In recent years, effective methods for predicting and suppressing electromagnetic interference over a broad band are required. In this paper, we focus on the prediction of electromagnetic (EM) radiation from a stripline structure with a ground thin wire by an equivalent circuit model. First, frequency responses of common-mode (CM) current on the printed circuit board and EM radiation are studied with finite difference time domain modeling. Secondly, an equivalent circuit model for predicting CM current is proposed. The equivalent circuit model for prediction is based on the concepts of CM antenna impedance, distributed constant circuit, and electric coupling between the power plane and the thin ground wire. Good agreement between the predicted and full-wave analysis results indicates the validity of the proposed equivalent circuit model. The frequency response of EM radiation from the stripline structure can be identified using our proposed model. In addition, the equivalent circuit model provides enough flexibility for different geometric parameters and can be used to develop physical insights and design guidelines.
 In terms of electromagnetic interference (EMI) in electronics devices, there are various potential radiators of electromagnetic (EM) waves called EMI antennas, such as a printed circuit board (PCB), slot, and chassis, in electronic devices [Paul, 1991]. Since power-ground planes in PCB have two parallel planes, it resembles a microstrip antenna. If decoupling and/or grounding in the power-ground plane pair does not work well, certain high frequency current could leak out with the largest amplitude at the resonance of the EMI antenna formed by the parallel plane pair. This kind of radiation from a multilayer PCB is considered as one of the main sources of EMI from electronic devices. Although many literatures have discussed the power integrity problems and power bus noise of the parallel plane pair [Fan et al., 2001; Wang et al., 2005; Engin et al., 2006, 2007; Leone, 2003; Nishida et al., 2007], effective design guidelines for suppressing the EM radiation still remain unclear. Effective methods for predicting and suppressing EMI are desirable, which can maintain signal integrity (SI) over a broad band and have a clear physical meaning at the same time.
 Although common-mode (CM) component can dominate the total EM radiation at low frequencies (approximately 30 MHz–1 GHz), it has been found that differential-mode (DM) component should be taken into account in predicting EM radiation at the gigahertz frequencies [Kayano et al., 2005]. Nevertheless, since the DM component can be predicted by transmission line theory, correct prediction of CM at low frequencies is the key to the prediction of the total EMI behavior. Some mechanisms by which DM could be converted to CM noise sources have been demonstrated in the works by Hockanson et al. , Kami and Tobana , Watanabe et al. , Wada , and Shim and Hubing . These mechanisms include the disturbance of the ground potential due to ground inductance (current-driven) and the formation of a CM current path due to capacitive coupling (voltage-driven) [Hockanson et al., 1996]. The current and voltage-driven mechanisms are models for lower frequencies (below the first resonance of EMI antenna). Some other methods for predicting EM radiation require measurements of the CM current at one point on a cable, and an antenna factor. The imbalance difference model [Watanabe et al., 2000; Wada, 2003] is another successful model for predicting the CM radiation from a PCB. The results calculated by the ground-inductance model (current-driven mechanism) and the imbalance difference model were in good agreement at the lower frequencies. Both models adequately represent the phenomenon of the CM generation. However, for mitigation of EMI problems, the identification of the primary coupling mechanisms is not possible in the imbalance difference model, making the method insufficient for developing design guidelines. It is necessary to accumulate a lot of data for the transfer impedance which would be related EMI coupling paths in PCBs, and to make some suggestions for the design guideline. Therefore, a new model for predicting EM radiation from practical multilayer PCBs is needed to overcome the various limitations of the conventional approaches.
 Full-wave analysis such as the finite difference time domain (FDTD) method and the Finite Element Method (FEM) is suitable for PCB geometries to estimate their EMI and SI performance in high-speed electronic designs. However, full-wave analysis is time and memory consuming and could be very inefficient due to complex geometry and mixed scales among traces, multilayer planes with gaps, and vias. Hence, an effective physics-based model for predicting SI/EMI, which needs to have clear physical meanings so that EMI and SI design guidelines can be developed, is preferred.
 In this paper, characteristics of EM radiation from a stripline with a thin wire is investigated and predicted by an equivalent circuit model. The frequency responses of CM current on the PCB and EM radiation from the stripline structure are studied with FDTD modeling to provide basic considerations for the realization of methods for predicting EM radiation from the stripline structure. Then, an equivalent circuit model for predicting the CM current is proposed from the FDTD results. The equivalent circuit model for prediction is constructed based on the concepts of CM antenna impedance and distributed constant circuit. Finally, the validity of the equivalent circuit model is demonstrated by comparing the predictions with the FDTD simulation results.
2. PCB Geometry Under Study
 The geometry of the PCB under study is illustrated in Figure 1. To discuss and compare the characteristics of EM radiation from a multilayer PCB, two different PCB cross sectional structures, as shown in Figure 2, were prepared. Figure 2a shows a surface-microstrip line structure (S-MSL). The PCB has two metal layers, with a signal trace in the upper layer and a ground plane in the bottom. The trace is located on top of a dielectric substrate, and the upper side of the trace is exposed to air. Figure 2b shows a stripline structure (SL). The PCB has three metal layers, with a signal trace in the middle layer and the upper and lower layers as power/ground planes. The trace was sandwiched by the floating conductive plane and the ground plane. In a typical power bus structure, decoupling capacitors are placed between the power and ground planes. But, commonly EM radiation from power-bus/multilayer PCB structure is caused by an incomplete decoupling and/or grounding. In this paper they are neglected for the worst case EMI studies of multilayer PCBs.
 The width w of the reference plane, the thickness h of the dielectric substrate, the width wt of the trace, and the relative dielectric constant of the dielectric substrate affect the characteristic impedance Z0 of the trace. The sizes of the PCB are 150 mm in length and 10 mm in width. The thicknesses of the dielectric substrate for the S-MSL and the SL traces are 1.53 mm and 3.06 mm, respectively, and the dielectric constant is ɛr = 4.5. Since the return current at the edges of the ground plane is denser than that at an infinite ground plane, the current flows in both longitudinal (x-axis) and transverse (y-axis) directions. In order to ensure longitudinal (x-axis) current, the PCB geometry under study has a high aspect ratio (l ≫ w) compared with typical PCB. The trace is 50 mm long and is in the center of the PCB. The trace width wt is designed so that its characteristic impedance Z0 is 50 Ω. In order to make the impedance matching, the trace was terminated with a 50 Ω SMT resistor.
 A thin wire of 150 mm long (lw) with a diameter of 0.6 mm, used to mimic the ground wire of an interconnect, is connected to the edge of the ground plane. The ground thin wire and the ground plane form an EMI antenna as a dipole antenna, and the ground thin wire extends electrical length of the EMI antenna.
3. FDTD Modeling
 The FDTD method is used to model the CM current on the PCB as well as the radiated electric fields. The computational domain is discretized into cells with non-uniform sizes. The sizes of the cells in coarse region are chosen to be Δx = Δy = Δz = 2.5 mm and those in fine region near the signal trace and dielectric substance are Δy = 0.7 mm for the “S-MSL”, or 0.65 mm for the “SL”, Δx = 1.0 mm and Δz = 0.765 mm. The time step is Δt = 1.47 ps determined from the Courant stability condition. Perfectly matched layers (PMLs) [Berenger, 1996], 12 cells deep, are used as the absorbing boundary condition. The trace and ground plane are modeled as perfect electric conductors (PEC). The SMT resistor is modeled as a one-cell lumped element in the PCB substrate. The PCB substrate is modeled as a dielectric two/four cells deep with a relative permittivity r = 4.5. The source is modeled as a voltage source with an internal resistance of 50 Ω to take into account the 50 Ω measurement system. A sinusoidally modulated Gaussian pulse voltage with a frequency range from 10 MHz to 10 GHz given by equation (1) is applied as the source voltage.
 The CM current is calculated from the loop integral of the magnetic field around the PCB. Equivalent electric and magnetic current distributions can be determined by the calculated values of the electric and magnetic fields on the closed surface surrounding the FDTD model. The far-field radiation is then obtained using the near-to-far-field transformation from the surface electric and magnetic current distributions [Luebbers et al., 1991].
4. CM Current and EM Radiation From PCB Structure
 The calculated results of EMI in the cases of “without” and “with the ground thin wire” are shown in Figures 3 and 4, respectively. In both figures, the frequency responses of ICM on the PCB at x = 0, the horizontal component of the far-electric field Ex, and the vertical component of the far-electric field Ez are shown in Figures 3a and 4a, 3b and 4b, and 3c and 4c, respectively.
 EMI mechanisms can be described by the “Source-Path-Antenna” model [Kayano and Inoue, 2009], and the strength of EMI can be expressed as the product of three factors. The “Source” in this study is the source voltage Vs, which is 0 dBm (107 dBμV) in the entire frequency range. The “Path” refers to the radiated coupling paths between the source and antenna. The “Antenna” for the horizontal component is the dipole type antenna due to the CM current flowing along the PCB. The far-electric field Ex at r = 3 m is formulated through the “Source-Path-Antenna” model. The CM dipole type antenna in the work by Paul  is for the electrically short PCB case, in which the magnitude of current on the antenna is constant. In this study, Paul's formula is expanded in order to study the EM radiation from an electrically large PCB. In the frequency domain, the CM component in the far-electric field is given as [Paul, 1991]
where the CM(f) are defined as the mean CM current on the PCB, to take into account the current distribution, and is given as
 The strength of the far-electric field Ex can be represented as a product of three factors as
where the source voltage is constant (0 dBm); and, is the conversion efficiency from the source to the CM current, which is determined by geometrical structure and is not yet analytically calculated. The mean current CM(f) in this case is calculated by FDTD modeling. The third factor in equation (4), , is the efficiency of the CM antenna, which is uniquely given by geometrical parameters as
 For the case of “without the thin ground wire”, as shown in Figure 3, the magnitudes of the CM current and the horizontal component Ex in the “SL” structure are much smaller than those in the “S-MSL” structure, and the difference in magnitude at lower frequencies is about 40 dB. There is a resonance at 860 MHz in both structures. This resonant frequency corresponds to the resonant frequency of the CM current on the PCB and is related to λ/2, where λ is defined as the wavelength corresponding to the total EMI antenna length, which is formed by the ground plane in this case [Kayano et al., 2004]. These results indicate that the ground plane of the PCB forms the dominant radiation factor for the horizontal component Ex at low frequencies as a dipole-antenna. There are different resonant frequencies in the vertical component Ez results of the two structures, and the difference in magnitude is small. The component Ez is dominated by the DM component and the DM current can be calculated by transmission line theory [Paul, 1991].
 As shown in Figure 4, for the case of “with the ground thin wire”, the magnitudes of the CM current and the horizontal component Ex are larger than those in the case of “without the ground thin wire”.
 The calculated results obtained from the Source-Path-Antenna model agree well with the FDTD results within 6 dB except at antiresonance frequencies, by using the current distribution obtained from the FDTD model. These results demonstrate the validity of the numerical results obtained from the Source-Path-Antenna model at higher frequencies, and the correct calculation of the CM component is the key to the prediction of the total EMI behavior of a multilayer PCB.
 In order to discuss the effect of the ground thin wire on the horizontal component Ex, additional cases are studied. As shown in Figure 6, four structures are used: Figure 6a is a stripline without the thin wire, Figure 6b is a stripline without the thin wire but having a symmetrical excitation to suppress the cavity modes, Figure 6b is a stripline with the thin wire,and Figure 6d is a stripline with the thin wire and having a symmetrical excitation.
 The calculated results of the horizontal component Ex for the four structures are shown in Figure 7. From the comparison between Figures 7a and 7b, it can be seen that the symmetrical excitation can suppress the cavity modes. Comparing the results between Figures 7a and 7c shows a dramatic increase in magnitude of the horizontal component Ex due to the thin wire (56 dB). In addition, the frequency response in the case of “with the ground thin wire” follows a slope of 6 dB/octave below the first resonant frequency. On the other hand, for the “symmetrical excitation” cases, the magnitude increase of the horizontal component Ex due to the thin wire is about 20 dB. EMC design guidelines commonly suggest that a stripline structure rather than a microstrip line structure should be used in PCB structure for high-speed signals to achieve lower radiation. However, the results here indicate that stripline could yield high level EMI when a ground wire is attached due to asymmetry. Consequently, it is estimated that large EMI is caused by the floating conductive plane. And hence, the electric-coupling between the power plane and the ground thin wire should be considered in an equivalent circuit model for prediction.
5. Prediction of CM Radiation
5.1. Circuit Model for Predicting CM Current
 An equivalent circuit model for predicting the CM current in the stripline structure without the ground thin wire has been proposed and shown in Figure 8. Concepts of CM antenna impedance and distributed constant circuit include the current and voltage-driven mechanisms [Kayano and Inoue, 2009]. The equivalent circuit is comprised of three parts: a source, a transmission line based on the TEM assumption in consideration of a ground inductance, and a termination (load). The antenna impedances and are used to account for the displacement current path of CM current. Although at low frequencies (below the first resonant frequency fr1), the impedance is capacitive [Hockanson et al., 1996], CM current at high frequencies can not be predicted by the capacitive impedance. Two CM EMI antenna models are used to explain the frequency response of CM current [Kayano et al., 2004]. One CM EMI antenna is comprised of the ground plane, and the other CM EMI antenna is comprised of the trace on the ground plane. These antennas correspond to antenna impedances and , respectively.
 Since the methodologies of modeling a surface-microstrip line structure have been studied [Kayano and Inoue, 2009], this paper will be focused on the effects of the stripline structure and thin wire. Previous study has demonstrated that a physics-based equivalent circuit model is available if the floating conductive (power) plate and the ground thin wire are not existed. Additionally, the surface-microstrip line structure with a ground thin wire can be modeled as a wire-antenna model to consider the effect of the thin wire. Therefore, an equivalent circuit model is established based on the previous microstrip line model and the wire antenna model by adding the electric-coupling (capacitor) between the power plane and the thin wire. The antenna impedance of current-driven is modeled as a dipole antenna. In order to quantify , the following calculation has been implemented. The ground plane of the PCB geometry was separated to two parts: the ground side and the signal side with 1 mm width gap at x = 0 (CM voltage source position), as shown in Figure 9a. The antenna impedances of the structure with and without the thin wire cases were calculated by FDTD modeling. On the other hand, the antenna impedance of voltage-driven is modeled as electric coupling (capacitor) between the conductive plane and the thin wire, as shown Figure 9b. The capacitance is calculated as 1.95 fF by a 3D-field solver (ANSOFT, Maxwell Q3D Extractor). The calculated results of the antenna impedances are shown in Figure 10. Although at lower frequencies (up to 300 MHz) is inversely proportional to frequency as a typical capacitive antenna impedance, the resonant and antiresonant frequencies vary depending on whether the thin wire is attached. By calculating from at 30 MHz, the values with and without the thin wire are 1.26 pF and 1.03 pF, respectively.
5.2. Results and Discussions on Predicted EMI
 Predicted results on the CM current on the PCB are shown in Figure 11a with the FDTD simulation as a benchmark. For the case of “without the ground thin wire”, the CM current due to the current-driven mechanism dominates the total CM current ICM in the frequency range of interest. Therefore the CM current follows a slope of 12 dB/octave below the first resonant frequency. The first resonant frequency of the CM current is 860 MHz which results from the half-wavelength resonance of the dipole antenna. The values of the per unit length ground inductance LCM in the equivalent circuit model, 0.61 [nH/cm] for the “S-MSL” case and 0.02 [nH/cm] for the “SL” are calculated by the formula in the work by Koledintseva et al. . The decrease of 40 dB compared with S-MSL can be predicted by considering decreases of the ground inductance. For the case of “with the ground thin wire”, the CM current increases dramatically (by 56 dB). In addition, the frequency response follows a slope of 6 dB/octave below the first resonant frequency. These phenomena can be explained by the electric-coupling (capacitor) between the conductive plane and the thin wire in the equivalent circuit model.
 Predicted results of the CM component (horizontal component Ex) of the far-electric field are shown in Figure 11b. Since the CM radiation is proportional to frequency, the magnitudes of the far-electric fields Ex in the cases without and with the ground thin wire increase at the rates of 18 dB/octave and 12 dB/octave, respectively, below the first resonant frequency. The predicted and FDTD calculated results below the first resonant frequency are in good agreement within 3 dB. Although there are discrepancies at the resonant and antiresonant frequencies due to the dipole-like and cavity modes, the envelopes of the frequency responses are predicted with accuracy suitable for engineering applications. Good agreement between the full-wave modeling and predicted results demonstrates the validity of the proposed approach. Thus, using the physics-based model, the frequency response of undesired EM radiation from stripline structures can be predicted. More importantly, the proposed equivalent circuit model can identify the primary coupling-mechanisms of CM generation. Identifying the primary coupling mechanisms facilitates the mitigation of EMI problems in PCBs.
 The characteristics of EM radiation from a stripline structure with a ground thin wire is investigated and predicted by an equivalent circuit model, which is constructed with the insights gained from the results of FDTD modeling and from a previously developed model for a microstrip line structure by taking into account the additional electric coupling between the power plane and the thin ground wire. The equivalent circuit model is validated by comparing the predicted results with FDTD simulation. The equivalent circuit model provides enough flexibility for different geometric parameters and can be used to develop physical insights and design guidelines. This study has successfully established a basic method to effectively predict EM radiation from a stripline structure.
 Future directions include further validation and improvement of the circuit extraction approach, study of engineering implications and development of design guidelines.
 The authors sincerely thank to Cyberscience Center of Tohoku University, and General Information Processing Center of Akita University for their support with computer resources. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), 19760190, and Telecom Engineering Center.