,fwd. Tv,fwd has a first-order behaviour and dominates atlow frequencies. On the other hand, Ti,fwd has a second-orderbehaviour, it has a higher DC-gain but starts falling off atlower frequencies before encountering a second pole aroundits unity-gain frequency. Hence, Ti,fwd dominates at higherfrequencies and consequently determines the AC-behaviour ofthe total closed-loop transfer function H . Note that the unity-gain frequency of Tv,fwd lies more than three times higher.On a Bode diagram, a parallel operation is readily per-formed and the total loop gain T can be constructed byjust looking at the Bode diagram. We conclude that thevoltage loop gain determines the low frequency gain of theemitter follower in the example, while the current loop gaindetermines its bandwidth. Both are required.Comparison with Spectre’s stability (stb) analysis: the stbanalysis built into Spectre calculates the loop gain as Tian’sloop gain, defined as [10]:Tt = Tfwd + Trev (7)Compared with (2), both definitions become increasinglymore different when the relative importance of the reverseloop gain increases, f.i. where elements cannot be consideredunilateral. Moreover, the relation of Tt to the closed-looptransfer function is not readily apparent, while that of Tinherently follows from the DT (1). In addition, the DT methodprovides all other transfer functions of interest. Note however,that both definitions satisfy for stability assertion [7], as thecharacteristic polynomial of the closed-loop transfer functionis equal in both cases. In the given example, since Trev is verylow, Tt is almost equal to T .Fig. 9. Third-level transfer functions of the emitter follower, showing thecomponents of the loop gain.V. CONCLUSIONThe described tool for Cadence can be of great help whentrying to gain insight into the workings of a circuit. Resultsobtained by hand analysis using fast ndi and dnti techniquescan be checked for correctness. Approximations for the handanalysis can be asserted a priori or validated a posteriori.During the design phase it can pinpoint the source of undesiredcircuit behaviour.ACKNOWLEDGMENTThe authors would like to acknowledge the Agency forInnovation by Science and Technology in Flanders (IWT).REFERENCES[1] R. D. Middlebrook, “The general feedback theorem: A final solution forfeedback systems,” IEEE Microwave Magazine, vol. 7, no. 2, pp. 50+,APR 2006.[2] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis anddesign of analog integrated circuits, 5th ed. John Wiley and sons,2010, ch. 8, p. 603.[3] R. D. Middlebrook, “Null Double Injection And The Extra ElementTheorem,” IEEE Transactions On Education, vol. 32, no. 3, pp. 167–180, AUG 1989.[4] ——, “Low-entropy expressions: the key to design-oriented analysis,” inProceedings. Frontiers in Education. Twenty-First Annual Conference.Engineering Education in a New World Order. IEEE; American Soc.Eng. Educ, pp. 399–403.[5] ——. The DT and the CT: The Dissection Theorem and theChain Theorem. “Ch 09.DT & CT.pdf”. [Online]. Available:http://www.rdmiddlebrook.com/D OA Rules&Tools/index.asp[6] ——, “The two extra element theorem,” in Proceedings. Frontiers inEducation. Twenty-First Annual Conference. Engineering Education ina New World Order. IEEE; American Soc. Eng. Educ, pp. 702–708.[7] F. Wiedmann. Loop gain simulation. [Online]. Available:http://sites.google.com/site/frankwiedmann/loopgain[8] V. Vorpe´rian, Fast analytical techniques for electrical and electroniccircuits. Cambridge, 2002.[9] R. D. Middlebrook, V. Vorperian, and J. Lindal, “The N Extra ElementTheorem,” IEEE Transactions On Circuits And Systems I-FundamentalTheory And Applications, vol. 45, no. 9, pp. 919–935, SEP 1998.[10] M. Tian, V. Visvanathan, J. Hantgan, and K. Kundert, “Striving forsmall-signal stability,” IEEE Circuits & Devices, vol. 17, no. 1, pp. 31–41, JAN 2001.

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Implementation of the dissection theorem in cadence virtuoso

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