a simulator for high-speed backplane transceivers
TRANSCRIPT
A Simulator for High-Speed Backplane Transceivers
Dianyong Chen, Bo Wang, Bangli Liang, and Tad Kwasniewski Department of Electronics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada
Abstract
A number of simulation methods dealing with
measured S-parameters have been developed for the simulation of high-speed backplane transceivers and printed circuit board signal integrity analysis. Although these methods implemented in many circuit simulators provide correct system-level and transistor-level simulations, they usually fail to give correct results for many challenging backplane channels. This paper discusses a simulator that we developed which can give correct simulation results for those channels. It can also process those channel model files to allow commercial circuit simulators to give correct results. 1. Introduction
The demand for higher data rates and integration in modern communications equipment has increased the transmission rate over electrical backplanes and printed circuit board (PCB) to gigabits per second or even tens of gigabits per second [1]. The conventional lumped circuit models for backplanes and interconnections on PCB become inadequate. Measured S-Parameters are quickly becoming the standard method in PCB signal integrity analysis and backplane transceiver design. A number of techniques have been developed over time to use frequency domain S-parameters to simulate transient system response and design equalization circuits. These methods include convolution, rational approximation, and equivalent circuits. Each method has its own bias and makes compromises between accuracy and performance [2]. Many circuit simulators dedicatedly implemented all these methods and give correct simulations for high-speed backplane transceivers. However, none of these methods is based on the physical models of electrical backplanes. Many commercial simulators do not give correct simulation results for challenging backplanes. The left part of figure 1 shows the magnitude of S11 and S21 of a 56-inch backplane on FR4. The right part of figure 1 shows the simulated time domain response given by a
very popular commercial circuit simulator when a 100-picosecond rectangular voltage pulse is applied to the differential backplane through two 50-ohm load resistors; and another two 50-ohm load resistors are impedance-matched at the outputs. The magnitude of the input voltage source is 1-volt. The time domain pulse response is obviously incorrect because the channel is at least 56-inch long, even free-space light will take 4.74 nanoseconds to travel from one end to the other end. The simulation however gives an output almost without any delay. When rational interpolation method provided by the circuit simulator is tried, the simulator fails to give poles and zeros. Matlab rf toolbox fails to give rational fit results either.
Figure 1. S11 and S21 of a 56inch FR4 backplane and the simulated time-domain pulse response given by a commercial circuit simulator
When a discrete-time finite impulse response (FIR) pre-emphasis equalizer is implemented, the popular circuit simulators that we have used encounter more problems. One of the problems is that the outputs go to infinity, although there is no feedback. The root cause of the problems is the measured S-Parameters. The S-Parameters are experimentally measured and saved in a file, for example, a 4-port S-Parameters file in touchstone format (.s4p). The file however does not necessarily contain data at frequencies that are essential for correct simulation. A circuit simulator usually gets those missing data by performing interpolation and/or extrapolation using numerical methods such as polynomial line fit and rational fit. Despite the advancement of those numerical methods they sometimes lead to incorrect simulation results because they do not look into device physics of the backplane. Polynomial line fit does not guarantee
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UKSim 2009: 11th International Conference on Computer Modelling and Simulation
978-0-7695-3593-7/09 $25.00 © 2009 IEEE
DOI 10.1109/UKSIM.2009.88
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causality and passivity of a backplane. Rational fit uses poles and zeros to fit the transfer function derived from the measured S-Parameters. The magnitude of the transfer function of a pole is proportional to the reciprocal of frequency if the frequency is much higher than the pole. This does not follow the natural trend of a backplane. The magnitude of the transfer function of a piece of backplane however decreases almost exponentially with frequency [3]. In addition, measured S-Parameters contain noises and system errors of the experimental setup that is used to measure the S-Parameters, especially at frequencies when the attenuation of the backplane is very large. This phenomenon can be clearly observed in figure 1.
We have developed a simulator that can correctly perform behavioral simulation for very challenging channels. As a cheap and practical solution, it can also properly process the channel model files to allow commercial simulators to give correct simulation results. This paper first briefly reviews the transmission line theories of backplanes. The section that follows explains why rational fit is inefficient for backplane channels. The remaining sections discuss the design and simulation of a 2X oversampled FIR pre-emphasis backplane transceiver. The last section compares the simulation results given by our simulator, a popular commercial circuit simulator and experimental results. 2. Transmission line theories of backplanes
A piece of backplane is usually some strip lines connected with some connectors. According to transmission line theory, the strip line can be described with distributed serial impedance and shunt admittance in equation (1) and equation (2).
σμπμπ 0
210222 fjggfjz += (1)
10 /)tan(2 gjfy reffr δεεεπ += (2)
where z and y are impedance per unit length and admittance per unit length, respectively, g1 and g2 are parameters decided by the transmission line geometries and their expressions are given in [4] and [5], respectively, εr is relative permittivity of the dielectric,
effrε is the effective dielectric constant of the substrate,
σ is the conductivity of the strip, tanδ is the loss tangent of the substrate, and ε0 and μ0 are permittivity and permeability in free space, respectively. The propagation constant γ=α+jβ and the characteristic impedance Z0 can be derived from equation (1) and (2). It can be proved from the equations and it has been
wide accepted that at low frequencies skin effect dominates the attenuation factor α and it is approximately proportional to the square root of frequency. At high frequencies, the loss tangent of substrate dominates α and it becomes approximately proportional to frequency. 3. Inefficiency of rational fit
Rational fit is very effective for curve fit for broadband data. Equation (3) defines a rational fit function.
fDelayjn
k k
k eDAfj
CfjF π
ππ 2
1 2)2( −
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
−= ∑ (3)
Where Ak, Ck, D, and Delay are constants, and f is frequency. For each pole Ak, the magnitude of the transfer function of this pole is approximately Ck/Ak when the frequency is much lower than the pole, and it is proportional to the reciprocal of frequency if the frequency is much higher than the pole. The magnitude of the transfer function of a piece of backplane, as discussed in section 2, however decreases almost exponentially with frequency and can be expressed in equation (4).
xvfjf eefjF
παπ2
)2(−− ⋅∝ (4)
where α is attenuation factor, v is the electromagnetic wave phase velocity in the backplane, and x is the length of the backplane. Therefore, for backplane, it is not efficient to use rational fit. It is not uncommon to use hundreds of poles to fit a highly lossy backplanes. It becomes more inefficient when connector discontinuity, influence of via holes and system noise and errors of the measurement setup are not trivia in the measured S-parameters file. 4. Extrapolate and interpolate data
Therefore, we stay with convolution methods that use inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT). To use FFT and IFFT to get the time-domain transient response, we need to extrapolate S-Parameters to very highly frequencies and very low frequencies, because the measured S-parameters file contains no data at those frequencies. We may also need to interpolate the measured S-Parameters. However, this is relatively rare, because the measured S-parameters usually have very fine frequency steps. From equation (4) we notice that if we take the logarithm of the transfer function, it becomes a linear function of frequency. Although
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crosstalk, discontinuities, and system noises make the transfer function not a perfect line, it is much more efficient to perform curve fit after we take the logarithm of the transfer function.
There are important rules to follow in extrapolating and interpolating the measured S-parameter files. Firstly, the loss at DC and very low frequencies should be negligible; secondly, the backplane is a passive component, the extrapolated S-parameters and the interpolated S-parameters must satisfy passivity; thirdly, the S-parameters must satisfy causality; in addition, there should be a method to reduce system noises and system errors in the S-Parameters file.
After extrapolating and interpolating the measured S-parameters, we can use them to derive system responses in the frequency-domain and time-domain. Once the system responses are known, we can use them to design various kinds of transceivers, including FIR pre-emphasis equalizer and decision feedback equalizer. The convolution methods used in our simulator are also used in commercial circuit simulators. Therefore, many commercial circuit simulators can give correct transistor level simulation results for backplane transceivers when using the S-parameters files processed by our simulator. 5. Equalizer design principles Fig.2 shows a popular backplane transceiver structure. A hybrid analytical model to describe this transceiver is given in Fig.3 [7].
Data Source
Channel Encoder
Tx Equalizer
ReferenceCMU
Rx Buffer
Clock and Data Recovery
Tx DriverPISO
Data Sink
Channel Decoder
Rx EqualizerSIPO
Clock Data
Figure 2. Block diagram of a popular backplane transceiver
The design of equalizer of a backplane transceiver is to reduce the side effects of various impairments introduced by the channel, especially inter-symbol-interference (ISI). From a system’s point of view, a successful transmission means the received sequence â(k) is a delayed version of the original sequence a(k). This is analytically expressed as
⎩⎨⎧
=≠
=+0
0
nn when ,nn when ,0
)(α
φnTh (5)
where h is the time domain channel response, n0T+φ is the delay, and α is the amplitude of the received
sequence. When expressed in the frequency-domain, it becomes
αωω =−⋅ ∑∞
−∞=
)(10
kkH
T (6)
In equation (6) we use a new coordinate where the time origin is at n0T+φ. The reason is that if no sample is made at the time origin, the conversion from a continuous-time Fourier transform (CFT) to a discrete-time Fourier transform (DFT) will be unnecessarily complicated. In practice, some systems may apply nonlinear operations such as modulus in duobinary or digitization in decision feedback equalization. In those systems the equivalence between equation (5) and (6) does not hold. A good method applied in those systems is to define a target response for the rest of the system except for the nonlinear parts.
Figure 3. A hybrid analytical model of a popular backplane transceiver
The equalizer design can be carried out in the time-domain or the frequency-domain. However, the time-domain simulation is indispensable for example to show eye-diagrams and to track error propagation in DFE.
6. Design of A 2x Oversampled FIR Pre-emphasis NRZ equalizer
The first step to design a 2X oversampled FIR pre-emphasis NRZ equalizer is to convert the measured S-Parameters into voltage or current transfer function. In order to do this, we need to define the transceiver circuit topology. We use two baud rate FIR pre-emphasis branches to form a half-rate structure. Therefore, for a 10-gigabit/second transmission our rectangular waveform is still 100 picoseconds. The conversion from S-Parameter matrix to a transfer function is done as follows. First, the voltage and current at a given port (assume at port n) has a relationship as:
)()()()( nZnInEnV S⋅−= (7)
where ZS(n) is the internal impedance of the stimulus E(n) at port n. The voltage V(n) and current I(n) can be expressed with the incident wave a(n) and reflected wave b(n) as follows:
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[ ])()()( 0 nbnaZnV +⋅= (8)
[ ] 0/)()()( ZnbnanI −= (9)
where Z0 is the characteristic impedance of the backplane. Solving the equations, we can express a(n) and b(n) in terms of V(n) or I(n). If we substitute a(n) and b(n) with V(n) in the definition of S-Parameters, we get the voltage at any port; If we substitute a(n) and b(n) with I(n), we get the current at any port. We can define the voltage transfer function from port m to port n as
)(/)( mVnVVTF mn = (10)
For differential backplane we usually define
[ ] [ ])()(/)()( nEmEkVjVVTF mnjk −−= (11)
Once the voltage or current transfer function is obtained from the measured S-parameters file, we need to extrapolate and extrapolate the transfer function to low frequencies and high frequencies and filter out measurement errors. To do this, we need to look into the device physics of backplane, which has been discussed in section 2 and section 4.
The second step is to get the time-domain pulse response of the backplane channel. In this step we should check if the transfer function is causal. We should force it to be causal if it is not [6]. In addition, we should take into account the rectangular waveform as part of the channel. We define the frequency-domain product of the transfer function of the channel and the continuous-time Fourier transform of the rectangular waveform as “channel response”.
The third step is to define the target response of the 2X oversampled FIR pre-emphasis NRZ equalizer. As an example, the target response is defined as a raised cosine function.
⎩⎨⎧
<=ΩΩ+=Ω
1 when ,)cos(1otherwise when ,0
)(π
Fg (12)
where Ω is the normalized frequency relative to the data rate. Once the circuit topology is fixed, Fg(Ω) should be multiplied with a constant to reflect maximum output power constraint. In the fourth step we define a time offset of the peak of the channel pulse response to the time origin of the target response. We need it to avoid unnecessary complicated operations to convert CFT to DFT. We also need this offset to guarantee a causal FIR equalizer. This time offset is added to the frequency-domain channel response as a positive phase shift.
In the fifth step we should sample the continuous-time channel response. This is necessary if we use time-domain methods to derive the filter coefficients. One of such time-domain method is to sample both the time-domain channel response and the target response. The filter coefficients are derived by de-convoluting the channel response and the target response. Another time-domain method is least mean square (LMS). The first method only takes very limited number of samples of the time-domain channel response. It may fail to give optimum results for very challenging channels. The LMS method may ends up with large residual errors if the first tap is the main tap and this tap has the maximum coefficient. The reason is that it is unable to remove pre-cursor ISI.
Frequency-domain methods to derive the filter coefficients have many advantages over time-domain methods. The DFT of the channel response (HD) is a folded version of its CFT (HC).
)(1)( 0ωωω ∑∞
−∞=
−⋅=k
CD kHT
H (13)
The frequency-domain filter response is the division of the target response by the folded channel response.
)()()(
ωωω
DHTgFilter = (14)
When we perform inverse discrete Fourier Transform of the frequency-domain filter response, we get the coefficients of the filter. The coefficients are usually very long and should be truncated near the peak coefficient. For a 2X oversampled equalization system, it should meet the famous Nyquist’s first criterion if it is decimated to 1X. 7. Results on a 2X oversampled FIR pre-emphasis
Figure 4. Simulated pulse responses given by a popular circuit simulator using s4p files before and after processed with our simulator
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Figure 4 shows the simulated time domain output pulse responses given by a very popular commercial circuit simulator. The channel and stimuli are the same as discussed in section 1. The waveform using the measured S-parameters file that has been processed by our simulator shows correct pulse response. The very small ripples near the time origin are not non-causal responses. In fact, they reflect the measurement errors in the S-Parameters file. The measurement errors were generated in the output ports. Therefore, the ripples need not to have a delay that is larger than the minimum delay time of the backplane. The errors can be suppressed in our simulator and the processed s4p file if we set a threshold and a frequency range of the errors. They are deliberately left here to reflect system imperfection.
Figure 5. Output eyediagram simulated by a popular circuit simulator
Figure 5 shows a commercial circuit simulator simulated eyediagram of the received signals of a highly lossy 40-inch backplane transceiver with a 2X oversampled FIR pre-emphasis equalizer. The bit rate is 10Gb/s. The input data pattern is PRBS9. The maximum output power constraint is a maximum voltage swing of 600mV over 50ohm load resistors. Figure 6 shows the simulation given by our simulator. An experimentally measured eyediagram is shown in figure 7. The 3 figures show similar results.
Figure 7. Measured eyediagram, 20ps/div, 20mV/div
0 50 100 150 200
−50
0
50
Time (ps)
Vol
tage
(m
V)
Figure 6. Output eyediagram simulated by the simulator proposed in this paper 8. Conclusion
We have introduced some advanced techniques to improve mixed signal high-speed transceiver simulation. Our simulator can give correct simulation results for many channels that some advanced commercial circuit simulators fail. It can also process those channel model files to allow commercial circuit simulators to give correct results. This simulator when used together with commercial simulators gives correct system-level and transistor-level simulation results. References [1] K. Krishna, D. A. Yokoyama-Martin, A. Caffee, C.
Jones, M. Loikkanen, J. Parker, R. Segelken, J. L. Sonntag, J. Stonick, S. Titus, D. Weinlader, and S. Wolfer, “A multigigabit backplane transceiver core in 0.13-μm CMOS with a power-efficient equalization architecture,” IEEE J. Solid-State Circuits, vol. 40, no. 12, Dec., 2005, pp. 2658-2666.
[2] V. Heyfitch, V. D. Zdorov, G. L. Pratt, and S. Azgomi, “Fast Time-Domain Simulation of 200+ Port S-Parameter Package Models, ” in DesignCon 2006.
[3] J. H. Sinsky, M. Duelk and A. Adamiecki, “High-speed electrical backplane transmission using duobinary signaling,” IEEE Trans. MTT, vol.53, no. 1, pp. 152-160, Jan. 2005, pp. 152-160.
[4] K. C. Gupta, R. Garg and I. J. Bahl. Microstrip Lines und Slotlines. Dedham, MA, Artech House, 1979.
[5] H. A. Wheeler, “Transmission line properties of parallel strips separated by a dielectric sheet,” IEEE Trans. Microwave Theory Tech., vol. MTT-13, pp. 172-185, 1965.
[6] P. A. Perry, and T. J. Brazil,"Forcing causality on S-Parameter data using the Hilbert transform," IEEE mircowave and guided wave letters, Vol. 8, No. 11, Nov. 1998, pp.378-380.
[7] J.W. Bergmans. Digital Baseband Transmission and Recording. Kluwer Academic Publishers, ISBN:0792397754
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