S and Z Parameters for PDN Measurements and Simulations
Istvan Novak Outline
• Background • Network matrices • Linearity and time invariance • Impedance matrix • Scattering matrix • Making scattering matrix work for PDN measurements • Scattering parameters in simulations • Conclusions Background
• Simulation and measurement of PDNs raise multiple questions － Time-domain vs. frequency domain － Time duration and frequency range － Linearity vs. nonlinearity － Time variance vs. time invariance － Causality － Passivity • Network matrix vs. circuit description of PDN Network Matrix Description of PDNs
• Black box approach, any number of ports: 1…N
i1 i3
v1 v3 Port1 Port3 Linear and Time-invariant Circuit
i2 i4
v2 v4 Port2 Port4
Scattering matrix Impedance or admittance matrix b = S a v = Z i, or i = Y v 1 푣 + 푍 푖 1 푣 − 푍 푖 푎 = 푖 0 푖 , 푏 = 푖 0 푖 푖 2 푖 2 푅푒{푍0ሽ 푅푒{푍0ሽ Network Matrix Description of PDNs • Network matrices assume that the DUT is － Linear － Time invariant • On a macro level, these assumptions are mostly valid • On a micro level, many DUTs can be more or less nonlinear and/or time variant • Sufficiently linear and time invariant: connectors, cables, PCB planes, traces, vias • Can be slightly nonlinear and/or time variant: certain types of capacitors • Can be more nonlinear and time variant: DC sources and noise sources
PCB Package Noise PDN Connectors, Bulk planes, HF and source DC cables capacitors traces, capacitors package and source vias capacitors victim Slight Nonlinearity in PDNs
• High-density ceramic capacitors with DC and AC bias sensitivity
Current through the capacitor
푑푉 I = 퐶 푑푡
Linear voltage ramp
Murata GRM219R60G476ME44 dynamic model simulated with LTSPICE.
See: “What You Need to Know About Bypass Capacitor S-Parameter Models,” http://www.electrical-integrity.com/Quietpower_files/QuietPower-51.pdf Potentially Stronger Nonlinearity in PDNs
LTM4604 simulated step load response [V] • DC sources, 1.56 0-4A clipping 1.54 0-3A 0-2A － Intentional 1.52 0-1A Load attack 0-0.5A － Unintentional 1.50
1.48 • Noise sources Load release 1.46
1.44
1.42 Current 1.40
1.38 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 Time [s] Impedance Matrix: Why and When
Noise • Power rails are expected to excitation Noise be voltage sources >> current I voltage unloaded output voltage n (i) (j) • Noise sources are expected to be much higher Noise PDN network PDN network Load V source Z (f) n impedance than the power Zijij(f) rail >> current sources • Users need noise voltage as a result of noise current >> self or transfer impedance 푉푛(푗) 푍푗푖 = 퐼푛(푖) Impedance Matrix: Pros and Cons
Pros: • Frequency-domain PDN design process uses impedance • Easy to measure (at low frequencies and at not very low values), because we have － High-impedance voltage sensors － High-impedance current sources • Unused ports can be left open, no need for any termination • Impedance matrix size can easily be changed － Ports can be added or dropped without recalculating the rest
Cons: • Not easy to measure low impedances and/or at higher frequencies Scattering Matrix: Why and When
• Easier to measure at higher a1 a3
frequencies b1 b3 • Well-established a Linear N port a instrumentation 2 4 b2 b4 • Well-established calibration procedures Impedance magnitude [Ohm] 1.E+0
1.E-1
Power OFF 1.E-2
1.E-3 Power ON
1.E-4 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 Frequency [Hz]
“Accuracy Improvements of PDN Impedance Measurements in the Low to Middle Frequency Range,” http://www.electrical-integrity.com/Paper_download_files/DC10_12-TH3_Novak-Mori-Resso.pdf Scattering Matrix: Pros and Cons
Pros: • Frequency-domain instruments for scattering parameters are readily available • Can measure microohm AC impedances • Many models are available in Touchstone format
Cons: • Proper termination of all ports is necessary • Need extra steps to do low-impedance measurements with regular Vector Network Analyzer (VNA) • VNAs don’t measure at DC
Important note: when modeling a design, ports that were unused in the measurements need to be open circuited and reduced The Vector Network Analyzer
Basic construction:
• Swept-frequency sinewave source
• Synchronous receivers
• Directional couplers to separate transmitted and reflected signals
The VNA measures the ~ Scattering matrix:
푏 푏2 b 1 1 b2 푆11 = 푆22 = 푎1 푎2
푏2 푏1 a1 a2 푆21 = 푆12 = 푎1 푎2
Port 1 Port 2 How to Measure PDN with Scattering Parameters; Challenges
• Many PDN components and systems need cable(s) to connect to the instrument(s) • Typical calibration requires coaxial connection － Not available on PDNs • There is some interconnect beyond the coaxial calibration plane － Uncalibrated discontinuity • One-port reflection measurement is not accurate for low impedances • Two-port shunt-through connection helps
－ Throw away S11 and S22 and use S21 (or S12) • But it creates cable-shield ground loop － Requires extra components or circuitry
See “Preamplifier Options for Reducing Cable-Braid Loop Error,” http://www.electrical-integrity.com/Quietpower_files/QuietPower-48.pdf Measured S Parameters of PDNs; How to Use
Throw away reflection terms, use transmission term(s)
For small impedances: For large impedances: Two-Port Shunt-Through Two-Port Series-Through
Cable: ZVNA Cable: ZVNA Cable: ZVNA ZDUT Cable: ZVNA
VNA Port1 VNA Port2 VNA Port1 VNA Port2 ZDUT
ZVNA S21 1− S21 Z DUT = Z DUT = 2ZVNA 2 1− S21 S21 How to Simulate PDN with Scattering Parameters
• Mimicking measurement is needed only if correlation to actual measurement details is required • Simulate impedance directly • If you simulate with scattering parameters, you can － Use reflection elements of the matrix － Leave out connection discontinuities － Use floating source and/or receiver How to Simulate PDN with Scattering Parameters; Capacitor Models
Accurate S-parameter model of the GRM32ER60J476 capacitor in Shunt mode
1 2 3
1: One-port model 2: Two-port Shunt model 3: Two-port Series model How to Simulate PDN with Scattering Parameters
Measured and simulated impedance of Murata GRM32ER60J476
Impedance magnitude [Ohm] 1.00E+02 Capacitor S-parameter models are available in: 1.00E+01 • Two-port Sunt and • Two-port Series 1.00E+00 Shunt-S21 Shunt-S11 connection schemes. Series-S21 1.00E-01 Series-S11 Measured-S21 1.00E-02
1.00E-03 1E+2 1E+4 1E+6 1E+8 1E+10 Frequency [Hz] How to Simulate PDN with Scattering Parameters; Normalization Impedance
S-parameter model S21 of the GRM32ER60J476 capacitor with different normalization impedances
• S-parameters need a
normalization impedance (Z0) • In SI, Z0 is selected to be close to the system impedance • In PI, there is no ‘typical’ impedance • In PI, system impedance can vary significantly with frequency • For a single component, normalization impedance can be in a wide range • For complex PDN systems, 0.1 Ohm normalization impedance is typical Summary and Conclusions
• We need to be familiar with both S and Z parameters • Use S parameters in measurements • Use Z parameters in design and validation • As long as linearity and time invariance is guaranteed, time and frequency domains are also interchangeable • Using S or Z parameters is just approximation • If needed, source and load conditions can be taken into account samtec.com/geekspeek [email protected]