1 Steady-State Methods UCB EE219A Oct 29 2002 Joel Phillips, Cadence Berkeley Labs Some artwork...
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Transcript of 1 Steady-State Methods UCB EE219A Oct 29 2002 Joel Phillips, Cadence Berkeley Labs Some artwork...
1
Steady-State Methods
UCB EE219A Oct 29 2002
Joel Phillips, Cadence Berkeley Labs
Some artwork thanks to: K. Kundert
2
Steady-State Methods: Goals
• Understand alternative way of analyzing differential equations
‑ Faster
‑ Application-Specific
• “Tie together” several numerical themes
‑ Circuit theory
‑ Solution of ODEs/DAEs
‑ Newton methods
‑ Iterative solvers & preconditioning
3
Review: Solution of ODEs/DAEs
1. Given an ODE/DAE
2. Start with an initial condition
3. Pick a next time point (discretize time)
4. Compute next solution (Newton etc.)
5. Go to 3. and repeat till done
),,( 1111
nnnnn
tuxfhxx),,( tuxf
dtdx
4
Good Questions For Transient Analysis
• How does the circuit behave
‑ When driven by a sinusoids (for a short time?)
‑ When driven by a step input?
‑ When driven by unstructured (i.e. a-periodic) inputs
• Other time-domain characteristics
‑ Delay, Risetime, Overshoot
5
Hard Questions for Transient Analysis
• How does the circuit behave
‑ When driven by sinusoid(s) for a very long time? (steady-state)
• How much noise does the circuit introduce to a signal?
‑ Where does the noise come from? Where does it go?
• Other frequency-domain questions
‑ Small-signal stability, Bode plot, pole-zero
7
Why steady-state methods?
• Speed
‑ E.g., AC
• Accuracy
‑ E.g., distortion
• Insight
‑ E.g., stability
• No choice!
‑ RF noise
8
Prototypical steady-state analysis: AC
• Linear Circuits
‑ Apply a sinusoidal source at single frequency
‑ Sweep the frequency of source (Bode/Nyquist plot)
Small-signalsource
Measuredresponse
+
–v1 i2 i4
+
–v5
+
–v3
9
AC via TRAN
• Apply source
• Solve IVP (Trap, Euler, etc.)
• Wait till steady state is reached
• Fourier-transform the output
10
Problem #1: Speed
• Consider a 1Hz sinusoid applied to a resonant RLC circuit
Transients mustdie out to avoid
corrupting steady-state
Each period requires20-50 timepoints?
Must simulate forminimum one second
past transients
11
Problem #2: Accuracy
• Widely used BAD method for Fourier analysis
‑ Interpolate onto uniform timepoints, apply FFT
• Problems
‑ Detecting “onset” of steady-state
‑ Polynomial interpolation creates high noise floors in Fourier analysis
‑ Truncation errors corrupt spectrum
‑ Aperiodicity/endpoint errors
12
How bad can it be?
• Experiment:
‑ Sample sinusoid at 256 random points
‑ Interpolate onto uniform 1024-point grid
‑ FFT
• Looks good so far!!!
14
AC Linear Analysis
• Consider linear problem
• Recall: In the frequency-domain
• Nice feature: work for all linear elements (e.g., transmission lines)
tjueAxdtdx
uAxxj )()(
)(AA
)()( ssxtxdtd xjx
dtd
Laplacetransform
Fouriertransform
Sinusoidalsteady-state
tjacextx )(
Sinusoidalinput u
FundamentalAC analysis equation
15
AC Small-Signal for Nonlinear Circuits
• Step 1: Find the DC operating point
‑ In a circuit, this means find a set of currents, voltages that satisfy Kirchoff’s voltage & current laws, with all capacitors deleted and all inductors shorted
),,( tuxfdtdx 0)0,,( dcuxf
(Recall that: DC operating point itself is very useful in circuit design…..)
16
AC Small-Signal for Nonlinear Circuits
• Step 2: Linearize around the DC operating point
‑ Assume the inputs are small perturbations around the DC point
‑ Assume circuit response is in turn a small perturbation around the DC point (use Taylor series)
0)0,,( dcdc uxf
tiacdc euutu )(
)()0,,(),,( dcdcdc xxxf
uxftuxf
17
AC Small-Signal for Nonlinear Circuits
• Step 3: Solve the AC analysis equation
• Note: Fourier portion of analysis is exact
‑ No truncation error
‑ No aliasing errors
‑ No periodicity errors
acx
uxxf
xjdc
)()(
18
More Linear Analyses: Pole-Zero
• Is the design stable? Poles all in left half-plane? Recall AC analysis:
• Poles occur at (complex) such that
• Solve eigenvalue problem by
‑ Direct methods: QZ; or Krylov methods: Lanczos, Arnoldi (similar to GMRES!)
0)(
sxxf
sIdcx
s
acx
uxf
sIsxdc
1
)(
19
Noise Analysis
• Lossy devices in circuit generate noise
• Noise is: Stochastic (random) unwanted signal
• Typical model:
‑ Stationary Gaussian process characterized by power spectrum
Thermal Noise
Source
Noise appearsat output
20
White Noise
•Noise at each time point is independent
‑ Noise is uncorrelated in time
‑ Spectrum is white
•Examples: thermal noise, shot noise
R(t,) S(f )
f
FourierTransform
Autocorrelation Spectrum
21
Colored Noise
‑ Noise is correlated in time because of time constant
‑ Spectrum is shaped by frequency response of circuit
‑ Noise at different frequencies is independent (uncorrelated)
Time correlation Frequency shaping
R(t,) S(f )
f
FourierTransform
Autocorrelation Spectrum
22
Noise Analysis
• Typical model
‑ Assume “small” small signal analysis
‑ Stationary Gaussian process characterized by power spectrum
• Small-signal analysis with noise sources
• Frequency-domain method:
‑ Compute transfer function from each noise source to the observation point (output) [Same transfer functions as computed by AC]
‑ Sum noise power contributions. Correlations will be correctly tracked.
dwwdwwdtxxf
dxdcx
21
Possibly correlated “white” Gaussian processes
23
AC-like Transfer Function Computation
LinearizedCircuit
Source 1
Source M
etc.
kuxxf
xj kac
k
x
k
dc
sourceeach for )()(
Output
• Each source requires a transfer function analysis
• Number of sources M number of devices
•Too many solves!
24
Adjoint Analysis
• Standard AC analysis to compute
‑ Solve (expensive)
‑ For any output desired, compute (cheap)
• Key observation:
• Adjoint analysis
‑ One c, lots of b (or many more b than c)
‑ Solve
‑ For all the inputs (sources), compute
bAx
bAcTk1
xcTk
cAbbAc TTT 1
cyAT ybTk
25
Forward Analysis: Circuit Interpretation
LinearizedCircuit
Output 1
Output 2
Output 3
Output 4
Input 1
Input 2
Input 3
For one input configuration, compute TF from to all possible outputs
26
Adjoint Analysis: Circuit Interpretation
LinearizedCircuit
Output 1
Output 2
Output 3
Output 4
Input 1
Input 2
Input 3
For one output configuration, compute TF from all possible inputs
27
Generalizations of Steady-State Analyses
• Mostly ways of dealing with LARGE signal effects
‑ i.e., NONLINEAR analysis
• Examples:
‑ Distortion
‑ Frequency Conversion
28
Distortion
• Consider amplifier with cubic nonlinearity
• Harmonic distortion
• Intermodulation distortion
+
-inv
3ininout BvAvv
tBatBaAvout 3sin4/sin4/3 33 tavin sin
tbtavin 21 sinsin ,)2(,3,3in termssinsin 212121 ttttbAtaAvout
29
Distortion
122
Real life is more complicated with nonlinearfrequency-dependent terms, higher-order nonlinearities, and more complex inputs.
30
Frequency-Translation
• Linear Mixer
• Common confusion: frequency translation itself is a linear process (not nonlinear)
‑ But all actual frequency-translating devices are nonlinear
tavin sin
tbvlo 2sin
ttab
vvv inloout
)cos()cos(2
2121
31
Simple Nonlinear Steady-State Problems
• Compute harmonic distortion in the amplifier
• Compute conversion gain in the mixer
• Compute noise in either
• All three are periodic-steady-state problems
‑ (or periodic steady-state + small-signal analysis)
*
*[intermodulation distortion is a quasi-periodic steady-state problem]
32
Periodic Steady-State
• Assume general excitation by periodic inputs
• In many cases, we expect a periodic solution [if we wait long enough…]
‑ Recall: periodic functions have a Fourier-series representation (sum of sine and cosines)
• Why not solve for the periodic solution directly?
33
Periodic Steady-State Computation
• Apply a sinusoid or other periodic input signal
• Find the periodic response
‑ Time-domain solution over one “fundamental” period
‑ Or spectrum: Fourier coefficients at fundamental + harmonics
• Sound familiar????
‑ Recall: in AC, we solved directly for the Fourier response (“fundamental”). No higher harmonics arise because system was linear.
• Need:
‑ Steady-state “equations” ala
‑ FAST & ACCURATE way of solving equations
acx
uxxf
xjdc
)()(
34
IVP vs BVP
• No problem specified with differential equations is complete without boundary condition
• Before: Solving an Initial Value Problem (IVP)
• What about:
• Example of a boundary-value problem (BVP)
),,( tuxfdtdx 0)0( xx
),,( tuxfdtdx )()0( Txx
35
Note on PBCs
• If solution to DAE is unique, then solution on one period determines solution for all time
‑ Both the shooting method and spectral interval methods (harmonic balance) use this fact
• From knowledge of solution at one timepoint, can easily construction solution over entire period by solving IVP
‑ We will exploit this in the shooting method
36
Enforcing PBCs
• Approach 1: Build BCs in basis function
‑ Example: Fourier series satisfy periodic boundary conditions
• Approach 2: Write extra equations
‑ PBC
tkbtkatxk
kk
k
00
10 cossin)(
)()0( Txx
37
PSS Algorithm #1: Harmonic Balance
• Periodic solution can be expressed in terms of Fourier series with fundamental frequency
• Pick
tkbtkatxN
kk
N
kk
00
10 cossin)(
N
Nk
tikkectx 0)(
0
kk cc
-- OR --
(real solutions please!)
38
PSS Algorithm #1: Harmonic Balance
• Pick
• Want to solve
• Clever trick: spectral derivatives!
N
Nk
tikkectx 0)(
N
Nk
tikkecik
dtdx
00
N
Nk
tikkecdt
dx0
kk cikc 0
utxfdtdx ),(
39
Spectral Differentiation
• Recall
• Works on any function
‑ With suitable technical conditions
• Spectral differentiation is exact for sinusoids!!!!
)()(
ssxdttdx
Occasional linearity confusion:Linear circuit sinusoids do not interact.Differentiation acts on signals, not the nonlinear functions, so Fourier analysisworks fine.
41
Aside on Weighted Residuals
• Many numerical methods you know are weighted residual methods
• General scheme to solve
‑ Pick basis functions
‑ Ansatz:
‑ Select weighting functions
‑ Force
uxF )(
N
kkkcx
1
k
j
0)(,0
N
kkkj cFu
42
Weighted Residuals: Examples
• Least-Squares
‑ GMRES
• Collocation:
‑ Original equations satisfied exactly at some “points”
‑ BDF collocates derivatives
• Galerkin:
‑ Residual orthogonal to basis space (or some other space)
‑ Krylov-based Model Reduction
)( jj t
jj
43
Harmonic Balance: Equation Formation
• Enforce
‑ Galerkin (true spectral method)
‑ Point collocation (“pseudo-spectral method”)
• Force at selected timepoints
‑ Which ones? Time for another trick…..
uxF )( ),()( txfdtdx
xF
uxF )(
44
Differentiation and the DFT
• Fourier transform (DFT) relates solution at discrete points to Fourier coefficients
Mx
x
x
KX
KX
2
1
0
0
F
)(
)(
[strictly speaking, we are evaluating the Fourier integral via quadratureusing the trapezoidal rule. Useful fact: trapezoidal rule is the best rule (spectrally accurate!) for quadrature on a circle. ]
45
Differentiation and the FFT
• Derivative in Fourier space:
• DFT-based differentiation formula
• We can evaluate a DFT fast using an FFT
• Suggests selecting timepoints to be evenly spaced
MM x
x
x
K
K
K
i
x
x
x
dtd
2
1
1-0
2
1
F1
F
)()( kXikkX
46
Equation Structure
• BVP becomes
• Jacobian with
),(
),(
),(
-F )( F),()(22
11
1-
MM txf
txf
txf
itxfdtdx
xF
M
2
1
1-0
g
g
g
-F
K
K-
F
iJ
xtxfg kkk /),(
(sorta looks like AC, doesn’t it???)
47
Equation Solution
• We need to solve
• These matrices are dense in either Fourier- or real- space LU factorization is bad news
• They are potentially very large
• Yet a matrix-vector product can be done fast
• Ideal candidate for iterative solution methods (GMRES!)
• Good preconditioners are necessary, but hard to construct
M
2
1
1-0
g
g
g
-F
K
K-
F
iJ
bJx
48
Historical Note about Device Evaluation
• Once upon a time…..microwave/RF simulators were purely frequency-domain…..like AC.
• Problem: This required frequency-domain transistor models.
• At some point it was noticed that the devices could be evaluated in the time-domain (with equations written in frequency domain) by using Fourier transforms.