**1. Introduction**

In high-speed printed circuit boards (PCBs) for digital communication systems, such as USB, HDMI, and PCI-Express technologies, electromagnetic interference during data transmission is a serious problem because the data rate of digital interfaces and the switching speed of processors continuously increase, but the voltage level continuously decreases. In recent high-speed PCB designs, simultaneous switching output noise in power delivery networks and near-field coupling noise in high-speed transmission lines notably undermine digital data transmission. To mitigate the electromagnetic interference effects, differential signaling is usually adopted. Given its balanced transmission, differential signaling is tolerant to both simultaneous switching output noise and near-field coupling noise. Hence, differential signaling is being increasingly adopted in high-speed PCBs by implementing a scheme using three conductor systems, namely positive signal line, negative signal line, and ground plane. Maintaining symmetry of those signal lines over the ground plane in high-speed PCBs is critical, because the noise robustness of differential signaling depends on the balanced scheme. However, it is difficult to prevent asymmetries in practical PCB designs. For instance, the unbalanced structures of meander delay lines, ball–grid–array escape routing, and asymmetric ground via configurations frequently occur during the practical design of high-speed PCBs. Moreover, current-strength mismatch of output driver circuits and trace-length mismatch in cables lead to unbalance in the differential scheme even if the PCB is symmetrical. Consequently, imbalance in differential signaling is inevitable in practical PCBs.

Unbalanced differential lines result in severe common-mode (CM) noise generation in high-speed PCBs [1–4]. Specifically, skewed signals in unbalanced differential lines generate wideband CM noise with high-order harmonics in the range of gigahertz and related electromagnetic interference problems. For intra-system electromagnetic interference, CM noise coupled to power delivery networks degrades the noise margin and timing budget of circuits mounted on the same board. Additionally, serious radio-frequency interference is produced when the CM noise flows into cables attached to PCBs. A practical example of radio-frequency interference induced by CM noise is presented in Reference [5], where experimental results show severe interference between a USB device and a 2.4-GHz wireless LAN device. Hence, CM noise caused by unbalanced differential structures should be mitigated in high-speed PCBs.

To suppress wideband and high-frequency CM noise in high-speed PCBs, metamaterial (MTM)-based techniques were proposed [6–11]. In Reference [6], differential transmission lines (DTLs), using a mushroom-type electromagnetic bandgap (EBG) structure with LTCC process technology, are presented. This type of EBG structure provides a wide stopband for even-mode propagation to suppress CM noise flowing along the differential lines, and the passband characteristics of odd-mode propagation ensure good differential-signal integrity during transmission. Then, CM noise suppression is predicted using dispersion analysis based on a lumped circuit model assuming a periodic structure, but prediction for differential signal transmission is not provided. In Reference [7], a complementary split ring resonator (CSRR) is employed to suppress CM noise of differential lines in high-speed PCBs. The CSRR etched in a ground plane forms an LC resonator to prevent even-mode propagation and behaves as an LC ladder network for odd-mode propagation. To predict selective CM noise suppression, a dispersion relationship is established from a lumped circuit model. However, the differential transmission characteristics are not estimated before performing full-wave simulations and measurements. As reported in References [8–11], a stepped impedance resonator is favorable for MTM-DTLs. In Reference [8], a stepped impedance resonator is implemented using various sizes for planar patches. A large patch and a narrow branch correspond to low and high characteristic impedances, respectively. CM noise suppression and differential signal quality are examined through full-wave simulations and experiments. In Reference [9], dual-type transmission lines are proposed to realize an MTM-DTLs. Impedance variations are implemented by alternating microstrip and strip lines, and a dispersion equation based on a periodic condition is derived, but it only predicts CM noise suppression. In References [10,11], a technique with ground planes vertically distributed in PCBs is proposed. The stepped impedance for odd-mode propagation is realized by variations of the vertical distances from differential lines to ground planes, representing a corrugated ground-plane electromagnetic bandgap (CGP-EBG) structure. Dispersion equations for odd-mode propagation are extracted to predict CM noise suppression only.

Therefore, recent research mainly focused on MTM-DTLs for CM noise suppression. It demonstrated wideband and high-frequency CM noise suppression with good differential signal integrity, simple implementation, and rigorous analysis based on dispersion relations. The key idea behind the development of MTM-DTLs is to provide structures handling different characteristics in the propagation modes of CM noise and differential-mode (DM) signal. To efficiently design and optimize MTM-DTLs, predicting and examining CM and DM characteristics of MTM-DTLs before their fabrication is needed. Although electromagnetic full-wave simulations can be employed, their prediction is computationally intensive in time and resources, thus being unsuitable for rapid design, testing, and prototyping of MTM-DTLs.

To overcome this problem, dispersion analysis with Floquet theory was adopted to estimate CM noise suppression. However, this analysis is limited to periodic structures, hindering the estimation of noise suppression characteristics when MTM-DTLs include a finite and small number of unit cells (UCs). Moreover, the periodicity of an MTM-DTL is not ensured in practical PCB applications. Likewise, dispersion analysis does not allow to accurately predict the suppression level, as its results only indicate whether a sufficient suppression will be provided in a stopband without retrieving a quantitative prediction. In addition, CM noise suppression characteristics in regions outside the stopband are not obtained, despite this information being critical in certain MTM-DTL applications. Moreover, the DM propagation associated with differential signaling characteristics was not estimated with a fast and simple approach in previous works. In Reference [12], the analytical method for the EBG structure employed in power delivery networks is presented. However, it is difficult to apply the method to DTLs because the segmen<sup>t</sup> model and recombined Z-parameter are limited to power delivery networks. Consequently, an efficient and accurate method to predict both CM noise and differential signaling characteristics of MTM-DTLs is still required for practical research and development of high-speed PCBs.

In this paper, an analytical model focusing on the MTM-DTL using a CGP-EBG structure is presented. In previous research [11], the CGP-EBG MTM-DTL was not fully characterized due to a limited method of CM noise prediction and a lack of estimation of differential transmission characteristics. Therefore, the research of the CGP-EBG MTM-DTL is extended by proposing an analytical model that efficiently provides rapid and accurate results for a nonperiodic array of CGP-EBG MTM-DTLs. The contribution of this paper is developing and verifying the analytical model, which simultaneously and quantitatively estimates the CM and DM propagation characteristics for the CGP-EBG MTM-DTL with a finite number of UCs.

#### **2. Analytical Model of CGP-EBG MTM-DTL**

#### *2.1. Description of CGP-EBG MTM-DTL*

In this section, a CGP-EBG MTM-DTL is briefly described, and the method for its analytical modeling is developed. Figure 1 shows a UC with its geometric parameters and the top and side views of the CGP-EBG MTM-DTL with a finite array size (three UCs). This CGP-EBG technique forms a stepped impedance resonator for even-mode propagation associated with CM noise and a constant impedance for odd-mode propagation associated with differential signal transmission by using the original distribution of ground planes. Basically, the characteristic impedances for odd- ( *Z*oo) and even-mode ( *Z*oe) propagations are determined by combining various geometric parameters of signal lines and ground plane. However, the CGP-EBG technique retrieves the desired *Z*oe and *Z*oo by only adjusting the vertical distance between signal lines and ground plane. The specific arrangemen<sup>t</sup> of ground patches, which are distributed in the different layers, provides varying *Z*oe but the same *Z*oo given in References [10,11]. The vertical distribution of ground planes efficiently achieves the decomposition of CM noise and differential signal propagation. The bandgap characteristics of the periodic CGP-EBG MTM-DTL were analyzed in References [10,11].

As shown in Figure 1, the UC of the CGP-EBG MTM-DTL consists of three parts, namely two DTLs with low *Z*oe (LZ-DTL) and length *d*L/2, and one DTL with high *Z*oe (HZ-DTL) and length *d*H. The vertical distance *h*L between signal transmission lines and a ground patch for LZ-DTL is relatively short compared to *h*H of HZ-DTL for obtaining low *Z*oe for LZ-DTL and high *Z*oe for HZ-DTL. Those ground planes are connected using plated-through-hole or blind vias for DC continuity. The via is separated from the center of the differential line by distance *S*v.

#### *2.2. Analytical Model*

The proposed model is based on an analytical expression for the impedance matrix (Z-matrix) of LZ- and HZ-DTLs and a segmentation method for recombining the Z-matrices. The procedure of the proposed method mainly consists of two steps, namely segmen<sup>t</sup> modeling and segmen<sup>t</sup> recombination, as illustrated in Figure 2. During segmen<sup>t</sup> modeling, analytical expressions for the Z-parameters of HZ- and LZ-DTLs are extracted using coupled-line theory. Moreover, an analytical model of the via, which is used for reference plane transition between HZ- and LZ-DTLs, is derived.

**Figure 1.** Schematic of a metamaterial differential transmission line (MTM-DTL) using a corrugated ground-plane electromagnetic bandgap (CGP-EBG) structure.

**Figure 2.** Procedure of proposed analytical method for predicting common-mode (CM) noise suppression and differential signal transmission of CGP-EBG MTM-DTLs.

The Z-parameters of the HZ- and LZ-DTLs are extracted using microwave theory for the four-port parallel-coupled lines that are represented in Figure 3a. The coupled line can be characterized by characteristic impedances of even and odd modes, a propagation constant, and a line length. In the parallel-coupled line for the HZ- and LZ-DTLs, the odd- and even-mode characteristic impedances at the *i*-th segmen<sup>t</sup> for recombination are denoted as *<sup>Z</sup>*oo(*i*) and *<sup>Z</sup>*oe(*i*), respectively, whereas the

propagation constant and length are denoted as *βi* and *di*, respectively. The voltage–current relationship for the parallel-coupled line of the *i*-th segmen<sup>t</sup> is given by

$$
\begin{pmatrix} V\_{1,\text{seg}(i)} \\ V\_{2,\text{seg}(i)} \\ V\_{3,\text{seg}(i)} \\ V\_{4,\text{seg}(i)} \\ \end{pmatrix} = \begin{pmatrix} Z\_{11,\text{seg}(i)} & Z\_{12,\text{seg}(i)} & Z\_{13,\text{seg}(i)} & Z\_{14,\text{seg}(i)} \\ Z\_{21,\text{seg}(i)} & Z\_{22,\text{seg}(i)} & Z\_{23,\text{seg}(i)} & Z\_{24,\text{seg}(i)} \\ Z\_{31,\text{seg}(i)} & Z\_{32,\text{seg}(i)} & Z\_{33,\text{seg}(i)} & Z\_{34,\text{seg}(i)} \\ Z\_{41,\text{seg}(i)} & Z\_{42,\text{seg}(i)} & Z\_{43,\text{seg}(i)} & Z\_{44,\text{seg}(i)} \end{pmatrix} \begin{pmatrix} I\_{1,\text{seg}(i)} \\ I\_{2,\text{seg}(i)} \\ I\_{3,\text{seg}(i)} \\ I\_{4,\text{seg}(i)} \end{pmatrix} \tag{1}
$$

where

$$Z\_{11, \text{seg}(i)} = Z\_{11, \text{seg}(i)} = Z\_{11, \text{seg}(i)} = Z\_{11, \text{seg}(i)} = -j/2 \left( Z\_{\text{osc}(i)} + Z\_{\text{osc}(i)} \right) \cot(\beta\_i d\_i),\tag{2a}$$

$$Z\_{12, \text{seg}(i)} = Z\_{21, \text{seg}(i)} = Z\_{34, \text{seg}(i)} = Z\_{43, \text{seg}(i)} = -j/2 \left( Z\_{\text{os}(i)} - Z\_{\text{os}(i)} \right) \cot(\beta\_i d\_i),\tag{2b}$$

$$Z\_{13, \text{seg}(i)} = Z\_{31, \text{seg}(i)} = Z\_{24, \text{seg}(i)} = Z\_{42, \text{seg}(i)} = -j/2 \left( Z\_{\text{osc}(i)} - Z\_{\text{osc}(i)} \right) \text{csc}(\beta\_i d\_i),\tag{2c}$$

$$Z\_{14, \text{seg}(i)} = Z\_{41, \text{seg}(i)} = Z\_{23, \text{seg}(i)} = Z\_{32, \text{seg}(i)} = -j/2 \left( Z\_{\text{os}(i)} + Z\_{\text{os}(i)} \right) \csc(\beta\_i d\_i). \tag{2d}$$

The analytical expressions for the Z-parameter of the *i*-th segmen<sup>t</sup> in Equation (2) are obtained from References [13,14]. The Z-parameter of the HZ- and LZ-DTLs are extracted by letting (*Z*oe(*i*) = *<sup>Z</sup>*oe,H, *<sup>Z</sup>*oo(*i*) = *Z*oo,H) and (*Z*oe(*i*) = *<sup>Z</sup>*oe,L, *<sup>Z</sup>*oo(*i*) = *Z*oo,L), respectively. The analytical expressions for the HZ- and LZ-DTLs at the *i*-th segmen<sup>t</sup> are obtained from Equations (2a)–(2d).

**Figure 3.** (**a**) Coupled line and (**b**) impedance parameter expression for a segmen<sup>t</sup> of CGP-EBG MTM-DTLs.

The physical transition from the HZ-DTL to the LZ-DTL is modeled to capture its inductive effect on return current. In particular, the return path for even-mode propagation is considered because the physical transition mainly contributes this propagation, as described in Reference [11]. The return path effect can be modeled as an effective partial inductance determined by the via diameter and length, the pitch between the via and the center of the signal transmission line, and the number of vias. In the proposed analytical method, only the via diameter and length are considered because they are the major parameters determining the effective inductance for the transitions. The inductive effect of the physical transition is analytically expressed as

$$L\_{\rm V} = \begin{cases} \ \mu\_0 \frac{h\_{\rm v}}{2\pi} \left[ \ln \left( \frac{2h\_{\rm v}}{r} + \sqrt{1 + \left( \frac{2h\_{\rm v}}{r} \right)^2} \right) - \sqrt{1 + \left( \frac{2h\_{\rm v}}{r} \right)^{-2}} + \left( \frac{2h\_{\rm v}}{r} \right)^{-1} + \frac{1}{4} \right] & \text{for even mode,} \\\ 0 & \text{for odd mode,} \end{cases} \tag{3}$$

where *μ*0 is the permeability of free space, and *h*v and *r* are the via length and radius, respectively, with *h*v = *h*H − *h*L. The analytical expression for the via inductance in Equation (3) is extracted from Reference [15].

Next, segmen<sup>t</sup> recombination for the proposed model is applied. In this step, a technique is presented for merging the Z-matrices of HZ- and LZ-DTLs connected to each other through a via inductance to obtain the Z-matrix of a finite array of CGP-EBG MTM-DTLs. Segment recombination consists of three parts, namely derivation of recombined Z-matrix, iterative recombination, and parameter conversion. To extract the recombined Z-matrix, the block diagram of the segmentation method is illustrated in Figure 4. Two segments are connected through via inductances. For the sake of generality, the segments are denoted as the *i*-th and (*i* + 1)-th segments. Each segmen<sup>t</sup> includes four ports, P1\_seg to P4\_seg. The Z-parameters of the segments are *<sup>Z</sup>*seg(*i*) and *<sup>Z</sup>*seg(*i*+1). Port P3\_seg of the *i*-th segmen<sup>t</sup> is connected to port P1\_seg of the (*i* + 1)-th segmen<sup>t</sup> through the via with inductance *L*v. Similarly, the other ports are connected through the via. Then, the recombined segmen<sup>t</sup> is conformed of Z-parameter and ports represented as *<sup>Z</sup>*rec(*i*), P1\_rec(*i*), P2\_rec(*i*), P3\_rec(*i*), and P4\_rec(*i*). Ports P1\_rec(*i*) and P2\_rec(*i*) of the recombined segmen<sup>t</sup> are associated with ports P1\_seg(*i*) and P2\_seg(*i*) of the *i*-th segment, whereas ports P3\_rec and P4\_rec of the recombined segmen<sup>t</sup> are associated with ports P1\_seg(*i*+1) and P2\_seg(*i*+1) of the (*i* + 1)-th segment. Using the segmentation method in Reference [16], recombined matrix Zrec(*i*) is extracted in terms of Zseg(*i*), Zseg(*i*+1), and *L*v with operator ⊕ representing Z-parameter recombination.

$$\begin{split} Z\_{\text{rec}(i)} &= Z\_{\text{seq}(i)} \oplus Z\_{\text{via}} \oplus Z\_{\text{seq}(i+1)} \\ &= Z\_{\text{pp}} + \left( Z\_{\text{pq}} - Z\_{\text{pr}} \right) \left( Z\_{\text{qq}} - Z\_{\text{qr}} - Z\_{\text{rq}} + Z\_{\text{rr}} + Z\_{\text{via}} \right)^{-1} \left( Z\_{\text{rp}} - Z\_{\text{qp}} \right), \end{split} \tag{4}$$

where

$$Z\_{\rm pp} = \begin{pmatrix} Z\_{11, \text{seg}(i)} & Z\_{12, \text{seg}(i)} & 0 & 0 \\ Z\_{21, \text{seg}(i)} & Z\_{22, \text{seg}(i)} & 0 & 0 \\ 0 & 0 & Z\_{33, \text{seg}(i+1)} & Z\_{34, \text{seg}(i+1)} \\ 0 & 0 & Z\_{43, \text{seg}(i+1)} & Z\_{44, \text{seg}(i+1)} \end{pmatrix} \prime \tag{5a}$$

$$Z\_{\rm Pq} = \begin{pmatrix} Z\_{13, \text{seg}(i)} & Z\_{14, \text{seg}(i)} \\ Z\_{23, \text{seg}(i)} & Z\_{24, \text{seg}(i)} \\ 0 & 0 \\ 0 & 0 \end{pmatrix} \tag{5b}$$

$$Z\_{\rm pr} = \begin{pmatrix} 0 & 0\\ 0 & 0\\ Z\_{31, \text{seg}(i+1)} & Z\_{32, \text{seg}(i+1)} 0\\ Z\_{41, \text{seg}(i+1)} & Z\_{42, \text{seg}(i+1)} \end{pmatrix} \tag{5c}$$

$$Z\_{\mathsf{qq}} = \begin{pmatrix} Z\_{\mathsf{33,seq}(i)} & Z\_{\mathsf{34,seq}(i)} \\ Z\_{\mathsf{43,seq}(i)} & Z\_{\mathsf{44,seq}(i)} \end{pmatrix},\tag{5d}$$

$$Z\_{\rm qr} = Z\_{\rm rq} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix},\tag{5e}$$

$$Z\_{\rm rr} = \begin{pmatrix} Z\_{11, \text{seg}(i+1)} & Z\_{12, \text{seg}(i+1)} \\ Z\_{21, \text{seg}(i+1)} & Z\_{22, \text{seg}(i+1)} \end{pmatrix} \tag{5f}$$

$$Z\_{\rm via} = \begin{pmatrix} j\omega L\_{\rm V} & 0\\ 0 & j\omega L\_{\rm V} \end{pmatrix} \tag{5g}$$

$$Z\_{\rm rp} = \begin{pmatrix} 0 & 0 & Z\_{13,\text{seg}(i+1)} & Z\_{14,\text{seg}(i+1)} \\ 0 & 0 & Z\_{23,\text{seg}(i+1)} & Z\_{24,\text{seg}(i+1)} \end{pmatrix} \tag{5h}$$

$$Z\_{\rm QP} = \begin{pmatrix} Z\_{\rm 31,seg(i)} & Z\_{\rm 32,seg(i)} & 0 & 0 \\ Z\_{\rm 41,seg(i)} & Z\_{\rm 42,seg(i)} & 0 & 0 \end{pmatrix} . \tag{5i}$$

**Figure 4.** Block diagram to derive recombined Z-parameter using segmentation method.

In the next part, recombination of the Z-parameter is iteratively applied. Suppose that a CGP-EBG MTM-DTL containing *N* UCs is given. The original problem of the MTM-DTL is replaced with subdomain problems constituting the *N* UCs further divided into HZ-DTLs, LZ-DTLs, and via transitions. The solutions of the subdomains are obtained using the expressions in Equations (1)–(3). Applying the recombined Z-parameter method iteratively, the subdomain solutions are updated to finally obtain the Z-parameter of the CGP-EBG MTM-DTLs with *N* UCs as

$$Z\_{\rm rec}(1) = Z\_{\rm DTL}(Z\_{\rm co,L\prime}, Z\_{\rm co\prime}, d\_{\rm L}/2) \oplus Z\_{\rm via} \oplus Z\_{\rm DTL}(Z\_{\rm co\prime,H\prime}, Z\_{\rm co\prime}, d\_{\rm H}),\tag{6a}$$

$$Z\_{\rm rec}(2) = Z\_{\rm rec(1)} \oplus Z\_{\rm via} \oplus Z\_{\rm DTL}(Z\_{\rm oc,L\prime}, Z\_{\rm co\nu}, d\_L), \tag{6b}$$

$$Z\_{\rm rec}(2(N-1)) = Z\_{\rm rec}(2(N-1)-1) \oplus Z\_{\rm via} \oplus Z\_{\rm DTL}(Z\_{\rm co,H}, Z\_{\rm co}, d\_{\rm H}),\tag{6c}$$

$$Z\_{\rm MTM-DTL} = Z\_{\rm rec}(2(N-1)+1) \oplus Z\_{\rm via} \oplus Z\_{\rm DTL} \left( Z\_{\rm oc,L\prime}, Z\_{\rm co\prime}, \frac{d\_{\rm L}}{2} \right),\tag{6d}$$

where *<sup>Z</sup>*DTL(*Z*oe,L*, Z*oo*, d*L/2) denotes a four-port Z-parameter of the LZ-DTL with the length being half of *d*L. The first and last segments of the CGP-EBG MTM-DTL, presented herein, are supposed to be the LZ-DTLs with length *d*L/2. Still, note that the proposed analytical model for the MTM DTL is not limited to this configuration. From the four-port Z-parameter of the CGP-EBG MTM-DTL, the S-parameter is given as

$$S\_{\rm MTM-DTL} = \left(Z\_{\rm MTM-DTL} + Z\_0E\right)^{-1} \left(Z\_{\rm MTM-DTL} - Z\_0E\right),\tag{7}$$

where *Z*0 is the reference characteristic impedance, commonly taken as 50 Ω, and *E* is the identity matrix. The CM and differential characteristics are finally obtained using mixed-mode S-parameter theory [17]. Only the analytical expressions for *S*cc21 and *S*dd21 are presented due to the focus on CM noise suppression and differential transmission characteristics.

$$S\_{\rm cc21} = \frac{1}{2} (\mathcal{S}\_{\rm 31, \rm MTM-DTL} + \mathcal{S}\_{\rm 41, \rm MTM-DTL} + \mathcal{S}\_{\rm 32, \rm MTM-DTL} + \mathcal{S}\_{\rm 42, \rm MTM-DTL}), \tag{8}$$

$$S\_{\rm dd21} = \frac{1}{2} (S\_{\rm 31,MTM-\rm DTL} - S\_{\rm 41,MTM-\rm DTL} - S\_{\rm 32,MTM-\rm DTL} + S\_{\rm 42,MTM-\rm DTL}).\tag{9}$$
