Chapter 1 Introduction
1.4 Contributions
1.4 Contributions
In this dissertation, an analysis method for the multiconductor-transmission-line (MTL) system and three novel CMF designs are proposed to improve the results of conventional ones. The details of the main contributions of the work are as follows:
1. An analysis method for MTL systems by applying modal decomposition and Kirchhoff’s circuit laws is proposed. Compared with the conventional works, this method can be applied to the cases in which the number of conductor changes, or additional terminations are included. An analysis flowchart is built to obtain the modal current as well as the line voltage/current wherever in the system. With this analysis method, SI problems (crosstalk and mode conversion) and EMI problems (noise from current flow, or radiated power converted from guided modes) can be predicted well, and solution to them will be clearer.
2. Instead of working on the system-level package as the previous works, a CMF designed in the chips is proposed to suppress the CM noise inside the chip package and to prevent the CM noise coupling or radiating among the circuits. Based on a second-order T model, the proposed CMF is co-designed with through-silicon vias (TSVs) in three-dimensional integrated circuits (3-D ICs) to compensate the side effect from the parasitics of TSV. The performance of CM suppression is also benefited from the co-design and can be improved further by applying the stacking-chip technique.
3. To achieve wider stopband or to enhance the suppressing level of CM, a CMF with three transmission zeroes (TZs) is realized. With the proposed theory and designing method, the intermediate TZ (or roughly the center frequency of the stopband) can be easily determined, and the separation of the other TZs can be controlled farther
or closer to make the stopband with larger bandwidth or higher suppressing level, respectively, for different kinds of applications.
4. An absorptive CMF (A-CMF) is proposed to dissipate the CM noise by the resistive elements. This design can prevent radiation caused by both the transmitted and reflected CM wave, which is necessary if the discontinuities exist before and after the CMF. Different from the conventional work realized in the integrated passive device (IPD) process, the proposed A-CMF can be implemented in the cost-efficient printed-circuit board (PCB) and have a better DM-to-CM-cutoff ratio.
Chapter 2
pter 2 Analysis of Current Distribution and Excited Radiation in Multiconductor- Transmission-Line Systems
In this chapter, an analysis method for multiconductor-transmission-line (MTL) systems by applying modal decomposition and Kirchhoff’s circuit laws is proposed. By defining the input impedance matrix in each section of the structure, this method can be applied to the cases in which the number of conductor changes, or additional terminations are included. Following the flowchart, the modal current as well as the line voltage/current can be all obtained. This analysis method can be applied to estimate voltage/current distribution, crosstalk, and mode conversion, which is helpful to solve the signal integrity (SI) and electromagnetic interference (EMI) problems when designing.
2.1 Theory of the Analysis Method
2.1.1 Modal Decomposition
As shown in Fig. 2.1, for a MTL system with N+1 conductors along z-direction, by choosing one of them as the return path (noted as #0) and numbering the others from #1
to #N, the per-unit-length loop inductance matrix L (N by N) and capacitance matrix C (N by N) can be defined. It should be noted that the off-diagonal terms in the per-unit-length capacitance matrix are negative, so a minus symbol is added for the mutual capacitance. And the ith diagonal term in that matrix is
1
The line voltage column vector V (N by 1) and line current column vector I (N by 1) which are composed of voltages and currents on conductors #1 to #N, respectively, will follow Telegrapher’s equations:
where the conductor loss and dielectric loss are both neglected here.
For such a system, there are N TEM modes which can propagate along the lines and decoupled to each other. The modal voltage column vector Vm (N by 1) and modal Fig. 2.1. A multiconductor-transmission-line system (N = 3 as the example) and its per-unit-length equivalent circuit model.
current column vector Im (N by 1), whose compositions stand for the magnitude of each mode, can be related to V and I by modal decomposition [47], [48]:
v m
where Mv and Mi are called modal transformation matrix for voltage and current, respectively. Since Mv and Mi are both independent of z at each cross section, the equations obtained by substituting (2.3) into (2.2) can be simplified as
( )
where Lm and Cm are called per-unit-length modal inductance matrix (N by N) and capacitance matrix (N by N), respectively, and must be diagonal since all the modes are decoupled. From the product of LmCm and CmLm,
it is obvious that Mv and Mi are the eigenmatrices of LC and CL, respectively. Besides, the relationship between the two transformation matrixes can also be found:
1
2.1.2 Wave Propagation in MTL System
Following (2.4), the wave equations of the propagating modes can be derived:
( )
in which (jω)2LmCm are equal to (jω)2CmLm since both Lm and Cm are diagonal. Then propagation constant matrix γ (N by N) whose components are propagation constant for each mode is defined by
( )
For a low-loss MTL system, the conductor loss and the dielectric loss can be taken into account by rewriting the propagation constant matrix as γ2 = (Rm+jωLm)(Gm+jωCm) ,where Rm and Gm are per-unit-length modal resistance matrix (N by N) and conductance matrix (N by N), respectively.
Then the solution to (2.7) can be expressed as
z z
The solution for line voltages and line currents can be obtained by substituting (2.9) in (2.3): Then by combing (2.2) and (2.11), the relationship between line voltages and line currents are obtained with p= L Cm m :
where Zc is called characteristic impedance matrix (N by N). It should be noted that Zc
will not be diagonal unless the MTL system is originally decoupled (i.e. with neither inductive nor capacitive coupling). The expression for Zc can be further simplified as
1 1
2.1.3 Discontinuities in MTL System
When the signal flows to a discontinuity (from the 1st section to the 2nd section), both reflected and transmitted waves are generated. From the continuities of line voltage and line current, the reflected wave can be expressed in term of incident wave as
(
2 1) (
−1 2 1)
− = + = − + −
I ΓI Z Z Z Z , (2.14)
where reflected coefficient matrix for current Γ is defined here and Zi (i = 1 and 2) can be either characteristic impedance matrix for transmission lines or load impedance matrix for terminations. Then the reflected modal current can be obtained:
1
m i i m
− = − +
I M ΓM I . (2.15)
If there are more than one discontinuities in the system, multi-reflection will occur, which will raise the difficulty to calculate the line voltages and line currents. To solve this problem, the concept of the input impedance is applied here, which can characterize the relation of total voltage and current at the discontinuity. As illustrated in Fig. 2.2, the input impedance matrix at the kth discontinuity Zin,k can be derived from (2.11) as well as (2.15), and be written as
where the subscript k stands for the kth section and lk is the physical length of the kth section. Zm, known as mode impedance matrix (N by N), is diagonal and can be expressed as:
1
m= m m−
Z L C . (2.17)
And the equivalent reflected coefficient at the k+1th discontinuity Γin,k+1 can be obtained with the characteristic impedance matrix of the kth section Zc,k and the input impedance matrix at the k+1th discontinuity Zin,k+1 as
( ) (
1)
It should be noted that if lumped elements or terminations are added in the discontinuity, some of the compositions in the input impedance matrix should be Fig. 2.2. The illustration of an MTL system with M discontinuities. The source is defined as the 0th section and the load is defined as the Mth section. The kth discontinuity is between the k-1th and kth sections.
modified accordingly. More details about this will be discussed with some examples in Section 2.3.
2.1.4 Distribution of Voltage and Current
Once the input impedance matrices at all the discontinuities are found, line voltages and line currents at each discontinuity can be obtained with Kirchhoff’s circuit laws. Then the forward modal current vector in kth section can be calculated with
(
1)
1where Iin,k is the line current vector flowing into the kth section. For a cross section which is in the kth section and with a distance d to the k+1th discontinuity, the forward and backward line current vectors can be expressed respectively as
( )
Then the line voltage vector there can be obtained, too:
( )
d = +( )
d − −( )
d = c⎡⎣ +( )
d − −( )
d ⎤⎦V V V Z I I . (2.21)
Equations (2.19) and (2.20) implies that there is no need to divide the MTL system into many segments for each point to observe, which will significantly increases the computing time and complexity. The observing points with the identical cross section can be grouped in the same section, and the line voltage and current vector can be acquired just by substituting different d into (2.19) and (2.20).
2.1.5 Flow of Analysis
The flowchart of the proposed analysis method is plotted in Fig. 2.3, and the major steps of the analysis flow are summarized as follows:
Step 1: Segment the MTL system into sections and number them from 0 to M (0th section for source including source impedance, and Mth for the load).
Step 2: Input the per-unit-length matrices L and C for the transmission lines in each section. Then calculate the characteristic impedance matrix Zc with (2.13), transformation matrices Mv as well as Mi with the eigen method, and the modal characteristics with (2.4), (2.8), and (2.10).
Step 3: Start from k = M – 1, calculate the equivalent reflected coefficient matrix Γin,k+1
with (2.18) and the input impedance matrix Zin,k with (2.16). Then modify Zin,k
if lumped elements or terminations are added into the kth discontinuity.
Step 4: Repeat step 3 for other values of k.
Step 5: Start from k = 1, calculate the input line current vector Ik from Kirchhoff’s circuit laws, the modal current vector Im,k with (2.19), and the distribution of line voltages and line currents with (2.20) and (2.21).
Step 6: Repeat step 5 for other values of k.
2.2 Prediction of Current/Voltage Distribution
2.2.1 Case 1: Coupling between Channels
The first structure for validation is two microstrip-line channels with coupling between them. The thickness of the substrate is 1.6 mm and the dielectric constant is 4.2.
As illustrated in Fig. 2.4, a differential pair (Lines A and B), which is expected with a differential-mode impedance of 85 Ω and common-mode impedance of 30 Ω, is designed with line width wd of 2.91 mm and separation sd of 1.1 mm. The total length of the differential pair is 100 mm. A single-ended line (Line C) with line width ws of 3.13 Fig. 2.3. The flow chart of the proposed analysis method to calculate the voltage and current distribution.
mm (for a characteristic impedance of 50 Ω) has some coupling with the differential pair (with a separation s of 1.54 mm) from z = 25 mm to z = 75 mm. Ports 1P and 1N are excited by voltage sources with a source impedance of 50 Ω, frequency of 2 GHz, amplitude of 1 V, and phase difference of 180°. All the other ports (2P, 2N, 3, and 4) are all respectively terminated to a load of 50 Ω.
Following the analysis flow, this structure is divided into 5 sections (0th for the source at z = 0, 1st for z = 0 to 25 mm, 2nd for z = 25 mm to 75 mm, 3rd for z = 75 mm to 100 mm, and 4th for the load at z = 100 mm). Since the differential pair is decoupled from the single-ended line at the 3rd section, the input impedance matrix (3 by 3) for this MTL system including Line C at the 3rd discontinuity (z = 75 mm) can be written as:
, 3
Fig. 2.4. The top view of the first structure to analyze where a single-ended line is near to a differential pair.
A
1stsection 2ndsection 3rdsection 0thsection
(source)
4thsection (load)
where Zin,AB3 is the input impedance matrix (2 by 2) for the differential pair at z = 75 mm, and Zin,s3 stands for the input impedance of the singled-ended line seen from the 3rd continuity of this system to Port 4.
At the 2nd discontinuity (z = 25 mm), the conductor number is changed from 3+1 to 2+1; therefore, matrix reduction method is applied here. From Kirchhoff’s circuit law, the current on Line C (noted as I3) can be related to the current on Line A (noted as I1) and Line B (noted as I2):
( ) (
1)
3 in s, 2 in2,33 in2,31 1 in2,32 2
I = − Z +Z − Z I +Z I , (2.23)
where Zin,s2 is the input impedance of the singled-ended line seen from the 2nd continuity to Port 3, and Zin2,ab stands for the element value of row a and column b in the input impedance matrix Zin2 (3 by 3) at z = 25+ mm. Then the input impedance matrix (2 by 2) for the differential pair (noted as Zin,AB2) at the 2nd discontinuity (z = 25 -mm) can be obtained:
The calculated current distribution on the three lines is compared with the simulated ones with circuit simulation tool (ADS) and full-wave simulation tool (HFSS), as shown in Fig. 2.5, where the good agreement of the results can be observed. Table 2.1 lists the analyzed modal current received at Ports 2P and 2N, where a similar case without Line C is compared. Although the differential-mode current almost is unchanged, the common-mode current is excited due to the nearby Line C. This is known as mode conversion, which will happen on an imbalanced differential channel even though there is no excitation on Line C.
2.2.2 Case 2: Guard Trace between Two Lines
The second structure to analyze is a guard trace between two single-ended microstrip lines on a substrate with a thickness of 1.6 mm and dielectric constant of 4.2.
As illustrated in Fig. 2.6, Line A (aggressor) and Line B (victim) are both with line width w1 of 3.13 mm (for a characteristic impedance of 50 Ω) and length of 100 mm.
The separation between these two lines s1 is 1 mm. To reduce the crosstalk noise, a guard trace (Line C) with both ends shorted (at z = 15mm and z = 85 mm) is placed between them. The line width of Line C (w) is 0.4 mm and the separation to the other Fig. 2.5. Simulation results of current distribution for Case 1.
DM current CM current
w/o Line C 9.935 mA 0 mA
w/ Line C 9.906 mA 0.190 mA
Table 2.1. The analyzed modal current for Case 1.
lines (s2) is 0.3 mm. A 10-Ω resistor in series is placed on Line C at z = 40 mm. Port 1 is excited by a voltage source with a source impedance of 50 Ω, frequency of 2 GHz, and amplitude of 1 V. The other ports (2, 3, and 4) are all respectively terminated to a load of 50 Ω.
This structure is segmented into 6 sections (0 to 5), where the 2nd and 4th discontinuities (z = 15 mm and z =85 mm) here are similar to the 2nd and 3rd ones in Case 1, respectively. Since the guard trace is short to ground at both ends, the input impedances Zin,s3 in (2.22) and Zin,s2 in (2.23) as well as (2.24) can be set to 0. At the 2nd discontinuity (z = 40 mm), the input impedance at z = 40- mm will change due to the series resistor, and can be expressed as:
Fig. 2.6. The top view of the second structure to analyze where a guard trace (short to ground at both ends) with a 10-Ω resistor (block at z = 40 mm on Line C ) on it is placed between two single-ended microstrip lines.
A
1stsection 2ndsection 3rdsection 5thsection (load)
'
Fig. 2.7 shows the analyzed current distribution and simulated ones with the conventional tools. It is obvious that the results by proposed method match well to the full-wave simulation by HFSS, while some discrepancy occurs at the results from ADS.
In addition to voltage/current distribution in certain frequency, this method can also be applied to estimate the frequency response. The voltages versus frequency at z = 0, 50 mm and 100 mm on Line B are plotted in Fig. 2.8, which tells that the guard trace can suppress the crosstalk problem in certain frequencies, but if it is not designed well, higher noise due to resonance may occur in other bands.
Fig. 2.7. Simulation results of current distribution for Case 2.
Responses among terminal (S-parameters) are also simulated, and the results in terms of both magnitude and phase are shown in Fig. 2.9 (S11 and S21) and Fig. 2.10 (S31
and S41). Good agreement in both magnitude and phase between proposed method and full-wave simulation tool can be seen except for some discrepancy at higher frequencies.
This is because the L and C matrices are extracted at 1 GHz, and can be improved by applying dispersive L and C when sweeping the frequency. The full-wave simulation is expected to give the most accurate prediction of the electrical behaviors, but on the other hand, the computing time and complexity (about 1.5 hours and 850000 meshes by a PC with a CPU of i7-2600 and with 16-GB dynamic RAM modules) is much more than the other two methods (within several seconds). To sum up, the proposed method provides a good accuracy on the prediction and takes only little computing resource, which can reduce the cost when designing the channels in MTL systems.
Fig. 2.8. Simulation results of the crosstalk voltage waves on Line B in different frequencies for the second case.
0
(a)
(b)
Fig. 2.9. Simulated S11 and S21 for Case 2. (a) Magnitude in dB. (b) Phase in radian.
(a)
(b)
Fig. 2.10. Simulated S31 and S41 for Case 2. (a) Magnitude in dB. (b) Phase in radian.
2.3 Prediction of EMI and RFI from Mode Conversion
2.3.1 Introduction of Antenna Mode
It has been introduced previously that a (N+1)-conductor transmission-line system can support N guided TEM modes. For a differential channel implemented as microstrip couple lines, there will be two signaling conductors and one reference plane; That is, N
= 2. The guided TEM modes are known as differential mode (DM) and common mode (CM), and by guided-wave theory, neither of them will radiate. But from the experience, if CM noise exists and flows through a discontinuity in this system, there will be unwanted radiation power and cause EMI or RFI problems.
To explain this phenomenon, a radiating mode named antenna mode (AM, also called as secondary common mode) is introduced in [16]-[19]. This mode occurs when the reference plane is not the real system ground; instead, the system ground may be the earth or shields of electrical devices. As shown in Fig. 2.11, the net current flow in the three conductors is unidirectional, which will totally return in the system ground. Since the return current is far from those on the three conductors, it is expected that the radiation will occur. To estimate the radiation power, current division factor (h) which depends on the cross section is defined before and after the discontinuity. Then the driven source that excites the radiation can be predicted by the difference of the current division factors. But there are some drawbacks in the method. Firstly, this method can be applied to only the local discontinuity but not the whole system. Secondly, the formulas derived depend on the structure and the number of conductors. That is, it can be applied in neither channels with multiple differential pairs such as HDMI nor channels with more conductors such as C-PHY. In addition, the capacitance-matrix-based h factor may be inaccurate in higher frequency ranges.
Hence, the MTL analysis method is applied here to characterize the radiation source for these kinds of system. The details and analysis flow will be introduced in the following sections.
2.3.2 Radiation Prediction by MTL Analysis
Fig. 2.12 shows the differential channel to be analyzed. It’s a 2-layer PCB with a pair of differential lines on the top layer and a reference plane at the bottom layer. The substrate is 0.3 mm thick and composed of FR4 with a dielectric constant of 4.0. The differential lines are separated (s) by 0.24 mm and with the same thickness of 35 μm and width (w0) of 0.53 mm to achieve the DM impedance of 85 Ω and CM impedance of 30 Ω. The reference plane is also with a thickness of 35 μm and can be departed into three segments. The 1st and the 3rd segments have the same length of l1 = l3 = 15 mm and the same width of w1 = w3 = 10 mm. The length of the 2nd segment (l2) is 20 mm, and the width (w2) can be either of 1.5 mm, 3.0 mm, or 4.5 mm to form 3 cases.
Applying the proposed MTL analysis method, the number of conductors in this system can be seen as 3+1. The system ground is #0; the signal lines are noted as #1 and Fig. 2.11. Net current flows of the (3+1)-conductor system under different modes.
#2; the reference plane is #3. To extract the per-unit-length L and C matrices, a PEC plane (acting as the system ground) with a width of 100 mm is put 50 mm away under the reference plane, as illustrated in Fig. 2.13.
Fig. 2.12. The top view of the differential channel to be analyzed.
Fig. 2.13. The simulation environment to extract the per-unit-length L and C matrices.
Once the L and C are extracted, current sources with a frequency of 1.5 GHz are used to excited the structure among the conductors at z = 0. To excite the DM incident power, the magnitude of current will be +0.02 A, -0.02 A, and 0 on the conductor #1, #2, and #3, respectively. As to CM, the magnitude of current on the three conductors (in the same order) will be +0.02 A, +0.02 A, and -0.04 A, respectively. The time-average power of DM and CM excited under this way will be both 0.02 W. At z = 100 mm, there is a 50-Ω lumped resistor placed between conductor #1 and #3 as well as between
Once the L and C are extracted, current sources with a frequency of 1.5 GHz are used to excited the structure among the conductors at z = 0. To excite the DM incident power, the magnitude of current will be +0.02 A, -0.02 A, and 0 on the conductor #1, #2, and #3, respectively. As to CM, the magnitude of current on the three conductors (in the same order) will be +0.02 A, +0.02 A, and -0.04 A, respectively. The time-average power of DM and CM excited under this way will be both 0.02 W. At z = 100 mm, there is a 50-Ω lumped resistor placed between conductor #1 and #3 as well as between