Controlling Power-Distribution-Plane Resonance in ... - CiteSeerX

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PAPER

Special Issue on Recent Progress in Electromagnetic Compatibility Technology

Controlling Power-Distribution-Plane Resonance in Multilayer Printed Circuit Boards Takashi HARADA† , Hideki SASAKI† , and Yoshio KAMI†† , Members

SUMMARY This paper describes the mechanisms of powerdistribution-plane resonance in multilayer printed circuit boards and the techniques to control the resonance. The powerdistribution-plane resonance is responsible for high-level emissions and circuit malfunctions. Controlling the resonance is an effective technique, so adequate characterization of the resonance is necessary to achieve control. The resonance characteristics of four-layer printed circuit boards are investigated experimentally and theoretically by treating the power-distribution planes as a parallel-plate transmission line with decoupling circuits. Analysis of the forward traveling wave shows that the resonance frequency is determined by the phase delay due to wave propagation and by the phase progress of interconnect inductance in the decoupling circuit. Techniques to control the resonance characteristics are investigated. The resonance can be shifted to a higher frequency by adding several decoupling circuits adjacent to the existing decoupling capacitor or by increasing the number of via holes connecting the capacitor mounting pads to the power-distribution planes. key words: EMC, EMI, printed circuit board, power distribution plane resonance, transmission-line theory

1.

Introduction

As the clock frequencies in digital circuits have increased, radiated emissions and circuit malfunctions due to a voltage bounce in the power-distribution planes which consisting of a power plane and a ground plane in multilayer printed circuit boards have become serious problems. The emissions increase and circuit malfunctions occur when the resonance frequency of the power-distribution planes coincides with one of the clock harmonics. Shifting the frequency away from the clock harmonics thus eliminates such problems [1]. Decoupling strategies are effective in controlling the resonance. To achieve the desired decoupling effects, we must clarify the behavior of decoupling circuits consisting of a capacitor and its parasitic components associated with the capacitor, pads, and via holes connecting the power plane and the ground plane. To incorporate resonance control into the board design, we need to quantify the decoupling effects. At frequencies lower than about 100 MHz, power-distribution planes can be treated as a lumped element circuit [2], Manuscript received July 1, 1999. Manuscript revised October 1, 1999. † The authors are with the Device Analysis Technology Laboratories NEC Corporation, Kawasaki-shi, 216-8555 Japan. †† The author is with the University of ElectroCommunications, Choufu-shi, 182-8585 Japan.

[3], which makes the relation between the decoupling circuit and the resonance characteristics easier to understand. At frequencies higher than about 100 MHz, however, the power-distribution planes must be treated as a distributed element circuit [3], [4], so that the effectiveness of the decoupling circuit on the resonance is rather complicated. Since voltage and current distributions in a distributed circuit are not uniform, the resonance depends not only on the parameters of the decoupling circuit but also on its location. Several researchers have analyzed the resonance characteristics of power-distribution planes by using computing tools or methods, such as FDTD [5], SPICE [6], MPIE [7], and Method of Moment [8]. Although these techniques are convenient for investigating complex board structures, they do not give clear intuitive images of the resonance mechanisms. A comprehensive model has been described that treats the powerdistribution planes as a parallel-plate transmission line [3]. This transmission-line model has been used to investigate the characteristics of planes like those shown in Fig. 1. Decoupling circuits are treated as interconnect circuits. Because transmission-line theory can treat only one-dimensional wave propagation, the application of this model is restricted to oblong-shaped simple printed circuit boards. However, it does describe the resonance mechanism intuitively [9]. Moreover, transmission-line treatment enables the behavior of the decoupling circuit to be expressed quantitatively. This paper investigates power-distribution-plane resonance by using the transmission-line model and develops techniques for controlling the resonance. Several multilayer printed circuit boards of different sizes are analyzed by experiment and by calculation. Each board contains two internal planes and pairs of capacitormounting pads connected to the planes. As we are interested in engineering techniques supported by a rigorous analysis, we also choose to investigate the resonance characteristics by forward traveling waves transmitting

Fig. 1

Transmission-line model.

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Fig. 3

Fig. 2

Measurement setup.

Printed circuit boards: (a) top views, (b) cross section.

in the planes. One of the main advantage of this technique is that it gives a quantitative approach of the phenomena, as it appears that study of phase progress is one of the key point to understand the resonance mechanism. At last, based on the proposed technique and keeping into account practical requirement, we develop techniques to shift the resonance frequency. 2.

Measurement of Power-Distribution-Plane Resonance

The measurements of power-distribution-plane resonance were performed by using three multilayer printed circuit boards designed and fabricated for this study. • Board A: 260 mm × 260 mm • Board B: 260 mm × 100 mm • Board C: 260 mm × 25 mm Each board was 1.6 mm thick and composed of four layers. The second and third layers were conducting planes devoted in the entire layers. These planes correspond to the ground and power planes in a conventional board. The top view of each board is shown in Fig. 2(a). These boards contained pairs of pads that mounted capacitors used to interconnect these two planes with via holes. These pads were located at intervals of 40 mm (x and y directions). The cross section of these boards is shown in Fig. 2(b). The space between the two internal planes was about 1 mm. Two SMA connectors were used to connect the board to the measurement equipment. The inner and outer conductors of the connectors were joined to the second and third planes, respectively. These connectors were mounted 10 mm away from the board edge. The power-distribution-plane resonance can be directly related to the measured S11 at a point or to the S21 between two points of the planes [2], [10]. Since our objective was to investigate the transmission behavior of the planes, we measured S21 . The measurement

Fig. 4 Measured S21 responses of the planes with no capacitor: (a) magnitude, (b) phase.

setup is shown in Fig. 3. A vector network analyzer was used in this measurement. 2.1 Resonance Characteristics without Decoupling Circuits Figure 4 shows the measured magnitude and phase characteristics of the S21 as a function of the frequency when the boards had no capacitor. The magnitude of each board reaches their peaks at approximately 270 and 540 MHz. The phase becomes π and 0 at those frequencies. This means that the board length corresponds to half a wavelength and one wavelength, respectively. The resonance frequency of those boards was calculated approximately by using

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Calculated resonance frequencies.


150 fresonance [MHz] = √ (m/lx )2 + (n/ly )2 εr m = 0, 1, · · · , n = 0, 1, · · · ,

(1)

where lx and ly are the board lengths in x and y directions, and m represents the mth mode associated with the x-dimension, n represents the nth mode associated with the y-dimension [11]. The calculated results with Eq. (1) when εr = 4.5 are shown in Table 1. The measured resonance frequencies, 270 and 540 MHz for each board, nearly coincide with the calculated frequencies when m = 1 or 2, and n = 0. When n = 0, there is no wave propagation towards transverse direction (y direction), so the resonance frequencies depend only on the board length. Therefore, for a board with no capacitor, the resonance characteristics can be evaluated by treating these planes as a parallel-plate transmission-line as far as only considering the length-directional (x direction) wave propagation, even if the board is square. Other peaks are found at 380 MHz for Board A and at 750 MHz for Board B. These frequencies approximately meet the calculation results when m = 1 and n = 1 for each board. The peak at 700 MHz for Board B is considered as the resonance frequency when m = 0 and n = 1. These resonances are related to the wave propagation in the transverse direction (ydirection). For simplicity, however, we will treat only length-direction propagation here. 2.2 Resonance Characteristics with Decoupling Circuits

Fig. 5 Measured S21 responses of the planes with two capacitors: (a) magnitude, (b) phase.

9.5 nF, as measured with an impedance analyzer. The steep drop in magnitude at appoximately 40 MHz is due to the series resonance of the capacitor and the interconnect inductance, which is composed of seriesinductance components associated with the capacitor, via holes, and pads. At frequencies lower than the series resonance, the S21 responses for all boards are nearly the same. This means that the characteristics of power-distribution planes are determined mainly by the mounted capacitors. At frequencies higher than the series resonance, the S21 responses vary between boards. The magnitude for Board A reaches peaks at 90 and 280 MHz, with corresponding phases of 0 and π respectively. The same peculiarity is observed in the other two boards. The magnitude reaches peaks with phases of 0 and π at 140 and 300 MHz for Board B and at 210 and 400 MHz for Board C. The 0 and π in the phase signify the occurrence of line resonance. In spite of interconnecting the same circuits at the same positions, the resonance frequencies differ depending on the board width. To investigate these results quantitatively, we analyzed the line characteristics by means of calculation. 3.

Figure 5 shows the measured S21 results when two chip capacitors were mounted on pads P1 and P2 , which are circled in Fig. 2(a). Their capacitance was about

Model for Analyzing Power-Distribution Planes

The power-distribution-plane characteristics are ana-

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Fig. 6

S21 calculation model.

lyzed by transmission-line model. In the case of mounting the decoupling circuits, Eq. (1) cannot be used. The model we used to calculate the resonance is shown in Fig. 6. The two planes were treated as a parallelplate transmission line. To take into account the placement of the SMA connectors 10 mm away from the board edges, the board length of 260 mm was divided into three lines; a 240-mm main line and two 10-mm open-circuited transmission lines. The characteristic impedance of the line is given approximately by d , Zp ≈ 20π √ εr W

(2)

where W is the width of the board [12], d is the distance between two planes, and εr is the relative permittivity of the dielectric. We found that the characteristic impedance for Board A, Board B, and Board C calculated using Eq. (2) were about 0.7, 1.7, and 7.0 Ω, respectively. Two interconnect circuits composed of interconnect inductance Li , and series resistance Ri were connected between the planes at P1 and P2 . Such circuits act as decoupling circuits in practical multilayer printed circuit boards. The interconnect inductance Li was 1.6 nH; it was determined by the capacitance Cd of the decoupling capacitor and series resonance frequency fsr : Li =

1 . (2πfsr )2 Cd

(3)

The series resistance Ri was 0.1 Ω obtained from manufacturer’s data. Since the frequency range of interest was higher than the series resonance frequency, the decoupling capacitance Cd could be ignored here. The S21 characteristics were calculated using S21 =

2Z0 , AZ0 + B + CZ02 + DZ0

(4)

where Z0 is the impedance of the measurement system, A, B, C, and D are elements of the F matrix:   A B C D         = Fˆ0 Fˆ (l1 ) Fˆd Fˆ (l2 ) Fˆd Fˆ (l3 ) Fˆ0 , (5) where l1 , l2 and l3 are the distances between the port #1 and P1 , P1 and P2 , and P2 and  #2, respec port  ˆ tively (see Fig. 6). The matrices F (l) and Fˆ0 in

Fig. 7 S21 calculation results of the planes with two capacitors: (a) magnitude, (b) phase.

Eq. (5) can be expressed as     cos(kl) jZp sin(kl) ˆ F (l) = , jZp−1 sin(kl) cos(kl)

(6)

   Fˆ0 =

 1 0 , (7) j tan(kledge )/Zp 1 √ where k = 2π εr /λ0 , λ0 is the wavelength in free space, ledge is the distance between the connector and board edge. The connectors are positioned 10 mm away from the board edges. This matrix means that opencircuited transmission lines of length ledge are connected at both ports, #1 and #2. For simplicity, the lines were assumed to be lossless.     1 0 ˆ Fd = (8) (jωLi + Ri )−1 1 is the F matrix of the interconnect circuit. The calculated magnitude and phase characteristics of the S21 as a function of the frequency are shown in Fig. 7. Between 40 and 350 MHz, these results agree well with the experimental ones. Line resonance appears at 100 and 280 MHz for Board A, at 140 and 300 MHz for Board B, and at 210 and 400 MHz for Board C. The small deviation in magnitude is due to the assumption that the lines are lossless. The resonance frequency does not depend on the impedance Z0 , as

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long as Z0 is larger than the characteristic impedance Zp and Z0 consists of only real components. The calculated results deviate from the experimental ones at frequencies less than 40 MHz because the capacitance Cd is ignored. At frequencies above 350 MHz, the S21 characteristics for Board A and Board B do not agree with the experimental ones due to the wave propagation in the transverse direction. It could be understood as the power-distribution planes cannot be treated as a one-dimensional transmissionline for wider board, such as Board A or Board B. According to transmission-line theory, the resonance frequency depends on the board length. Line resonance generally occurs when the line length is a multiple of the half wavelength: λ/2, λ and so on. The first resonance occurs when the line length corresponds to the half wavelength. The phase at the right edge in this case is π. Adding interconnect circuits, however, shifts the resonance frequencies. This shift depends not only on the parameters of the decoupling circuit but also on the board width. Moreover, the phase at the first resonance frequency is 0. This means that the line length corresponds to zero or one wavelength. To understand the mechanism of these frequency shifts and phase behaviors quantitatively, we investigated the characteristics of the line resonances by using the forward traveling wave. The forward traveling wave is directly related to S21 and describes the line behavior intuitively. 4.

nitude and phase characteristics of forward traveling wave along the line at 142 MHz, which is the lowest resonance frequency for Board B. The magnitude for Board B is larger than those of the other boards. This means that a large amount of power is supplied to the line under the resonance condition. Since the line loss is ignored, the magnitude is constant along the line beside the interconnect circuits. However, this doesn’t affect our investigation because we are mainly concerned with the phase analysis. The phase is delayed along the line due to the wave propagation. The delay slope is proportional to the wave-number k(= 2π/λ), so that the delay depends only on frequency. It becomes larger as the frequency increases. At the interconnect circuits, however, the phase progresses noticeably due to the interconnect inductance. This phase progress depends on the board width. The narrower the board, the larger the progress. The phase also progressed little at x = 0 due to an impedance mismatch between the impedance Z0 and input impedance Zin , as seen looking toward the line (see Fig. 8). As a result of the phase delay and the progress, the phases at the right edge for each board are about −π/2 for Board A, 0 for Board B and about π/3 for Board B, respectively. The zero phase for Board

Analysis of Resonance by Using Forward Traveling Wave

The forward traveling wave a along a line defined in x axis is calculated as a(x) =

V (x) + Zp I(x)  , 2 Zp

(9)

where V (x) and I(x) are the voltage and current distributions along the line [13]. To simplify the calculation, we rearranged the model as shown in Fig. 8. The voltage source was connected at the left end through impedance Z0 , and output impedance Z0 was connected at the right end. This rearrangement does not affect the resonance frequency or the phase characteristics. The V0 we used in the calculation was 1 V for normalization purpose. The calculation of V (x) and I(x) is described in Appendix. Figure 9 shows the mag-

Fig. 8

Power-distribution plane analysis model.

Fig. 9 Forward traveling wave characteristics along the planes at 142 MHz: (a) magnitude, (b) phase.

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Fig. 11

Equivalent-circuit model of interconnect inductance.

Therefore, θp depends on the frequency, characteristic impedance Zp , and interconnect inductance Li . It is also a function of voltage V2 and current I2 . For simplicity, the length of the right hand line is assumed to be infinite or terminated by matched load. Since there is no reflected wave, V2 = Zp . I2

Fig. 10 Phase of forward traveling wave characteristics along the planes at 278 MHz.

B means that the line is under the resonance condition. The phase progress at interconnect circuit also depends on the circuit location. The progress at the two decoupling circuits are almost the same for Board B. However, those for Board A and Board C differ; the progress at the center circuit, P2 , is larger than that at P1 . Figure 10 shows the phase of traveling wave along the line at 278 MHz, which is the second resonance frequency of Board A, the widest board. The phase at the right edge is −π(= π) for Board A due to the phase delay of the line and the progress. The board length electrically coincides with a half-wavelength, so only Board A is in the resonance condition at this frequency. These results indicate that the phase progress at an interconnect circuit is an important factor in determining the resonance frequency. We thus investigated the phaseprogress characteristics in more detail. Figure 11 shows the interconnect circuit model. At frequencies higher than the series resonance, this circuit can be represented only by interconnect inductance Li because ωLi  1/ωCd and ωLi  Ri . The a1 and a2 denote the forward traveling wave at Port A and Port B, respectively. Phase progress θp is calculated to obtain the ratio of a1 and a2 , a2 V2 + I2 Zp = . a1 V1 + I1 Zp

(10)

Substituting V1 = V2 and I1 = I2 + V1 /jωLi into Eq. (10), we obtain   −1 1 a2 Zp = 1+ . (11) a1 jωLi 1 + I2 Zp /V2 The phase progress θp is obtained using   Im(a2 /a1 ) θp = tan−1 . Re(a2 /a1 )

(12)

Substituting Eq. (13) into Eq. (11), we obtain  −1 a2 Zp = 1+ . a1 j2ωLi The phase progress θp can then be expressed as   Zp . θp ∼ = tan−1 2ωLi

(13)

(14)

(15)

The phase progress depends on characteristic impedance Zp which is related to the board width: as the width becomes narrower, Zp becomes larger. Consequently, a narrower board has a larger phase progress at the same frequency. Since the phase delay does not depend on the board width, the narrowest board has the highest resonance frequency, resulting in a larger phase delay that sets off the phase progress. Since V2 and I2 vary with the board location, the phase progress also depends on the location of an interconnect circuit. As described above, θp depends on the interconnect inductance: the smaller the inductance, the higher the resonance frequency. In other words, the resonance frequency can be controlled by varying the inductance. 5.

Controlling the Resonance Frequency

When the resonance frequency coincides with one of the clock harmonics, high-level emissions and circuit malfunctions are liable to occur. Shifting the frequency away from the clock harmonics can avoid such problems. As shown previously, the resonance frequency can be controlled by changing the board width or interconnect inductance. We attempted to shift that frequency by using two techniques to decrease the interconnect inductance. In a manufacturing approach, as the board width is one of the fixed requirement, and is not changeable at will.

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(a)

Fig. 12 Board configuration to reduce inductance component of the interconnect circuit.

(b) Fig. 14 Board and pad configuration used to investigate interconnect inductance reduction: (a) board, (b) pad and via hole.

Fig. 13 Magnitude of measured S21 responses of the planes for each interconnect circuit configuration.

5.1 Adding Decoupling Circuit The first technique is to add several decoupling circuits adjacent to the existing one. This reduces the total inductance component due to the parallel connection. This effect was confirmed by adding two 9.3 nF capacitors to the pads near the two existing capacitors at P1 and P2 , as shown in Fig. 12. Since only the wave propagation in the x direction needs to be considered at lower frequencies, the three capacitors mounted in a transverse line were regarded to be located at the same point. The interconnect inductance was expected to drop to 1/3. The measured S21 response when the capacitors are added to Board B was plotted in Fig. 13 with a bold solid line. A bold broken line in this figure represents S21 response when only one capacitor is mounted. By adding the capacitors, the first and the second resonance frequencies shift from 142 MHz to 200 MHz, and from 302 MHz to 370 MHz, respectively. The calculated S21 results, when the interconnect inductance of each two circuits is 0.53 nH, is shown with a fine solid line. This inductance is 1/3 of one circuit’s inductance, 1.6 nH. The calculated result agrees well with the measured one. When using this technique in practical applications, the capacitance values and interconnect inductance values must be kept constant for each added circuits to avoid parallel resonance.

Fig. 15 Magnitude of measured S21 responses of each pad configuration.

5.2 Reducing Inductance Associated with Pads and Via Holes The second technique is to reduce the inductance associated with the pads and via holes. Such inductance reduction should be realized by enlarging the area of the pads or increasing the number of via holes. Four types of capacitor mounting pads were fabricated on boards. The top view of the board is shown in Fig. 14(a) and the configurations of the pads and via holes are shown in Fig. 14(b). The board is 260 mm in length and 25 mm in width, to avoid the effect of transverse wave propagation. Each board contains a pair of pad. Pad A has a conventional configuration that consists of a 1.4 mm-square pad and a 1.2 mm-diameter via hole. The pad area of Pad C and Pad D is twice the pad size of Pad A. Pad B and Pad D have two via holes for each pad. Measured S21 responses for each pad configurations are shown in Fig. 15. The series resonance frequencies are approximately 42 MHz for Pad A

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Calculated interconnect inductance. [3]

[4]

and Pad C, and 50 MHz for Pad B and Pad D. From these frequencies and the capacitance of 9.3 nF, the interconnect inductance for each pad was calculated. The results are shown in Table 2. The interconnect inductance for Pads B and D is less than that for Pad A by 0.5 nH. Adding via holes reduces the inductance, but enlarging the pad area does not. This technique was applied for Board B. S21 characteristic were calculated with an interconnect inductance of 1.1 nH instead of 1.6 nH, by using the model shown in Fig. 8. We confirmed that the first and second resonance frequencies shifted from 142 MHz to 178 MHz and from 300 MHz to 330 MHz, as expected. 6.

[5]

[6]

[7]

[8]

Conclusion [9]

The characteristics of power-distribution-plane resonance have been investigated quantitatively by using the transmission-line model. The forward traveling wave characteristics show that the resonance frequency is determined by phase delay due to wave propagation and phase progress of the interconnect inductance in the decoupling circuit. This phase progress depends on the board width. A narrow board causes larger phase progress than a wide board. The resonance frequency of a narrower board must therefore be higher to achieve a larger phase delay to set off this phase progress. The problem of high-level radiated emissions and circuit malfunctions due to power-distribution-plane resonance can be avoided by shifting the resonance frequency away from the clock harmonics. The resonance frequency can be shifted to a higher range by reducing the interconnect inductance. Adding several decoupling circuits or increasing the number of via holes connecting the capacitor mounting pads are two ways to reduce the inductance. The transmission-line model used here is applicable for one-dimensional wave propagation. To analyze a more general structure, we have to treat a towdimensional wave propagation case. Such an analysis is actually our concern and is ongoing research. References [1] R.W. Dockey and R.F. German, “New technique for radiating printed circuit board common-mode radiation,” Proc. 1993 IEEE International Symposium on EMC, pp.334–339, Aug. 1993. [2] T.H. Hubing, J.L. Drewniak, T.P. Van Doren, and D.M. Hockason, “Power bus decoupling on multilayer

[10]

[11] [12]

[13]

printed circuit boards,” IEEE Trans. Electromagn. Compat., vol.37, no.2, pp.155–165, May 1995. T. Harada, H. Sasaki, and Y. Kami, “Investigation on radiated emission characteristics of multilayer printed circuit boards,” IEICE Trans. Commun., vol.E-80-B, no.11, pp.1645–1651, Nov. 1997. J. Franz and W. John, “An approach to determine decoupling effects of Vcc and Vbb structures in multilayer technique,” Proc. 1994 International Symposium on EMC, EMC ’94 Sendai, 17P208, pp.56–59, Aug. 1994. S.V. Berghe, F. Olyslager, D.D. Zutter, J.D. Moreloose, and W. Temmerman, “Study of the ground bounce caused by power plane resonances,” IEEE Trans. Electromagn. Compat., vol.40, no.2, pp.111–119, May 1998. C.B. O’Sullivan, L.D. Smith, and D.W. Forehand, “Developing a decoupling methodology with SPICE for multilayer printed circuit boards,” Proc. 1998 IEEE International Symposium on EMC, pp.652–655, Aug. 1998. H. Shi, J. Fan, and L. Drewniak, “Modeling multi-layered PCB power-bus design using an MPIE based circuit extraction technique,” Proc. 1998 IEEE International Symposium on EMC, pp.647–651, Aug. 1998. O. Ueno, D. Iguchi, H. Arakaki, H. Itoh, and T. Ozawa, “Three-dimensional noise current distribution on power and ground planes in printed circuit boards,” Proc. 1998 IEEE International Symposium on EMC, pp.1136–1141, Aug. 1998. T. Harada, H. Sasaki, and Y. Kami, “Investigation on power distribution plane resonance in multilayer printed circuit boards using a transmission-line model,” Proc. 1999 International Symposium on EMC, EMC’99 Tokyo, 18P106, pp.21–24, May 1999. H. Shi, F. Sha, J.L. Drewniak, T.P. Van Doren, and T.H. Hubing, “An experimental procedure for characterizing interconnects to the DC power bus on a multilayer printed circuit board,” IEEE Trans. Electromagn. Compat., vol.39, no.4, pp.279–285, Nov. 1997. R.F. Harrington, Time-harmonics Electromagnetic Fields, sec.4 McGraw-Hill, New York, 1961. S. Ramo, J.R. Whinnery, and T.V. Duzer, Fields and Waves in Communication Electronics, sec.8 John Wiley & Sons, New York, 1965. R.N. Ghose, Microwave Circuit Theory and Analysis, sec.9.2, McGraw-Hill, New-York, 1963.

Appendix The voltage V (x) and current distribution I(x) along the line were calculated using the model shown in Fig. A· 1. First, the voltage and current at the left edge, V (0) and I(0) were calculated using V (0) =

I(0) =

(A Z0 + B  )V0 Zin V0 =  2 Zin + Z0 C Z0 + (A + D )Z0 + B 

V0 (C  Z0 + D )V0 =  2 Zin + Z0 C Z0 + (A + D )Z0 + B  (A· 1)

where Z0 =50 Ω and A , B  , C  , and D are the factors of the F matrix:

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  −1  V (0) ˆ · F (lc1 ) x > x2 I(0) (A· 5) The voltage and current distributions along Board B at its lowest resonance frequency, 142 MHz are shown in Fig. A· 2. Fig. A· 1

Calculation model.

Takashi Harada received his B.Eng. and M. Eng. degrees in electrical engineering from Tokyo Metropolitan University, Tokyo, Japan, in 1981 and 1983, respectively. He joined NEC Corporation, Kawasaki, Japan, in 1983 and has been engaged in research on electromagnetic compatibility. He is now a principal researcher of the EMC Engineering Center, Device Analysis Technology Laboratories. He is also a graduate student of the University of Electro-Communications.

Fig. A· 2 Voltage and Current distributions for Board B (142 MHz): (a) magnitude, (b) phase.



A C

B D



      = Fˆ (lc1 ) Fˆd Fˆ (lc2 ) Fˆd Fˆ (lc3 ) . (A· 2)

The two types of matrices are expressed as     cos(kl) jZp sin(kl) ˆ F (l) = and (A· 3) cos(kl) jZp−1 sin(kl)    Fˆd =

1 (jωLi + Ri )−1

0 1

 .

(A· 4)

Then, V (x) and I(x) were calculated successively using     −1  V (0) V (x) x ≤ x1 = Fˆ (x) I(0) I(x)   −1  −1  −1  V (0) Fˆd Fˆ (lc1 ) = Fˆ (x−x1 ) I(0) x 1 < x ≤ x2  −1  −1    −1 Fˆd Fˆ (lc2 ) Fˆd = Fˆ (x − x2 )

Hideki Sasaki was born in Kanagawa, Japan on April 11th, 1968. He received his B.E. and M.E. degrees in electronics engineering from Nihon University, Chiba, Japan in 1991 and 1993, respectively. He joined NEC Corporation, Kawasaki, Japan in 1993, where he is a Research Engineer in the EMC Engineering Center, Device Analysis Technology Laboratories. He has been engaged in the development of measurement methods for investigating electromagnetic compatibility problems.

Yoshio Kami was born in Kagoshima, Japan in 1943. He received his B.E. degree from the University of Electro-Communications, the M.E. degree from Tokyo Metropolitan University, and the D.E. degree from Tohoku University in 1966, 1970, and 1987, respectively. From 1970 to 1987 he was with the Junior College of Electro-Communications. Since 1987 he has been with the University of Electro-Communications, Chofu, Japan. He is currently a Professor at that university. He has been engaged in research in the area of filters, transmission lines, microwave components, and electromagnetic compatibility.