This paper presents a compact structure composed of an upper high-impedance transmission line, a middle extended parallel coupled line, and a pair of inter-coupled symmetrical stepped impedance stubs. Detailed investigation into this structure based on an equivalent circuit analysis reveals that this proposed structure exhibits a quasi-elliptic low-pass filtering response with three transmission zeros. Moreover, the positions of the three transmission zeros can be tuned and reallocated flexibly by choosing the proper circuit parameters. Finally, the design concept is validated through the design, fabrication, and measurement of two exemplary low-pass filters (LPFs) with one single unit and two cascaded asymmetric units. The measured results agree well with the simulated results. In addition, in the range of 1.42 f_c to 7.03 f_c , the fabricated quasi-elliptic LPFs experimentally demonstrate a very wide upper-stopband of 20 dB using a compact size of only 0.0089 λ _g ² , where λ _g is the guided wavelength of a 50 Ω transmission line at the central frequency. Microwave low-pass filters (LPFs) are frequently required in many types of wireless communication systems to suppress unwanted harmonics and spurious signals. Conventional LPFs adopting a stepped-impedance configuration cannot meet the requirements of modern communication systems because of their large size and narrow stopband [1] . Therefore, new LPFs with a compact size, low insertion loss, and wide stopband are highly desirable today. To address the above mentioned problems, many methods have been recently proposed. To obtain wide stopband lowpass filtering responses, different electromagnetic band-gap (EBG) structures are employed in [2] . However, it is well known that the periodic characteristic of an EBG causes a very large size. Defected ground structures (DGSs) have also been used in [3] and [4] to obtain a performance-improved LPF. However, a minimum air-space volume beneath the ground plane is required for the structure to function effectively. Additionally, these DGS structures with an aperture etched on the ground plane bring about many other disadvantages such as complex configuration and fabrication difficulties. Different structures with multiple transmission zeros are also employed to create an elliptic-function low-pass response with a wider stopband. In [5] , elliptic-function LPFs based on microstrip stepped-impedance hairpin resonators have been presented by creating two coupling routes between non-adjacent sections. However, the filter transition in [5] is still gradual since the only transmission zero is far from the cutoff frequency. Different LPF topologies with two transmission zeros have also been investigated in [6] - [8] . However, since only two transmission zeros are provided, these filters are unavailable for the application when a wide and high-rejected stopband are required. Though multi-section structures composed of two asymmetrical cascaded units are employed in [7] and [8] to expand the stopband, this will greatly increase the filter size and design complexity. To generate more transmission zeros, a tightly coupled hairpin unit [9] and a capacitive loaded anticoupled line [10] are presented to realize an LPF with three transmission zeros. However, the narrow zero-separation of the LPF in [9] and the larger size in [10] seriously limit their application. More than three transmission zeros using a stubloaded coupled-line hairpin resonator are generated in [11] . However, several fluctuations in the stopband result in a significant deterioration of the stopband-rejection performance of an LPF. In this paper, a compact structure is proposed to construct LPFs with an improved stopband performance. The investigation of this structure based on an equivalent circuit analysis reveals that this proposed structure exhibits a quasielliptic low-pass filtering response with three transmission zeros. Moreover, the mechanism for reallocating and adjusting the transmission zeros is also explained in detail. It was found that a larger zero separation can be easily achieved by choosing the proper circuit parameters. A simultaneously sharpened, high-rejected, and expanded stopband can thus be obtained. Finally, two LPFs using a single unit and two asymmetrical units, respectively, were optimally designed. The measured results agree well with the simulation results. Both the simulations and measurements of the two fabricated filters exhibit a good quasi-elliptic low-pass response including a sharp skirt, a lower passband insertion loss, an expanded stopband, and a compact size. Figure 1 shows a schematic of the proposed LPF. Assuming the structure is lossless that and all discontinuity effects can be neglected, the proposed structure can be divided into three shunt networks: one is the upper high-impedance transmission line, shown in Fig. 2 ; another is a mid-extended parallel coupled line, shown in Fig. 3 ; and the lower one is a pair of inter-coupled symmetrical stepped-impedance stubs, shown in Fig. 4 . As shown in Fig. 2 , the upper single transmission line with a characteristic impedance of Z _0u and a length of l_u can be modeled as an equivalent L-C π-network. For this upper lossless transmission line, the ABCD matrix is given by (1) $$\{\begin{array}{l}{A}_u=\cos ({\beta }_u{l}_u),\\ B_u=j{Z}_{0u}\sin ({\beta }_u{l}_u),\\ C_u=j{Y}_{0u}\sin ({\beta }_u{l}_u),\\ D_u=\cos ({\beta }_u{l}_u),\end{array}$$ where β_u and Y _0u = 1/ Z _0u are the phase constant and characteristic admittance of the single transmission line, respectively. In addition, the equivalent inductance and capacitance of the lumped π-network circuit are given by [5] (2) $$L_u=\frac{Z_{0u}\sin ({\beta }_u{l}_u)}{{\omega }_{}},$$ (3) $$C_u=\frac{1-\cos ({\beta }_u{l}_u)}{\omega Z_{0u}\sin ({\beta }_u{l}_u)}.$$ For the mid-extended parallel coupled line, the total ABCD matrix is obtained as [1] (4) $$\{\begin{array}{l}{A}_m=A_1^2{A}_2+A_1{B}_1{C}_2+A_1{C}_1{B}_2+B_1{C}_1{A}_2,\\ B_m=2A_1{B}_1{A}_2+B_1^2{C}_2+A_1^2{B}_2,\\ C_m=2A_1{C}_1{A}_2+A_1^2{C}_2+C_1^2{B}_2,\\ D_m=A_1^2{A}_2+A_1{B}_1{C}_2+A_1{C}_1{B}_2+B_1{C}_1{A}_2,\end{array}$$ where The mid-section can also be represented by the lumped π-network circuit in Fig. 3 , and the equivalent capacitance of C_mg and C_mp can be given by (5) $$C_{mg}=1/(j\omega B_m),$$ (6) $$C_{mp}=(A_m-1)/(j\omega B_m).$$ The lower section in Fig. 4 consists of one cascaded structure composed of two bilateral high-impedance transmission lines and one symmetric parallel coupled-line. For the bilateral transmission line with a characteristic impedance of Z _0l and a length of l_{l 1} , the ABCD matrix is given by (7) $$\{\begin{array}{l}{A}_{l1}=\cos ({\beta }_{l1}{l}_{l1}),\\ B_{l1}=j{Z}_{0l}\sin ({\beta }_{l1}{l}_{l1}),\\ C_{l1}=j\sin ({\beta }_{l1}{l}_{l1})/Z_{0l},\\ D_{l1}=\cos ({\beta }_{l1}{l}_{l1}),\end{array}$$ where β_{l 1} is the phase constant of the bilateral transmission line. This transmission line can be represented by the lumped π-network circuit in Fig. 4 . In addition, the equivalent inductance and capacitance [5] are given by (8) $$L_{l1}=\frac{Z_{0l}\sin ({\beta }_{l1}{l}_{l1})}{\omega },$$ (9) $$C_{l1}=\frac{1-\cos ({\beta }_{l1}{l}_{l1})}{\omega Z_{0l}\sin ({\beta }_{l1}{l}_{l1})}.$$ For the symmetric parallel low-impedance coupled-line with odd- and even-mode characteristic impedances of Z _0ol and Z _0el , and a length of l _l2 , the ABCD matrix of the lossless parallel coupled-line [1] , [5] can be expressed as (10) $$\{\begin{array}{l}{A}_{l2}=\frac{Z_{0el}+Z_{0ol}}{Z_{0el}-Z_{0ol}},\\ B_{l2}=-j\frac{2Z_{0el}{Z}_{0ol}\cot ({\beta }_{l2}{l}_{l2})}{Z_{0el}-Z_{0ol}},\\ C_{l2}=j\frac{2\tan ({\beta }_{l2}{l}_{l2})}{Z_{0el}-Z_{0ol}},\\ D_{l2}=\frac{Z_{0el}+Z_{0ol}}{Z_{0el}-Z_{0ol}},\end{array}$$ where β_{l 2} is the phase constant of the lower coupled lines. The equivalent π-network lumped model can be obtained as shown in Fig. 4 , and the capacitances of the π-network are found [5] as (11) $$C_{l2g}=\frac{Z_{0el}-Z_{0ol}}{2{\omega }_0{Z}_{0el}{Z}_{0ol}\cot ({\beta }_{l2}{l}_{l2})},$$ (12) $$C_{l2p}=\frac{1}{{\omega }_0{Z}_{0el}\cot ({\beta }_{l2}{l}_{l2})}.$$ Furthermore, if the discontinuity effects are ignored, the lumped equivalent circuit of the proposed structure is shown in Fig. 5 by combining the above derived equivalent circuits of the three single sections, where L ₁ = L_u , C ₁ = C_mg , L ₂ = L_{l 1} , C ₂ = C_{l 1} + C_{l 2 p} , C ₃ = C_{l 2 g} , and C ₄ = C_u + C_mp + C_{l 1} . To gain insight into this structure, we first assume C ₃ = 0, which is a reasonable approximation since C ₃ << C ₂ for a loose coupled-line. When C ₃ is removed, the model in Fig. 5 can be considered as a three-pole LC LPF. In addition, as observed from the simplified model, two transmission zeros positioned at f_{z 1} and f_{z 2} are generated as the result of the parallel resonance between L ₁ and C ₁ and serial resonance between L ₂ and C ₂ . Clearly, f_{z 1} and f_{z 2} can be derived as (13) $$f_{z1}=1/(2\text{π}\sqrt{L_1{C}_1}),$$ (14) $$f_{z2}=1/(2\text{π}\sqrt{L_2{C}_2}).$$ Let f_{z 1} > f_{z 2} by choosing the appropriate L ₁ , C ₁ , L ₂ , and C ₂ , and the transmission zero at f_{z 2} then creates a sharpened transition, and the transmission zero at f_{z 1} contributes to the stopband expansion. Clearly, a larger transmission zero separation will help to sharpen and extend the stopband. However, the stopband rejection between two transmission zeros is also deteriorated with an increase in the zero separation since only two transmission zeros exist. To obtain a sharpened, extended, and high-rejected stopband, C ₃ is introduced to create an extra transmission zero between f_{z 1} and f_{z 2} . To gain insight into the effect of C ₃ on the structure’s response, the odd- and even-mode analysis method is employed to investigate this filter by utilizing the symmetry of the equivalent model. The equivalent odd- and even-mode circuits are given in Figs. 6(a) and 6(b) , which were obtained from the replacement of the reference plane with magnetic and electric walls, respectively. (15) $$Y_{\text{ino}}=\frac{2}{j\omega L_1}+2j\omega C_1+j\omega C_4+\frac{1}{j\omega L_2+\frac{1}{j\omega C_2+2j\omega C_3}},$$ (16) $$Y_{\text{ine}}=j\omega C_4+\frac{1}{j\omega L_2+\frac{1}{j\omega C_2}}.$$ Clearly, C ₃ exerts an effect on Y _ino , but has no effect on Y _ine . Network theory [1] relates the transmission coefficient of the filter to the odd- and even- mode input admittances through the following formula: (17) $$S_{21}=Y_0\frac{Y_{\text{ino}}-Y_{\text{ine}}}{(Y_0+Y_{\text{ino}})(Y_0+Y_{\text{ine}})},$$ (18) $$S_{11}=\frac{Y_0^2-Y_{\text{ino}}{Y}_{\text{ine}}}{(Y_0+Y_{\text{ino}})(Y_0+Y_{\text{ine}})},$$ where Y ₀ = 1/ Z ₀ is the input and output microstrip line characteristic admittance. By letting S ₂₁ = 0, the transmission zeros are positioned at the frequencies where (19) $$Y_{\text{ine}}=Y_{\text{ino}}.$$ Based on the above theoretical analysis, the theoretical transmission responses of the LPF with different C ₃ varying from 0 pF to 0.3 pF are as shown in Fig. 7 . In each of these cases, L ₁ = 18 nh, C ₁ = 0.1 pF, L ₂ = 2.5 nh, C ₂ = 6 pF, and C ₄ = 1.5 pF. As illustrated in Fig. 7 , only two transmission zeros appear on the S ₂₁ curve when C ₃ = 0. In fact, (19) will be inevitably satisfied when f = f_{z 1} or f_{z 2} , which coincides well with (13) and (14). When C ₃ increases from 0, though the two transmission zeros on the left and right sides are both drawn to the lower area, the zero separation, that is, the stopband bandwidth, remains almost unchanged. Moreover, since the left transmission zero shifts to the passband with an increase in C ₃ , a sharpened filter transition is easily obtained. Furthermore, owing to the introduction of C ₃ , an extra transmission zero appears between the original two transmission zeros. This additional transmission zero also moves to the upper area with an increase in C ₃ . Therefore, the stopband rejection can be effectively improved by choosing a proper C ₃ to properly relocate the zeros. Additionally, the change in C ₃ hardly alters the filter’s passband, which largely facilitates the filter design and adjustment. Figure 8 shows the geometry of the LPF using one of the proposed structures. The filter was fabricated on a 0.5 mm substrate with a relative dielectric constant of 2.65. The optimized dimensions of the LPF are arranged as follows: l_u = 24.6 mm, w_u = 0.2 mm, w ₅₀ = 1.35 mm, w_m = 0.2 mm, w_{l 1} = 0.2 mm, l_{l 1} = 1.2 mm, w_{l 2} = 8.1 mm, l_{l 2} = 7.9 mm, s_l = 0.4 mm, l_{m 1} = 2.7 mm, l_{m 2} = 10.8 mm, and s_m = 0.3 mm. The simulated and measured results, which show a good agreement, are illustrated in Fig. 9 . This LPF has a measured 3 dB cutoff frequency of f_c at about 0.85 GHz. The measured passband insertion loss is within 0.5 dB including the SMA connector loss in the frequency range up to 0.7 GHz. The three transmission zeros can be observed to be located at 1.65 GHz with 54.5 dB, at 2.65 GHz with 67 dB, and at 3.17 GHz with 36.25 dB, respectively. The transmission zero located at 1.65 GHz contributes to a sharper transition; the one at 3.17 GHz extends the stopband; and the one at 2.65 GHz provides additional rejection between the lower and upper transmission zeros. Owing to the co-operation of these three transmission zeros, the rejection band is extended from 1.31 GHz (1.54 f_c ) to 3.23 GHz (3.8 f_c ) over 20 dB, resulting in a 20 dB stopband bandwidth of 2.26 f_c . Moreover, the filter occupies a compact size of 16.6 mm × 14.65 mm corresponding to 0.0042λ _g ² (0.068λ _g × 0.061λ _g ), where λ _g is the guided wavelength of a 50 Ω transmission line at f_c . To obtain an even broader stopband and strengthen the stopband attention, a LPF with two cascaded asymmetrical proposed units is designed, fabricated, and tested. Figure 10 shows the LPF developed from the filter shown in Fig. 9 by cascading the second unit with different dimensions, corresponding to different cutoff frequencies and positions of the transmission zeros. The two cells are connected by a short high-impedance line with a narrow width of 0.16 mm and a length of 6.5 mm. The dimensions of the second smaller unit are l_ur = 12.8 mm, w_ur = 0.2 mm, w_mr = 0.2 mm, w_{l 1 r} = 0.2 mm, l_{l 1 r} = 1.1 mm, w_{l 2 r} = 4.5 mm, l_{l 2 r} = 5.6 mm, s_lr = 0.3 mm, l_{m 1 r} = 2.1 mm, l_{m 2 r} = 4.7 mm, and s_mr = 0.32 mm. This smaller-sized unit is designed to extend the stopband and suppress the harmonic resonance at around 4 GHz, as shown in Fig. 9 . The smaller optimized unit has three transmission zeros at 2.68 GHz, 4.72 GHz and 5.75 GHz. The HFSS simulation and measurement are shown in Fig. 11 . As shown in Fig. 11 , this LPF exhibits a good low-pass response with a measured f_c at about 0.875 GHz. The filter exhibits a good passband performance with a return loss of larger than 22 dB. In addition, multiple transmission zeros can be clearly observed to be positioned at 1.62, 2.48, 3.14, 4.79, and 6.18 GHz. As a result of the effects of these multiple transmission zeros, the 20 dB rejection stopband is extended from 1.275 GHz (1.457 f_c ) to 6.3 GHz (7.2 f_c ) over 20 dB, indicating a very wide 20 dB stopband bandwidth of 5.743 f_c . The attenuation rates at the passband to stopband transition knees are 87 dB/GHz (the measured attenuations being 5 and 65 dB at 0.96 GHz and 1.65 GHz, respectively). Furthermore, the filter’s size is only 32.4 mm × 14.65 mm, corresponding to 0.0089λ _g ² (0.14λ _g × 0.0635λ _g ), where λ _g is the guided wavelength of a 50 Ω transmission line at 0.875 GHz. Table 1 shows a performance comparison of the proposed LPF with some recently reported compact and high-performance LPFs. As shown in the table, the proposed LPF in this paper exhibits a good return loss, compact size, and significantly wider stopband than the other LPFs. This paper presented a compact structure with three inherent transmission zeros and its application in a compact and high-performance LPF design. A theoretical analysis reveals that the stopband performance of the LPFs based on the proposed structure can be effectively upgraded by adjusting the circuit’s electrical parameters to generate and reallocate the multiple transmission zeros. Finally, after a single-unit LPF was extensively studied, an improved LPF with two cascaded asymmetric units was developed. The measured results of the cascaded LPF demonstrate many desirable features as compared with a conventional LPF, particularly its ultra-wide 20 dB rejection stopband and compact size. lilin_door@hotmail.com Lin Li received his BS in electrical engineering from Zhejiang University, Hangzhou, China in 1999, his MS in electronic science and technology from Zhejiang University, Hangzhou, China in 2005, and his PhD in electromagnetic field and microwave technology from Shanghai Jiaotong University, Shanghai, China in 2009. His current research interests include microwave integrated circuits and antenna technologies. baojia@zstu.edu.cn Jia Bao received her BS in applicative electron technology from Anhui Normal University, Wuhu, China in 2000, and her MS in power electronics and power transmission from South West Jiaotong University, Chengdu, China in 2003. She is currently a PhD candidate of mechanical engineering at Zhejiang Sci-Tech University, Hangzhou, China. Her current research interests include embedded systems, network applications, and protocol development. Jingjdu@163.com Jing-jing Du received her BS in electronic and information engineering from Harbin Engineering University, Harbin, China in 1999, and her MS in circuits and systems from Hangzhou Dianzi University, Hangzhou, China in 2006. She is currently a PhD candidate of mechanical engineering at Zhejiang Sci-Tech University, Hangzhou, China. Her current research interests include EDA, image processing, and information fusion. ywang@zist.edu.cn Yaming Wang earned his PhD degree in biomedical engineering from Zhejiang University, China. He is currently a professor of computer science at Zhejiang Sci-Tech University, Zhejiang, China. He was a visiting researcher and visiting scientist at Hong Kong University of Science & Technology, HKUST. His current title is Dean of the School of Information Science and Technology at Zhejiang Sci-Tech University. His research interests include computer vision, pattern recognition, signal processing, and medical image processing.