Invited Tutorial: Analog & Mixed Signal Verification Kevin D Jones

An Apology I owe you (collectively) an apology! Paper accompanying this talk is not in proceedings Kevin D. Jones Rambus Inc.

3 Overview Audience Calibration Digital and Analog The Analog Design (& Verification) Process Mixed mode Design (& Verification) Process Formal Verification of A(MS) Systems An Analog Example The State of the Art A Mixed Mode Example Open Problems - A Challenge Summary and Conclusions

4 Audience Calibration Think about what you see in the following picture

5 If you saw this: Digital world view - An approximation of a 0 to 1 transition

6 If you saw this: Analog world view - An approximation of a linear transfer function

7 If you saw this: OUT = tanh(IN)‏ Theory world view - A representation of the hyperbolic tangent function High class analog world view

8 Digital and Analog Different worlds Think differently Different mindset, tools, approaches Occasionally forced to coexist in mixed signal designs My assumptions for this audience: People understand digital design and verification Analog is the novel part Main focus on the “front end”

9 Digital Circuits Verify this two-input AND gate

10 Digital Abstraction States are discrete and countable Must verify the properties at all states

11 Digital Verification Practice today: (Exhaustive) Simulation Directed pseudo-random simulation Coverage based Formal methods BDD based exhaustive state analysis Symbolic simulation Tools: Specman, SystemVerilog, Equivalence Checkers, Model Checkers, Theorem Provers, … Quite tractable for Formal Methods and lots of research has been done

12 Analog Circuits Verify this op-amp BA

13 Aside: AMS Verification is becoming a Significant Problem More and more systems have analog components In recent DAC Keynote, Justin Ratner pointed out the world is fundamentally analog Fast becoming a leading cause of SoC failure

14 Analog Design (& Verification) Process

15 Analog Design Fundamentally schematic based Draw the circuit Very little abstraction Not too many equations in practice Theory and equations in school Simulation in practice Draw it, Spice it, Repeat

16 Analog Design Process Identify circuit class Pick a topology Initial sizing Simulation Observe results Repeat until acceptable “Verify”

17 Topology For the desired class of circuit current mirror, amplifier, oscillator, PLL,... Go to the text book and pick the topology that you believe has the best possibility of meeting the critical specs Topology means arrangement of fundamental components transistors, capacitors, resistors

18 Sizing The fundamental parameters for transistors are Width and Length Physical dimensions effecting speed (delay), strength, power and load of the circuits Initial sizing involves an “educated guess” as to the lengths and widths that will deliver the desired critical specification elements

19 Simulation Unlike the digital world, simulation is a fundamental part of the design process many parameters are unknown until after simulation SPICE DC-analysis AC-analysis Transient

20 Design Loop Simulate Eyeball (or measure) result Tweak parameters Repeat until results are acceptable for critical parameters measure remaining parameters Write spec

21 Aside: Fancier Tools Cadence Neocircuit Automates the sizing loop Reduces the human effort Very expensive in terms of machine effort

22 End of Design Loop A completed schematic Essential parameters are shown to be in spec (for process parameters used for simulation)‏

23 Verification of Analog Circuits Transistor level models Circuit simulator (Spice)‏ DC simulation AC simulation Parameter sweeping Statistical variation (Monte Carlo)‏ Tools: SPICE (HSPICE, PSPICE, Spectre, eldo, …)‏ Transient (time based) simulation is very expensive Simulation inputs based on designers’ understanding/experience/intuition i.e. verification is the same as design simulation

24 Verification No real notion of verification as a separate process using different tools and approaches Design is exercised not verified Everything is based on simulation Analog Verification today is roughly equivalent to Digital Verification in 1990

25 Statistical Verification Monte Carlo simulation simulators support statistical variation of certain input parameters Hits some variant spaces classical “corners”, random variation Very expensive Each simulation may take a long time Overall result is as strong as the number of simulations

26 Mixed Mode Real systems are always “Mixed Mode” or “Mixed Signal” Some digital components Some analog components Communicating across boundary One tool AMS Simulator

27 Verification (should) != Simulation 25+ Years of research in verification of digital systems has shown that simulation is not a sufficient tool for verification For analog systems, this is even more true since (transient/time based) simulation is very expensive Look for an abstraction that allows analysis

28 Linear Abstraction If   -   it’s an op-amp with gain |  | Choose an Operating Point Apply small-signal perturbations to the OP (  A,  B)‏ Derive the relation  Y = · A + · B “Linearizing the system at the OP” Operating Point (OP)‏ A B

29 AC Analysis is a Formal Method for Analog AC analysis in SPICE provides the transfer function (TF)‏ TF: a complete description of the linear system in frequency domain No further need to simulate circuits in transient simulation as long as the input signals remain small The circuit is formally verified

30 Non-linear Domains The “formal” techniques for analog circuits work very well in linear domains Many (most) circuits are not linear in the voltage or current domains PLLs are obviously very non-linear in V All the simulator AC analysis tools work in V or I This seems to imply they are not useful for most circuits!

31 Variable Domain Transformation Most circuits are not linear in voltage or current but are linear in some domain: phase, frequency, delay, duty-cycle, …

32 Domain Translators V   ≈ Phase Detector and   V ≈ Phase Mixer Must propagate perturbations correctly

33 Domain Transformation gives “FV” Use domain translators to map the linear domain to V or I Verilog-A is a good vehicle for developing translators Perform analysis in this domain using AC analysis tools Use inverse translator to map back to original linear domain for results All the benefits of linearity together with all the benefits of V/I AC simulation techniques A “classic” analog approach to Formal Verification

34 Limitation of Linear Analysis Linear methods verify only LOCAL properties Most circuits are nonlinear Don’t forget these! AB Must make sure that the circuit does operate at the assumed OP

35 Bugs in Analog/MS Systems Parametric bugs E.g. Jitter too high The usual focus of designers effort Requires better models and simulation techniques for improvement Functional bugs E.g. PLL doesn’t lock! Very difficult to find due to long simulation times and large spaces

36 An Example from the “Real world” The example is extracted from an actual design failure Some issues were only found in measurement of fabricated test chips The design was validated as well as any analog designs are in practice today The scale of the example is reduced to make it approachable from an academic perspective but all of the issues are still present

37 An Even Stage Ring Oscillator Ring oscillators are a common component used in a variety of analog applications The obvious design uses an odd number of inverter stages to produce oscillating “0” and “1” signals at a frequency dependent on the inverter delay Even (2 or 4) stage oscillators are used when quadrature clocks are useful

38 The Complete Description * sring - a ring oscillator with bridges * * Libraries, parameters and models * protect *.lib '/user/cad/process/tsmc65/model/spice/g+/v1.0/local/fets.li b' TT.lib./cln90g_lk.l TT.unprotect.model pch pmos.model nch nmos * * Operating point * temp 30.param supplyVoltage=1.5V * * Initial conditions and stimuli * IC V(A2)='supplyVoltage*1.' *.IC V(B2)='supplyVoltage/2.' Vdd vdd gnd supplyVoltage DC * * Simulation controls * tran 25ps 5000ps UIC.option post.plot v(A2))‏ * * Simulation netlist * param lp=1.0u ln=1.0u.param wnbridge= 2u wpbridge=4u * Circuit ceases to oscillate above this ratio *.param ratioChainToBridge = * *Circuit ceases to oscillate below this ratio *.param ratioChainToBridge = 0.31 *.param wnchain = 'wnbridge*ratioChainToBridge'.param wpchain = 'wpbridge*ratioChainToBridge'.global vdd gnd.subckt INV in out psource nsource Mpullup out in psource vdd pch l=lp w=wp Mpulldown out in nsource gnd nch l=ln w=wn.ends INV.subckt CELL inp inm outp outm Xinv1 inp outm vdd gnd INV wp='wpchain' wn='wnchain' Xinv2 inm outp vdd gnd INV wp='wpchain' wn='wnchain' Xinv3 outp outm vdd gnd INV wp='wpbridge' wn='wnbridge' Xinv4 outm outp vdd gnd INV wp='wpbridge' wn='wnbridge'.ends CELL.subckt CHAINOFCELLS2 ina inb outa outb XCell1 ina inb oa1 ob1 CELL XCell2 oa1 ob1 outa outb CELL.ends CHAINOFCELLS2 XChainOfCells2 A2 B2 B2 A2 CHAINOFCELLS2.end sring

39 Challenge for Formal Methods The “obvious” digital abstraction doesn’t hold It has interesting failure modes It is very sensitive to the exact sizing of the ring and bridge transistors It has sensitivities to initial conditions for some sizes The challenge: Show that this circuit oscillates for all initial conditions for some sizing Extra credit: Show the sizing ratios regions for the ring and bridge transistors

40 Good

41 Bad

42 Ugly

43 Some Formal Approaches The ESRO example has been tackled by a number of different groups Methods used differ dramatically Recent publication in FAC 08 and surrounding conferences Following section gives an overview of a variety of approaches to this problem and some other noteworthy approaches on a similar scale

44 Stability analysis UBC “Pencil and paper” approach Use monotonicity of fundamental components to find DC equilibria, VT derivative to analyse stability Unstable => no DC steady state => no lock up GLSVLSI 2008

45 iSpice CBL Formulate example as an SMT problem using arithmetic intervals as underlying theory Proves properties based on DC, Transient and PSS simulations Shows regions of stability FAC 2008

46 Discrete Abstraction University of Frankfurt Divide continuous space into discrete regions for model checking Analog specification language DATE 08

47 Petri Net Models University of Utah Translating AMS circuits into LHPN Using this model as a basis for both BDD and SMT model checking FAC 08

48 Hybrid Automata Verimag Using dense time automata to model behaviors of analog circuits Analog assertions and monitors FMCAD 04, FAC 08

49 Reachability Analysis CMU Forward and backward reachability analysis on over- approximated partitions DATE 06

50 Bond Graph Abstraction Concordia Abstract the properties of analog circuits into a representation based on Bond Graphs Constraint solving to reason about safety and reachability properties FAC 08

51 Example 2: Moving up the AMS food chain Phase Locked Loops are critical components of most high speed PHYs They are made up of subcomponents Phase detector Charge pump Linear filter Voltage Controlled Oscillator Divider There are many possible failure modes Simulating the locking behavior of a PLL at the transistor level in the time domain is very expensive

52 PLL locking bugs

53 PLL Locking Bugs (II) PLL Locking Bugs (II)‏

54 PLL Locking Bugs (IV)‏

55 PLL Locking Bugs (V) PLL Locking Bugs (V)‏

56 A Verification Nightmare Individual components can be wrong Individual components can be fine but assumptions between components can be wrong Most of these issues are not visible if we assume “correct initialization” i.e. we start simulation from a locked state, as we do for most parametric simulation It takes a very (very, very, …) long time to simulate, using sufficiently accurate models, from any arbitrary initial state to reach lock Bugs like these make it through to production

57 Open Analog Problems (A Challenge)‏ Avoiding transient simulation Establishing that operating point assumptions are valid Establishing that all initial conditions result in correct behavior Dealing with non-linearity Good candidates for FM approaches

58 Conclusions and Future Work Analog and digital are different Different mindsets, different tools, different problems There are problems in the analog space that are really looking for solutions Different points of view yield valuable approaches The FV community is just beginning to come to grips with this problem =>Lots of interesting opportunities We can provide “realistic” examples to interested parties Some are small, representative and tractable If any one really wants it, we have a software → digital → analog → physics problem (ms → ns → ps → fs) in a 20GB SERDES system

59 Some Further Reading General Analog Design Gray & Meyer: Analysis and Design of Analog integrated Circuits, Wiley. B. Razavi: Design of Analog CMOS Integrated Circuits, McGrawHill Verification issues for Analog/MS Thomas Sheffler, Kathryn Mossawir, Kevin Jones: PHY Verification - What's Missing?, DVCon 2007 More information on Domain Transformation Jaeha Kim, Kevin D. Jones, Mark A. Horowitz: Variable domain transformation for linear PAC analysis of mixed- signal systems. ICCAD 2007: The State of the Art for Formal Verification of Analog Circuits Proceedings of FAC 08