DesignCon 2013
A Fast and Inexpensive Method for PCB Trace Characterization in Production Environments
Peter J. Pupalaikis, Teledyne LeCroy
[email protected] Dr. Kaviyesh Doshi, Teledyne LeCroy
[email protected]
Abstract A fast and cost-effective method for performing loss measurements on differential traces in a printed circuit board manufacturing environment utilizing two-port structures and measurement instruments is provided. The method is an alternative to the SET2DIL method presented by Intel two years ago at DesignCon.
Patent Disclosure Portions of this document are the subject of patents applied for.
Biography Pete Pupalaikis was born in Boston, Massachusetts in 1964 and received the B.S. degree in electrical engineering from Rutgers University, New Brunswick, New Jersey in 1988. He joined Teledyne LeCroy (formerly LeCroy Corporation), a manufacturer of high-performance measurement equipment located in Chestnut Ridge, New York in 1995 where he is currently Vice President of Technology Development. He currently manages integrated-circuit development, signal processing technology development and intellectual property. Prior to LeCroy he served in the United States Army and has worked as an independent consultant in embedded systems design. Mr. Pupalaikis has twenty nine patents and numerous publications in the area of measurement instrument design and is a member of Tau Beta Pi, Eta Kappa Nu and the IEEE signal processing, instrumentation, and microwave societies. In 2013 he was elevated to IEEE Fellow for his contributions to high-speed waveform digitizing instruments. Kaviyesh Doshi received his Ph.D. in electrical engineering from University of California, Santa Barbara in June 2008. He joined Teledyne LeCroy, a manufacturer of high-performance measurement equipment located in Chestnut Ridge, New York in August 2008, as a Research & Development Engineer. At Teledyne LeCroy, he has been involved in design and development of SPARQ – a TDR based instrument for measuring s-parameters. He has filed patents in the area of de-embedding and signal processing for time domain network analyzer.
Introduction ATELY, A LOT OF PERFORMANCE is demanded from critical traces on a printed circuit board (PCB).
L
These performance demands require differential traces, low loss, and controlled impedance. PCB manufacturers produce high performance boards by offering improved materials and tighter dimensional tolerances. This allows board designers and engineers to design faster and denser on-board communications channels. The traditional language of communicating specifications between design and development engineers and the board manufacturers is the language of materials and dimensions. This means that boards are specified with various material stackups dictating material and thickness along with dimensions for widths, lengths and diameters of holes, pads and traces. As such, the specifications are mechanical and material in nature. As boards carried faster and faster signals, design engineers became focused on impedance and loss. Since these specifications are functions of materials and dimensions, it was incumbent upon them to make the correlation between impedance and loss and other electrical characteristics and translate these specifications into materials and dimensions that get communicated to board manufacturers. The board manufacturer might not know why or even care why certain materials, dimensions, and dimensional tolerances were being specified; they just knew that they were being relied upon to meet the specs. In the ideal world, from an electrical engineers perspective, the electrical characteristics of the board are specified to the board manufacturer. This is unrealistic in that this is not the board manufacturers area of expertise. To come closer to this ideal world, the design of the board comes from the board designer and certain electrical characteristics are measured and verified. Previously, this was done only in a design phase with the hope that the board manufacturer produces a consistent product. Lately, the goal has become to measure electrical characteristics on the production line as a board is being manufactured. Again, the main characteristics required are those of impedance and loss. Since the standard transmission medium today is differential, this requires the measurement of fourport trace structures. A reasonable plan is to produce these test structures in an area of each PCB panel produced. This area is called a test coupon and does not appear on the final circuit board. The testing requirements for boards produced in this manner create some difficult problems: 1. The popular method of measuring performance of microwave devices (which these differential traces have become) is to use the vector network analyzer (VNA). The VNA is a very expensive measurement instrument well suited for laboratory measurements. It is not well suited for production measurements. 2. The popular method of measuring performance of high-speed devices is to use time-domain reflectometry (TDR)1 . TDR instruments are less expensive instruments, but are also not well suited for production measurements. 3. The measurement of four-port structures is slow using TDR or VNA techniques. Several solutions are in use today for the purpose of impedance measurement. Impedance measurement requires a one-port or two-port (actually viewed as a one-port differential-mode) instrument. Impedance measurements can be performed easily with TDR. Today, the requirement has been extended to loss and the speeds have increased. High-speed loss measurements would generally require a four-port VNA. 1
We will use TDR and time domain transmission (TDT) interchangeably.
1
termination characterization structure termination structure
differential trace element .
trace characterization structure Figure 1: SET2DIL Characterization Structures
In 2010, a DesignCon paper was presented for a simple, cost effective method of accomplishing loss measurements in PCB traces called SET2DIL [1]. The single-ended TDR/TDT to differential insertion loss (SET2DIL) method provides for loss measurements on differential structures using two-port singleended measurements and enables TDR instruments to make these measurements. Since its inception, the SET2DIL algorithm has successfully undergone empirical round-robin testing and is included in an (formerly named) Institute for Printed Circuits (IPC) specification pertaining to PCB loss measurements [2]. The SET2DIL algorithm, while useful, is heuristic in nature and does not provide for mathematical insight into accuracy expectations and sensitivity to measurement uncertainty. This paper presents an algorithm that provides an alternate form of SET2DIL deriving from mathematical foundations based on s-parameters and network theory. Like SET2DIL, it provides for loss measurements that can be correlated with those made by a four-port VNA. The goal is for a rigorous absolute loss measurement like that which would be obtained from a four-port VNA measurement while retaining the two-port, inexpensive measurement characteristics of SET2DIL.
SET2DIL Test Structures The test structures recommended for use with SET2DIL are shown in Figure 1. Here we see two actual measurement structures, a termination characterization structure and a trace characterization structure. The idea is that the loop-back termination structure is contained identically in both characterization structures. Measurements of the termination characterization structure are used to determine the characteristics of the loop-back termination structure, and these characteristics are used in conjunction with measurements of the trace characterization structure to determine the characteristics of the trace. As such, the trace is the device under test (DUT) and the characteristics of the trace are the end goal of the measurement. SET2DIL is concerned with measurements of loss of this trace. 2
P
C
1 P
2 S
3
4
C
Figure 2: Problem Definition
. Measurement Problem Description We endeavor to solve the same problem as SET2DIL, but as an s-parameter measurement and model determination problem. While the VNA is the s-parameter measurement instrument of choice, many TDR instruments are also capable of measuring s-parameters. We will also be using the SET2DIL test structures. The algorithm derived here is therefore usable with any two-port instrument capable of measuring s-parameters. The s-parameter measurement problem is shown in Figure 2. Here we see block diagram representations of test structures shown in Figure 1. In Figure 2, the s-parameters of the probe (and probing structure), DUT and termination structure are reference by P, S and C respectively. Ignoring for the moment the problem of determining the characteristics of the probe P and assume for the moment that direct measurements are possible with probe de-embedding. This means that the measurement of the top network in Figure 2 is a direct measurement of C and the problem reduces to the determination of S given a measurement of the bottom network and the knowledge of C. The measurement of the bottom network is a two-port network measurement which contains four sparameters, or in other words, four equations. The unknown S is a four-port network and therefore contains sixteen unknown s-parameters. So there is an under-constrained system of four equations with sixteen unknowns. Figure 3 demonstrates the problem for a given frequency point from a signal-flow diagram standpoint. Given s-parameter measurements at nodes n1 , n2 , n3 , and n4 , one knows the relationship between reflected waves at n2 and n3 with respect to incident waves e1 at node n1 and e2 at node n3 . If M represents, for a given frequency, two-port s-parameter measurements of the trace structure where nodes n1 and n2 represent port 1 and n3 and n4 represent port 2, then M11 is the value at node n2 and M21 is the value at node n4 when e1 = 1 and e2 = 0. Similarly M12 is the value at node n2 and M22 is the value at node n4 when e1 = 0 and e2 = 1. This is what is meant by four equations. If the flow diagram is solved, these four values in M are found to be functions of the sixteen unknown s-parameters S and the presumed known s-parameters of the termination C. 3
e1
n1
S21
n5 S41
S11
S22
S31
n2 S13 e2
C11 n6
S12
S32 S42 C 12
S23
n3
n7 C22
S43 S24
S33
C21
S44
S14
n4
S34 n8
. Figure 3: Signal Flow Diagram Problem Definition
Mixed-mode Problem A first step that can be taken to solve the problem is to convert the problem from single-ended to mixedmode [3]. Without loss of generality, certain port numbering can be enforced so that certain equations can be directly employed. If, for a two-port network with single-ended s-parameters denoted by E, it is enforced that port 1 is the positive port and port 2 is the negative port associated with an equivalent mixedmode device whose mixed-mode s-parameters are denoted by M and whose port 1 is the differential-mode port and port 2 is the common-mode port, then these s-parameters are related by (1) and (2).
1 −1 −1 1 − 1 − → 1 1 −1 −1 → E =M 1 −1 1 −1 2 1 1 1 1 −1 1 −1 −1 1 1 1 1 −1 −1 − → → − 2 1 −1 1 −1 M = E 1 1 1 1
(1)
(2)
Similarly if, for a four-port network with single-ended s-parameters denoted by E, it is enforced that ports 1 and 2 are the left and right, respectively positive ports and that ports 3 and 4 are the left and right, respectively negative ports associated with an equivalent mixed-mode device whose mixed-mode s-parameters are denoted by M and whose ports 1 and 2 are the left and right, respectively differentialmode ports and whose ports 3 and 4 are the left and right, respectively common-mode ports, then these s-parameters are related by (3) and (4). 4
1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0
1 2 1 2
1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0
0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0
0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0
−1 0 1 0 0 0 0 0 −1 0 1 0 0 0 0 0
−1 0 1 0 0 0 0 0 −1 0 1 0 0 0 0 0
0 −1 0 1 0 0 0 0 0 −1 0 1 0 0 0 0
0 −1 0 1 0 0 0 0 0 −1 0 1 0 0 0 0
0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0
0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0
0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1
0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1
0 0 0 0 −1 0 1 0 0 0 0 0 −1 0 1 0
0 0 0 0 −1 0 1 0 0 0 0 0 −1 0 1 0
0 0 0 0 0 −1 0 1 0 0 0 0 0 −1 0 1
0 0 0 0 0 −1 0 1 0 0 0 0 0 −1 0 1
−1 0 −1 0 0 0 0 0 1 0 1 0 0 0 0 0
−1 0 −1 0 0 0 0 0 1 0 1 0 0 0 0 0
0 −1 0 −1 0 0 0 0 0 1 0 1 0 0 0 0
0 −1 0 −1 0 0 0 0 0 1 0 1 0 0 0 0
1 0 −1 0 0 0 0 0 −1 0 1 0 0 0 0 0
1 0 −1 0 0 0 0 0 −1 0 1 0 0 0 0 0
0 1 0 −1 0 0 0 0 0 −1 0 1 0 0 0 0
0 1 0 −1 0 0 0 0 0 −1 0 1 0 0 0 0
0 0 0 0 −1 0 −1 0 0 0 0 0 1 0 1 0
0 0 0 0 −1 0 −1 0 0 0 0 0 1 0 1 0
0 0 0 0 0 −1 0 −1 0 0 0 0 0 1 0 1
0 0 0 0 0 −1 0 −1 0 0 0 0 0 1 0 1
0 0 0 0 1 0 −1 0 0 0 0 0 −1 0 1 0
0 0 0 0 1 0 −1 0 0 0 0 0 −1 0 1 0
0 0 0 0 0 1 0 −1 0 0 0 0 0 −1 0 1
0 0 0 0 0 1 0 −1 0 0 0 0 0 −1 0 1
− → → − E =M
(3)
−1
− → → − M= E
(4)
We define a vectorize operator in (5) that serves to stack the columns successively in a vector such that: −−−−−−−−−−−−−−−−−−−→ S11 S12 · · · S1P S21 S22 · · · S2P .. .. .. = .. . . . . SP 1 SP 2 · · · SP P
S11 S21 .. . SP 1 S12 S22 .. . SP 2 .. . S1P S2P .. .
(5)
SP P One can obtain measurements of the termination structure and the trace structure by calibrating to the probe tips. This calibration can be performed as a second-tier calibration [4, 5] or preferably by using timedomain gating methods [6][7]. This allows measurements to the probe tip reference plane and resolves the probe de-embedding portion of the problem alluded to earlier. The problem then involves converting measurements of the trace structure and the termination structure to measurements of the unknown DUT. Figure 4 shows a block diagram of the problem once the measurements are converted to mixed-mode. Here we see a block diagram representation of the structures shown in Figure 1 where P refers again to the probe, SD and SC refer to the differential- and common-mode, respectively s-parameters of the DUT and CD and CC refer to the differential- and common-mode, respectively s-parameters of the termination structure. 5
CD P CC
P
1
SD
2
3
SC
4
CD CC
Figure 4: Mixed-mode Problem Definition In Figure .4, the top network represents two one-port networks consisting of the probe and the two-port termination structure separated into two mixed-mode one-ports with differential-mode s-parameters represented by CD and common-mode s-parameters represented by CC. It now exposes two one-port networks at the probe reference plane representing the fact that it is possible to obtain two one-port mixed-mode measurements of the aggregate network. The bottom network represents a two-port network consisting of the probe, a four-port DUT and a two-port termination, but now the four-port DUT is separated into two two-port mixed-mode devices representing the DUT with differential- and common-mode s-parameters represented by SD and SC. At the probe reference plane it now exposes two one-port networks at the probe reference plane consisting of a a differential-mode network consisting of a two-port differential-mode DUT connected to a oneport differential-mode termination with differential-mode s-parameters again represented by CD and a common-mode network consisting of a two-port common-mode DUT connected to a one-port commonmode termination with s-parameters represented by CC. The new mixed-mode problem is therefore split into two separate problems for the moment. The first problem is in determining the differential-mode s-parameters of the DUT and the second problem is determining, if desired, the common-mode s-parameters of the DUT. In the differential-mode case, one is provided with one-port s-parameters corresponding to the differential-mode measurement of the termination characterization structure and one-port differential-mode s-parameters corresponding to trace characterization structure. Both measurements are, through calibration, de-embedding, or other means, moved to the probe reference planes to form differential-mode measurements of the termination and the combination of the DUT and the termination. One may be provided similar information in the commonmode case. For each mode, the problem resorts to a signal flow diagram representation as shown in Figure 5. One may assume knowledge of C given a probe-de-embedded measurement of the termination and knowledge of M given a probe-de-embedded measurement of the combination of the DUT and the termination and that the four s-parameters corresponding to one mode are unknown. An assumption has already been made that mode-conversion is either not present or insignificant which 6
S21
1
S11 S22 M
C
S12
Figure 5: Two-port Signal Flow Diagram 1+ρ
L
ρ −ρ .
1−ρ
1−ρ −ρ ρ
L
1+ρ
Figure 6: Single Transmission Line Section is a reasonable assumption considering the goal of the measurement. If the DUT was totally arbitrary, this would not be a good assumption, but the DUT is constructed with dimensional symmetry and assumed homogeneity as it is intended to test properties associated with absolute and not relative dimensions and board properties. In other words, while the trace characterization being performed will be measuring effects of errors that can be, but are not limited to items like etch, plating thickness, trace thickness, board thickness and board material and dielectric properties, it is reasonable to assume that errors made in these areas are similar for each portion of the DUT allowing for symmetry to still hold. If this were not reasonable, the usage of a test coupon, which is intended to be representative of other traces in the board, would be of . limited use anyway. In any case, in subsequent discussion within this paper, opportunity is provided for determining the quality and correctness of these assumptions. A next step in reduction of unknowns is to assume reciprocity. This is a safe assumption because all passive networks are reciprocal and the goal here is to measure a passive network. Reciprocity means that the transpose of an s-parameter matrix equals itself, or for each port x and y we have Sxy = Syx . This reduces the number of unknowns to three. Finally, symmetry is assumed, for the same reason as the assumption of no mode-conversion. Symmetry is an extension of the fact that not only are each single-ended trace in the DUT required to be similar, but for each trace to look the same when viewed from either side. Symmetry in this case implies that S11 = S22 . This reduces the number of unknowns to two.
Transmission Line Models In dealing with these two unknowns, it is particularly useful to consider the DUT as a transmission line (for a given mode) as in the model shown in Figure 6. Here we see that a transmission line can be determined by two-unknowns at a given frequency: ρ and L. This model assumes that it consists of an impedance discontinuity at each end which is determined by ρ. This ρ relates the characteristic impedance of the line Zc and the arbitrarily chosen reference impedance Z0 , which is usually equal to 50 Ω, through (6): 7
1+ρ
1
ρ −ρ M
1−ρ
1−ρ
L
−ρ ρ L
C
1+ρ
Figure 7: Transmission Line Section in Measurement
ρ=
Zc − Z0 Zc + Z0
(6)
The model also assumes that in between the impedance discontinuity at the boundary of the line, that the line is homogeneous with a loss and delay characteristic L given by (7): L = eγ = eα+jβ = eα e−j2πf T
(7)
. In (7), eα contains the loss characteristic and β = −2πf T contains the phase or delay characteristic. α and β combine to form the complex thru coefficient L with the magnitude of L forming the loss characteristic and the argument of L forming the phase characteristic which, when considering the frequency forms the delay characteristic. The model in Figure 6 still contains two unknowns. When the model in Figure 6 is substituted into the diagram shown in Figure 5, the diagram shown in Figure 7 is obtained. If M is then solved for, (8) is obtained: M=
ρ2 C − ρ + L2 ρ − L2 C L2 ρ2 − L2 Cρ + ρC − 1
(8)
Let’s consider two cases. The first case has the line infinitely long so that L goes to zero2 . So: lim M =
L→0
ρ2 C − ρ ρ (ρC − 1) = =ρ ρC − 1 ρC − 1
(9)
The second case has the line terminated in C = ρ: lim M =
C→ρ
ρ3 − ρ + L2 ρ − L2 ρ ρ (ρ2 − 1) = =ρ L2 ρ2 − L2 ρ2 + ρ2 − 1 ρ2 − 1
(10)
One can see that if one could take a measurement of the system with either an infinitely long trace (causing L = 0) or with something to absorb all of the wave in the line after the interface or with a termination that matched the line characteristic impedance exactly (causing ρ = C), then one can take a direct measurement of ρ. There is a way to create this effect and that is by not considering the reflection 2
Certainly if there is loss in the line, the infinite line will dissipate the energy in all of the incident wave, but either way, the infinitely long line will cause the reflection to never return. This may not be readily apparent when looking in the frequency domain because there is an assumption that the incident wave has been present forever which means that even infinitely long lines cannot stop the reflection from eventually returning. But examining Figure 6 should convince you that the infinitely long line prevents the reverse transmission path, which is like setting L = 0 in the reverse path which causes the result to collapse to ρ.
8
amplitude
0.2
0
0.2 0
0.5
1
1.5
2
1
1.5
2
time (ns)
amplitude
0.2
0
0.2 0
0.5 time (ns)
.
Figure 8: Time-domain Waveforms
from the end of the structure. This is accomplished in the time domain by eliminating this reflection. To do this, one computes m = IDFT (M ), where IDFT refers to the inverse discrete Fourier transform. m is the time-domain equivalent of M which, for a homogeneous line shows an impulse near time zero due to the interface to the transmission line and multiple reflections thereafter at times that are multiples of twice the electrical length of the structure as shown in Figure 8. Two plots are shown in Figure 8. On top is a plot of the inverse discrete Fourier transform (IDFT) of the differential-mode measurement of the trace characterization structure with the probe de-embedded. Here one can see the trace showing a small bump downward at time zero, a relatively smooth area and a large downward bump at 1.175 ns. The small downward bump at time zero is indicative of an impedance discontinuity between the reference impedance (50Ω single-ended or 100Ω differential) and the differentialmode trace and the fact that it’s downward indicates that the differential-mode characteristic impedance is less than 100 Ω. The large downward bump at 1.175 ns locates the termination. Because the termination is for the most part a straight thru single-ended element, it represents a differential short, hence the large spike downward. On the bottom is a plot of the IDFT of the common-mode measurement of the trace characterization structure with the probe de-embedded. Here one can see the trace showing a tiny bump upward at time zero, a substantially smooth area and a large upward bump at 1.25 ns. The tiny upward bump at time zero is indicative of an impedance discontinuity between the reference impedance (50 Ω 9
single-ended or 25 Ω common-mode) and the common-mode trace and the fact that it’s tiny and upward indicates that the common-mode characteristic impedance is slightly more than 25 Ω. The large upward bump at 1.25 ns locates the termination. Because the straight thru single-ended element represents an open to a common-mode signal component, it creates the large upward spike. Note that the location in time of the termination is different for the differential- and common-modes which indicates a different mode propagation velocity and is indicative of coupling in the single-ended trace structure of the DUT. For a given mode, the termination can be grossly located by simply searching the waveform for the minimum value (for the differential-mode) or the maximum value (for the common-mode). Trimming the waveform to somewhat shorter than the termination location (like 80%) and computing the discrete Fourier transform (DFT) of the trimmed waveform provides a very good approximation of ρ as a function of frequency because it emulates the situation outlined in the limits of (9) and (10) because it simulates the condition in which the line is terminated in the line characteristic impedance and simultaneously simulates the condition in which the line is infinitely long. We call m ˆ the trimmed version of m. The four inch lines generally used with SET2DIL are fine for this approximation. Now that ρ has been determined in addition to C and M for a given frequency and mode, we return to (8) and compute L according to (11): (1 − ρC) (M − ρ) (11) (1 − ρM ) (C − ρ) To compute L, one needs to compute a square root of a complex number which must be performed carefully. The range of the principle branch of the complex square root is ±π/2 and the result should go between ±π so one needs some sort of phase unwrapping algorithm. The simplest way to deal with this is to use the value of Td which can be estimated from m and multiply the right side of (11) by ej2πf Td (remembering that Td is the time between the first and second reflection, but twice the electrical length), Td j2πf 2 to unwrap the phase on ±π boundaries, compute the square root, and then multiply again by e put the delay back in. With the knowledge of these unwrapping complexities and that other unwrapping methods can be used, one can write generally (12): L2 =
Td √ −2πf (1 − ρC) (M − ρ) 2πf T d 2 L≈e e (1 − ρM ) (C − ρ)
(12)
Better yet is to determining L for a double-length line. In SET2DIL it is customary to report loss for an eight inch line even though a four inch line is actually being measured. This sidesteps any issues with the complex square-root. To summarize the steps followed to this point, the process started with at least one-port measurements of one mode, usually the differential-mode of the trace structure. These measurements were taken for a number of frequency points. These measurements can be taken directly in the mode of interest or converted from single-ended or other measurements. These measurements were taken in a manner such that the probe was de-embedded or calibrated out of the measurement. The process also started with one-port measurements of one mode of the termination. These measurements were taken for a number of predetermined frequency points. These measurements were taken directly in the mode of interest or converted from single-ended or other measurements. These measurements were taken in a manner such that the probe was de-embedded or calibrated out of the measurement. Alternatively, the termination can be obtained from simulation, or idealized or obtained through some other method, but the point is that the s-parameters in the 10
mode of interest are determined. Given now one-port s-parameters of the trace structure and termination in a given mode at certain frequency points, the reflection coefficient for the mode of interest ρ at each frequency point is determined. The reflection coefficients are calculated at many frequencies and are calculated from the time-domain waveform corresponding to the one-port s-parameters of the trace structure by limiting the time length to remove the reflection caused by the termination and converting back to the frequency domain. Given now the reflection coefficient for each frequency point, the loss characteristic, which is the magnitude of L in (11) is computed for each frequency point. The loss and delay characteristics for the mode of interest encapsulated in the complex L at all of the predetermined frequencies are calculated using methods to preserve the phase information such as shown in (12). The determination of ρ and L for multiple modes allows the determination of a complete set of sparameters. The s-parameters for a given mode at a given frequency corresponding to the model shown in Figure 6 with a frequency dependent ρ and L are described by (13): 1 S= 1 − ρ2 L 2
(
ρ (1 − L2 ) L (1 − ρ2 ) L (1 − ρ2 ) ρ (1 − L2 )
) (13)
Note that (13) applies to any mode given that ρ and L are in that mode. If one defines ρd and Ld as describing the differential-mode transmission line and ρc and Lc as describing the common-mode transmission line, then the full mixed-mode s-parameters of the DUT are given by (14): S=
ρd (1−L2d )
Ld (1−ρ2d )
1−ρ2d L2d Ld (1−ρ2d )
1−ρ2d L2d ρd (1−L2d )
1−ρ2d L2d
1−ρ2d L2d
0
0
0
0
0
0
0 ρc (1−L2c ) 1−ρ2c L2c Lc (1−ρ2c ) 1−ρ2c L2c
0 2 Lc (1−ρc ) 1−ρ2c L2c ρc (1−L2c )
(14)
1−ρ2c L2c
Fitted Results for Improved Accuracy The mixed-mode s-parameters in (14) can be converted to single-ended s-parameters using (4). Because of the symmetries already imposed, these single-ended s-parameters will have the form in (15).
S11 S21 S31 S41
S12 S22 S32 S42
S13 S23 S33 S43
x1 S14 x2 S24 = S34 x3 x4 S44
x2 x1 x4 x3
x3 x4 x1 x2
x4 x3 x2 x1
(15)
(15) says that there are only four values x1 , x2 , x3 , and x4 that make up the single-ended s-parameters. As an optional step, these s-parameters can be fit to the measurement made of the trace characterization structure (with the probe de-embedded) to refine the calculation. To do this, we solve the flow diagram in Figure 3 to obtain the solution (16) for the nodes as a function of the unknown DUT s-parameters and the known termination element: 11
1 −x1 0 −x3 −x2 0 −x4 0
0 1 0 0 0 0 0 0
0 −x3 1 −x1 −x4 0 −x2 0
0 0 0 0 0 0 1 0 1 1 0 −C11 1 0 0 −C21
0 0 −x2 0 0 0 −x4 0 −x1 0 1 C12 −x3 1 1 C22
0 −x4 0 −x2 −x3 0 −x1 0
n1 n2 n3 n4 n5 n6 n7 n8
=
e1 0 e2 0 0 0 0 0
(16)
(16) can be expressed as (17): Gn = e
(17)
It is useful to reorder the nodal equations to arrange the nodes representing measured reflected waves n2 and n4 in the top two nodes and the measured incident waves n1 and n3 in the next two nodes. This is accomplished through the following permutation matrix in (18): 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 P= (18) 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 Using (18), one can see (19): Pn =
(
n2 n4 n1 n3 n5 n7 n6 n8
)T
(19)
Therefore, one can rewrite (17) as (20) PGPT Pn = Pe
(20)
(20) can be solved for the nodal values as in (21): [ ]−1 n′ = Pn = PGPT Pe Therefore one can write the solution in block matrix form as in (22): F 02 I [ ] = PGPT −1 I2 A 02 02 B
(21)
(22)
In (22), F is a 2 × 2 block matrix representing the measured s-parameters of the system, I2 is a 2 × 2 identity block matrix, 02 is a 2 × 2 block matrix of zeros and A and B are 2 × 2 block matrices representing incident and reflected waves on the termination structure. 12
Remember that G contains both the unknown s-parameters and the known termination structure sparameters. The unknown s-parameters can be solved in a nonlinear fashion using the Levenberg-Marquardt algorithm [8, 9, 10]. We define the column extraction operator which serves to extract column m as (23): S11 S12 · · · S1P S1m S21 S22 · · · S2P S2m (23) .. .. .. = .. . . . . . . . SP 1 SP 2 · · · SP P ∗m SP m We define the row extraction operator which serves to extract row m as (24): S11 S12 · · · S1P S21 S22 · · · S2P ( ) S S · · · S = .. .. . m1 m2 mP . .. .. . . SP 1 SP 2 · · · SP P m∗
(24)
We define the Hermitian operator as the conjugate transpose such that:
S11 S21 .. . SP 1
S12 · · · S22 · · · .. ... . SP 2 · · ·
S1P S2P .. .
H
=
SP P
∗ S11 ∗ S12 .. .
∗ S1P
∗ S21 ··· ∗ S22 · · · .. ... . ∗ S2P ···
SP∗ 1 SP∗ 2 .. .
SP∗ P
(25)
Given this defined notation, we define a function that returns what the measured s-parameters would ( )T be if x = x1 x2 x3 x4 represented the s-parameters of the unknown DUT. We also introduce the function G(x, C) = G which fills in the matrix G as a function of x and the s-parameters of the termination structure. In other words, given a set of s-parameters x, f (x) shown in (26) produces what the two-port measurement of the trace measurement structure would be: 02 [ ] T −1 I2 (26) f (x, C) = 02 PG (x, C) P 02 1 − → The goal is therefore to find the values of x such that f (x, C) differs from M by as little as possible (i.e. they match in a least-squares sense). Given this, we take a guess at what x (the s-parameters) might be and form the residual vector from the guess and the actual measurement M of the trace structure as in (27). The good news is that there is already a very good guess or estimate of x as calculated in the preceding discussion. −−−−→ − → r = f (x, C) − M
(27) [→ ] − Then we form a weights matrix that forms the weight of measurement M in the fit, where k, n ∈ k 1 . . . K and K = 4 representing the number of unknown s-parameters. The weights are arbitrarily chosen as in (28): 13
W = I4
(28)
Then we form the Jacobian matrix J as in (29) where Jk,n is the partial derivative of the kth element of −−−−→ f (x, C) with respect to the nth element of x: Jk,n
−−−−−−−−→ −−−−−−−−−−−−−−−−−−−→ [ ] [ ] ∂f (x, C) f (x + δ∗n , C) − f (x, C) = k ≈ k ∂xn ϵ
(29)
In (29), one may form an approximation to the partial derivatives, but an analytic form can be used as well. To form the approximation, we choose column n of δ as shown in (30) which is the 4 × 4 element identity matrix multiplied by some arbitrary, but generally small value of ϵ, like 0.001. δ = I4 ϵ
(30)
Then we form the approximate Hessian matrix as in (31): H = JT WJ
(31)
To implement Levenberg-Marquardt, we create a matrix D as in (32) which contains the diagonal elements of H, but is zero elsewhere: Dk,k = Hk,k
(32)
Levenberg-Marquardt steers the fit between Newton-Gauss convergence and a gradient walk through the use of the variable λ. When λ is zero, one has Newton-Gauss convergence. When λ is infinity, one has a gradient walk with an infinitesimally small step size. λ starts out as an arbitrary value, like 100. We define ∆x as the estimate of the amount that our guess at x is off by in (33): ∆x = [H + λD]−1 JT Wr
(33)
The plan is to subtract ∆x from x in the assumption that this will get closer to the correct value of x. In order to check this, we compute the original variance as (34): rH Wr σ = K One computes the new variance that we would get with the application of ∆x as (35): 2
2 = σnew
[−−−−−−−−−→ − →]H [−−−−−−−−−→ − →] f (x − ∆x, C) − M W f (x − ∆x, C) − M
(34)
(35) K If σnew < σ, then the step succeeded and we subtract ∆x from x and decrease λ to favor Newton-Gauss convergence on the next iteration. Usually, in this case, we divide λ by 10. Otherwise, if the step failed to reduce the variance, we keep the old value of x and increase λ to favor the gradient method with a smaller step size on the next iteration. With this new value of x which is a better estimate of the s-parameters, we go back to the residual calculation step with the new value of x and do it again. We stop when either the variance is low enough, the variance stops changing significantly, or the value of λ moves outside some predetermined upper and lower limit. 14
In this manner, one can refine the estimate of the s-parameters of the DUT. With this, the measure of the goodness of the assumptions of reciprocity and symmetry can be tested through the final value of σ which should ideally become zero if we obtained the values of the DUT perfectly such that f (x, C) = M. If the variance becomes zero, it would mean that one has found the DUT perfectly and that all of our assumptions of reciprocity and symmetry were perfectly correct. In practice, there will be slight errors in the assumptions of symmetry, in the measurement of M, and in the knowledge of C, so one will want to use an arbitrary threshold on σ to determine the goodness of the values obtained. At this point, one will have refined single-ended s-parameters of the DUT by applying our refined x to (15). One can convert these s-parameters back to mixed-mode using (3).
PCB Trace Characterization Although the goal of characterizing the DUT from an s-parameter measurement perspective has been accomplished, there are other goals of PCB measurements that are accomplished by converting the now obtained s-parameter data into measurement values used for pass-fail testing and other basic characterization. The differential- and common-mode DUT characteristic impedance can be calculated utilizing the impedance profile [6]. The impedance profile provides the characteristic impedance as a function of time (i.e. electrical length into the DUT). Choosing an arbitrary location allows calculation of the characteristic impedance, a useful characteristic The impedance profile can also be approximated by integrating the traces in Figure 8 and converting to impedance using (6). The loss is calculated directly in an impedance normalized environment and is already encapsulated in L. In other words, if L is plotted over frequency, it provides the true loss characteristic of the trace without the effects of the impedance discontinuity. It is useful to further fit the loss to a curve that is preferably a function of the square-root of frequency (to capture skin-effect type loss characteristics [11]) and a linear function of frequency (to capture dielectric type loss characteristics). This is done by, given N + 1 frequency points, n ∈ 0 . . . N where fn is the frequency in GHz of element n and Vn = 20 log (|Ln |) is the magnitude of the loss in decibels where Ln is the loss calculated in the preceding discussion at frequency fn . We then compute the values in (36), (37) and (38) [12]: Hn,0 = −fn Hn,1 = −
√
fn
[ ]−1 x = HT H HV
(36) (37) (38)
(38) is a vector such that x0 is the loss coefficient in dB per GHz and x1 is the loss coefficient in dB per root-GHz. If one knows the length of the DUT, these can be converted further to a loss per inch. The fitted loss is therefore calculated as: Vn′ = x0 fn + x1
√ fn
(39)
An example of loss √ fit is shown in Figure 9 where the differential-mode loss is calculated as 0.506 dB/GHz and 0.753 dB/ GHz and the common-mode loss is calculated as 0.586 dB/GHz and 0.649 15
Differential-mode Fit Comparison
magnitude (dB)
0
-5
-10
-15
-20 0
5
10 frequency (GHz)
15
20
15
20
Loss Calculated Fit
.
magnitude (dB)
0
Common-mode Fit Comparison
-5
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5
10 frequency (GHz)
Loss Calculated Fit Figure 9: Fitted Loss
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Figure 10: LeCroy SPARQ Measuring SET2DIL Structure Figure 11: Closeup of Probe Arrangement √ dB/ GHz. Any desired losses, like loss at a particular frequency for a given mode can be calculated from the fitted curve. The electrical length for each mode can be found by taking half of the values at the markers shown and if the length is known, propagation velocities for each mode can be found. These calculations can be refined using group delay calculations of the phase of L. Given propagation velocity, and impedance, one can calculate the inductance and capacitance per unit length and derive all sorts of board properties that are known to signal integrity (SI) engineers. One could also fit values of impedance and fitted loss characteristics as calculated in the preceding paragraphs to the measured characteristics of the DUT to fit an idealized model instead of the refined single-ended s-parameter calculation shown here. This fit would proceed in a similar manner with the values that were fitted for the single-ended s-parameters x replaced with the fitted or calculated values of ρ and L to obtain a least-squares fit to an ideal model.
Experimental Results A two-port setup as shown in Figure 10 was used to make measurements to validate our algorithm. A closeup of the probing setup is shown in Figure 11. Here we are using an Intel board which provides a number of SET2DIL test structures, including four-port structures to compare against. We show Cascade Microtech |Z| probes for our experimental results although results for the previous sections of this paper were taken with GGB Picoprobes. All of the SET2DIL measurements were made using a Tektronix Sampling Scope using the 50 GHz TDR head. All of the measurements to validate our algorithm were made using a Teledyne LeCroy SPARQ - a TDR based signal integrity network analyzer. The four-port measurements were made by Jeff Loyer at Intel using GGB Picoprobes and an Agilent E8363B PNA. The network analyzer in this case was calibrated at the end of the probes using calibration substrates. A comparison plot for microstrip traces is shown in Figure 12. The four-port microstrip trace is an eight inch long differential structure whereas the two-port SET2DIL microstrip trace is a four inch long differential structure. As mentioned earlier, the result of the SET2DIL algorithm and the proposed algorithm is modified to output the differential loss for an eight inch long differential microstrip trace. Figure 17
magnitude (dB)
0
-10
-20
-30
0
5
10 frequency (GHz) SD2D1 VNA (four-port) SET2DIL Proposed Algorithm (SPARQ)
15
20
Figure 12: Microstrip Four-port VNA Measurement vs. Two-port Loss Calculations
magnitude (dB)
0
-10
-20
-30
0
5
10 frequency (GHz) SD2D1 VNA (four-port) SET2DIL fitted Proposed Algorithm fitted (SPARQ)
15
20
Figure 13: Microstrip Four-port VNA Measurement vs. Two-port Fitted Loss Calculations
18
0.02
difference (dB/inch)
0.015 0.01 0.005 0 -0.005 -0.01
.
0
5
10 frequency (GHz)
15
20
Figure 14: Microstrip Loss Measurement Difference Between SET2DIL Calculation and Proposed Algorithm
12 shows the differential insertion loss of the four-port measurement, and the differential insertion loss calculated by the SET2DIL algorithm and our proposed algorithm. As mentioned in section PCB Trace Characterization, one can use a least squares fit to obtain a smooth value for loss L. The differential insertion loss can be now calculated from this fitted L and (13) on page 11. Figure 13 shows a comparison plot between the differential insertion loss from the four-port measurement and the differential insertion loss obtained by fitted L for both the SET2DIL algorithm and our proposed algorithm. The results of our proposed algorithm and the SET2DIL algorithm indicate good agreement. For clarity the difference between the insertion loss calculated by the two different methods is shown in Figure 14. Since the differential insertion loss will increase with the length of the trace, the parameter of interest for PCB manufacturers is insertion loss per inch of the trace. Hence the error shown in Figure 14 is divided by the length of the trace (eight inches). The difference in dB per inch between the two algorithms is quite small. The curve is bent because our fit is not linear as explained in section PCB Trace Characterization. A similar comparison of differential insertion loss for a stripline structure is shown in Figure 15, Figure 16 and Figure 17. Figure 18 is a plot of the phase obtained from the four-port measurement for the stripline trace and that obtained by the proposed algorithm and Figure 19 is a plot of group delay comparison. Note that the phase has been unwrapped and the nominal delay of 181 ps/inch has been removed to generate the plot in Figure 18. The phase and group delay are normalized by the length of the trace to show a phase per inch and a group delay per inch value. Here we see small disagreement in the phase due to the fact that our algorithm assumes linear phase. We see group delay variation at low frequency where we suspect that the VNA measurement’s lack of DC measurement capability is the culprit. As mentioned previously, our proposed algorithm produces a complete set of s-parameters for the trace measured. In order to confirm the generated model validity with respect to return loss, we show the corresponding differential-mode impedance profile for the stripline trace generated in Figure 20. Here we see 19
magnitude (dB)
0
-10
-20
-30
0
5
10 frequency (GHz) SD2D1 VNA (four-port) SET2DIL Proposed Algorithm (SPARQ)
15
20
Figure 15: Stripline Four-port VNA Measurement vs. Two-port Loss Calculations
magnitude (dB)
0
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-30
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10 frequency (GHz) SD2D1 VNA (four-port) SET2DIL fitted Proposed Algorithm fitted (SPARQ)
15
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Figure 16: Stripline Four-port VNA Measurement vs. Two-port Fitted Loss Calculations
20
0.02
difference (dB/inch)
0.015 0.01 0.005 0 -0.005 -0.01
0
5
10 frequency (GHz)
.
15
20
Figure 17: Stripline Loss Measurement Difference Between SET2DIL Calculation and Proposed Algorithm
300
5
0
5
.
.
group delay (ps/inch)
phase (degrees/inch)
10
0
5
10 15 frequency (GHz)
200
100
0
20
SD2D1 VNA (four-port) Proposed Algorithm (SPARQ)
0
5
10 15 frequency (GHz)
20
SD2D1 VNA (four-port) Proposed Algorithm (SPARQ)
Figure 18: SD2D1 Phase Calculated
Figure 19: SD2D1 Group Delay Calculated
21
150
impedance (Ω)
1.45 ns 100
50
0 .
86 Ω
0
0.5
1 1.5 time (ns) Two-port SET2DIL Structure Measurement Four-port Calculated
2 .
Figure 20: Stripline Differential Impedance Profile that the calculated impedance profile indicates an electrical length of 1.45 ns and an impedance of 86 Ω3 . The impedance profile is compared to that obtained from the two-port measurements of the SET2DIL structure (converted to differential-mode) where we see very good agreement up to the termination structure. Note that our proposed algorithm is capable of generating full common-mode s-parameters as well, but these results have not been shown.
Considerations of Results The experimental results shown in the previous section indicate very good correlation between the fourport measurements, our proposed algorithm and the SET2DIL algorithm. To achieve such good correlation we had to carefully consider certain details that we originally got wrong in the algorithm. The first concerns the calculation of ρ from the two-port measurement. As explained in section Transmission Line Models, the impulse response obtained as the time domain equivalent of the two-port SET2DIL structure is truncated and zeroed starting from just prior to the reflection from the termination. Remember, this was to emulate the condition where the trace is terminated in a termination that is matched to the characteristic impedance. This was necessary to obtain ρ. Originally, unlike the trace shown in Figure 20, we found poor agreement between the expected impedance profile and that obtained because we had not properly considered the impulse response points that occur before time zero. Keep in mind that the IDFT is used for obtaining the time-domain response and that the IDFT contains both positive and negative time. Although we expect the results to be causal, there is some residual response prior to time zero due to sampling effects [13] and due to small errors made in de-embedding the probe. We found that the retention of a small amount of time (approximately 2 ns) before time zero was required to achieve the proper agreement. The second consideration for the proposed algorithm regards probe de-embedding. As explained in the 3
The trace is advertised as 85 Ω but we had no way of comparing to the VNA measurement because could not easily generate a trustworthy impedance profile due to both lack of a DC measurement point and unequally spaced points.
22
magnitude (dB)
0.2 0 -0.2 -0.4 -0.6 -0.8
.
0
5
10 frequency (GHz)
15
20
Probe Gating Probe Gating & Thru Measurement Figure 21: Probe Response
section Mixed-mode Problem, the algorithm always expects s-parameter measurements with the effects of probe removed. This is especially required based on our assumptions of symmetry. For our results we used time-domain gating to de-embed the probes. As explained in [6][7], for simplicity, the gating algorithm uses a linear loss model for the gated structure. For our case this proved to be a limitation. Figure 21 shows the insertion loss of the probe using two different techniques. One is generated by using the gating algorithm with a linear loss model of 0.2 dB per GHz per ns of probe length. Although the loss model is linear, the gating algorithm does consider the impedance. The non-monotonicity shown is due to this consideration. Using this simplistic technique gives incorrect results for the differential insertion loss of the DUT because the insertion loss of the probe is not perfectly linear. To solve this issue, we combined measurements of the probe’s response obtained using the measurement of the SET2DIL thru structure with the gating algorithm. The red trace in Figure 21 shows the insertion loss of the probe obtained by this modified method. Using this de-embedding technique provides for a calculated result that matches the differential insertion loss obtained by the four-port measurement. As a final note we would like to point out the difference between loss and differential insertion loss. The differential insertion loss would have small ripples at low frequency due to the nature of the return loss and low loss value at lower frequencies. The actual loss without considering return loss due to resistive and dielectric loss effects would be smoother in comparison. This is a nuanced topic. To understand the subtlety, consider that S21 in Figure 5 is not the same as L in Figure 7. Figure 22 shows the difference between the actual loss per inch and the insertion loss per inch, which is very small despite the fact that the line is not 100 Ω. 23
difference (dB/inch)
0.04 0.02 0 -0.02 -0.04 0
5
.
10 frequency (GHz)
15
20
Figure 22: Loss vs. Differential Insertion Loss
Conclusion An alternative to the SET2DIL algorithm was presented in the paper. Experimental results were presented that showed that the algorithm presented and the SET2DIL algorithm provided similar results and the results in both cases matched the four-port measurements. Our algorithm was found to require accurate deembedding of the probes while SET2DIL seems insensitive to probe effects - an advantage of SET2DIL. SET2DIL, however, does require careful attention to deskew. The calibration and probe de-embedding required for our proposed algorithm removes the deskew considerations. We demonstrated the capability of our proposed algorithm to generate a full set of s-parameters for the trace which does provide for the extraction of virtually any parameter of interest while also providing for use in simulation. In this paper we did not address the issue of sensitivity to measurement uncertainty. The mathematical formulation of the new algorithm makes it more conducive to the analysis of the effects of measurement uncertainty, which shall be the subject of a future paper.
Acknowledgements The authors would like to thank • Jeff Loyer at Intel for providing the test boards and the four-port measurements that were used to validate the accuracy of the algorithm and for the time spent in his lab making measurements. • Jeff Meyerson at GGB Industries for giving us a prototype Picoprobe designed for use with the SET2DIL structure. • Anthony Lord at Cascade Microtech for loaning us a |Z| probe designed for use with the SET2DIL structure.
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References [1] J. Loyer and R. Kunze, “SET2DIL: Method to Derive Differential Insertion Loss from Single-Ended TDR/TDT Measurements,” in DesignCon, IEC, February 2010. [2] —, IPC-TM-650 Test Methods Manual. IPC, 3000 Lakeside Drive, 309 S,Bannockburn, IL 60015, 2.5.5.12 ed., July 2012. Rev. A. [3] D. E. Bockelman and W. R. Eisenstadt, “Combined differential and common-mode scattering parameters: theory and simulation,” IEEE Transactions on Microwave Theory and Techniques, vol. 43, pp. 1530–1539, July 1995. [4] A. Blankman, “Using 2nd Tier Calibration for Cable Fixture De-embedding,” LeCroy Application Note, 2011. [5] A. Blankman, “Measurements with the LeCroy SPARQ and Cascade Microtech Probes Using WinCal XE Calibrations,” LeCroy/CascadeMicrotech Application Note, 2011. [6] P. J. Pupalaikis and K. Doshi, “Method For De-embedding in Network Analysis,” May 2011. [7] A. Blankman, “De-embedding Gigaprobes® Using Time Domain Gating with the LeCroy SPARQ,” LeCroy/CascadeMicrotech Application Note, 2011. [8] K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” The Quartlerly of Applied Mathematics, vol. 2, pp. 164–168, 1944. [9] D. Marquardt, “An algorithm for least-squards estimation of nonlinear parameters,” SIAM Journal on Applied Mathematics, vol. 11, pp. 431–441, 1963. [10] K. Madsen, H. Nielsen, and O. Tingleff, Methods for Nonlinear Least Squares Problems. Technical University of Denmark, 2 ed., 4 2004. [11] H. Johnson and M. Graham, High-Speed Signal Propagation: Advanced Black Magic, p. 734. Prentice Hall, 1 ed., Feb 2005. [12] G. Strang, Introduction to Applied Mathematics, ch. 1, p. 39. Wellesley-Cambridge Press, 1 ed., Jan 1986. [13] P. Pupalaikis, “The relationship between discrete-frequency s-parameters and continuous-frequency responses,” in DesignCon, IEC, February 2012.
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