Design equation of parallel and open-ended coupled microstrip line

A half-wavelength coupled transmission line resonator is used for the BPF design. These resonators are positioned parallel to each other to achieve maximum coupling by optimizing the space between the resonators as illustrated in Fig. 4. This structure provides a wider bandwidth to the end-coupled microstrip bandpass filter by suppression of harmonics. The proposed geometry of the BPF is designed over an FR-4 substrate with a thickness (h) of 1.6 mm and a relative permittivity (ɛ_r) of 4.4. The used substrate may be suitable for the TE mode propagation up to 25.89 GHz as per the relation f_TE = c/(4 × h × [√ɛ_eff − 1]) [Reference Awasthi, Singh, Sharma, Kumari and Verma23], where c is the speed of light (3 × 10⁸ m/s) and ɛ_eff is the effective relative permittivity.

Fig. 4. Transmission line model of the proposed UWB bandpass filter.

The design equations of the filter are given as follows. All the values regarding bandpass analysis are given in Table 1:

(1)$$\displaystyle{{J_{01}} \over {Y_0}} = \displaystyle\displaystyle{\sqrt {{\pi \over 2} \times {FBW \over g_0 g_1} }} = 50.08 $$

(2)$${{{J_{\,j,j + 1}}} \over {{Y_0}}} = {{\pi \times FBW} \over 2} \times {1 \over {\sqrt {{g_j} \times {g_{\,j + 1}}} }},\;\;j = 1\;{\rm{ to }}\;n - 1$$

Table 1. Element values for optimum bandpass filter for ripple 0.5 dB

Therefore, J _1,2/Y ₀ = J _4,5/Y ₀ = 50.08 Ω, J _2,3/Y ₀ = J _3,4/Y ₀ = 39.65 Ω:

(3)$$\displaystyle{{J_{n, n + 1}} \over {Y_0}} = \sqrt {\displaystyle{\pi \over 2} \times \displaystyle{{FBW} \over {g_ng_{n + 1}}}} $$

where g ₀, g _1, g ₂,…,g_n are the elements of a prototype bandpass filter with normalized cut-off Ω_c = 1 and FBW is a fractional bandwidth of the filter. J_j, _j+1 is the characteristic admittance of the J-inverter and Y ₀ is the characteristic admittance of the terminating transmission line. For the realization of J-inverter, equations (1)–(3) [Reference Hong and Lancaster24] are used and characteristic impedance of parallel-coupled microstrip line resonators for even and odd modes is obtained by using equations (4) and (5) as follows:

(4)$$( {Z_{0e}} ) _{\,j, j + 1} = \displaystyle{1 \over {Y_0}}\left[{1 + \displaystyle{{J_{\,j, j + 1}} \over {Y_0}} + {\left({\displaystyle{{J_{\,j, j + 1}} \over {Y_0}}} \right)}^2} \right] = 73\,\Omega , \;\;j = 0\,{\rm to}\,n$$

(5)$$( {Z_{00}} ) _{\,j, j + 1} = \displaystyle{1 \over {Y_0}}\left[{1-\displaystyle{{J_{\,j, j + 1}} \over {Y_0}} + {\left({\displaystyle{{J_{\,j, j + 1}} \over {Y_0}}} \right)}^2} \right] = 35\,\,\Omega , \;\;j = 0\;{\rm to}\;n$$

where Y ₀ = 1/Z ₀ = 1/50 mho.

The dimensions of the filter can be determined by even and odd mode characteristic impedance. The actual length of each coupled line can be determined by using equation (6):

(6)$$L_j = \displaystyle{{\lambda _0} \over {4\left({\sqrt {{( {\varepsilon_{re}} ) }_j \times {( {\varepsilon_{ro}} ) }_j} } \right)}}-\Delta L_j = 15.48\,\,{\rm mm}$$

where L_j is the equivalent length of open-ended microstrip line and, (ɛ_re)_j = C_e/C_e^a = 4.08, (ɛ_ro)_j = C_o/C_o^a = 3.82, C_e is the even mode capacitance of the dielectric for parallel-coupled microstrip line, C_e^a is the even mode capacitance in air, C_o is the odd mode capacitance of the dielectric, and C_o^a is the odd mode capacitance in air, for a coupled microstrip line.

Calculation of microstrip line width (w)

Width of open-ended and parallel-coupled transmission lines calculated by using equation (7) [Reference Awasthi, Singh, Sharma, Kumari and Verma23] is shown in Fig. 2(a):

(7)$$\displaystyle{w \over h} = \left\{{\displaystyle{{8e^x} \over {e^{2x}-2}}} \right\}, \;\;\;w/h \le 2$$

The width of the microstrip line (w) used to design the proposed BPF is 1.89 mm and the same is calculated by using the following equation:

$$x = \displaystyle{{Z_0} \over {60}}\sqrt {\displaystyle{{\varepsilon _r + 1} \over 2} + \displaystyle{{\varepsilon _r-1} \over {\varepsilon _r + 1}}\left({0.23 + \displaystyle{{0.11} \over {\varepsilon_r}}} \right)} = 1.39$$

where Z ₀ is the characteristic impedance of the microstrip transmission line, i.e. 50 Ω.

Calculation of length of parallel microstrip-coupled line (L_mcl)

The length of the line is calculated by using equation (8). L ₁ and L ₂ are the lengths of open-ended parallel transmission lines which are optimized by simulation, where L ₁ is equal to L ₂ due to symmetry as shown in Fig. 2(a):

(8)$$L_{mcl} = 2L_1 + 2L_2 = \lambda _{eff} = 21.78\,{\rm mm}$$

where the effective wavelength is $\lambda _{eff} = c/f_{\min }\sqrt {\varepsilon _{eff}} = 4.44\,{\rm mm}$ 21.78 mm and c is the speed of light in a vacuum and ɛ_eff can be calculated [Reference Awasthi, Singh, Sharma, Kumari and Verma23] by using equation (9):

(9)$$\varepsilon _{eff} = \displaystyle{{\varepsilon _r + 1} \over 2} + \displaystyle{{\varepsilon _r-1} \over 2}\displaystyle{1 \over {\sqrt {1 + 12( {h/w} ) } }}\approx 4.1$$

Calculation and analysis of MMR and DGS

The multi-mode resonance techniques offer low insertion loss and better matching with the transmission line. The proposed filter has mixed-shaped resonators (circular and Mickey) and DGSs (arrowhead and circular) as shown in Fig. 2:

(10)$$R_{dgs}( {\rm cm}) = \displaystyle{\,p \over {\sqrt {\{ {1 + ( {2h/\pi \varepsilon_rp} ) [ {\ln ( {\pi p/2h} ) + 1.7726} ] } \} } }} = 0.209\,{\rm cm} = 2.09\;{\rm mm}$$

DGS provides a low profile and better degradation/sharpness (dB/GHz) in stopband to the filter. The radius of circular stubs and slots (R_dgs) is calculated from equation (10) and approximately the same area (A ₁) is optimized for the triangular shape to increase required corner discontinuity [Reference Kumar and Pathak17].

In the above equation, $p =\! ( {( {8.791 \!\times\! {10}^9} ) /f_{min}} )\! \times\! \left({1\!/\!\sqrt {\varepsilon_r} } \right)\! =\! 1.4$, f_min is the lowest cut-off frequency of the proposed filter, i.e. 3.1 GHz.

The equivalent circuit of the proposed DGS is shown in Fig. 5 and the values of resistance R (Ω), capacitance C (pF), and inductance L (nH) can be calculated by using equations (11)–(13) [Reference Hong and Lancaster24], respectively. The inductance and capacitance are mostly influenced by the width of the coupling gap (g) and the neck length (L_n) of the DGS slot. The resistance R (Ω) shows the loss in the DGS which can be computed by using equation (11). Plate capacitance (C_p) is approximately 1835.68 nF by calculation:

(11)$$R( {\rm \Omega } ) = \displaystyle{{2 \times Z_0} \over {\sqrt {[ {1/{\vert {S_{11}} \vert }^2-{\{ {2 \times Z_0( {\omega_dC-( {1/\omega_dL} ) } ) } \} }^2-1} ] } }} = 41.66\,{\rm \Omega }$$

where ω_d is the design frequency of the filter, i.e. 6.85 GHz. The values of C (pF) and L (nH) can be computed by using equations (12) and (13):

(12)$$C( {\rm pF}) = 5\left({\displaystyle{{\,f_c} \over \pi }} \right)( {\,f_0} ) ^2( {\,f_c} ) ^2 = 2225.90\,{\rm pF}$$

(13)$$L( {\rm nH}) = \displaystyle{{250} \over C}\left({\displaystyle{\pi \over {\,f_0}}} \right)^2 = 23.6\,{\rm nH}$$

Fig. 5. Proposed DGS: (a) structure, (b) equivalent circuit and (c) equivalent circuit model of all DGSs.

Bandstop characteristic analysis

Open-ended microstrip lines that are quarter wavelength length provide bandstop characteristics. Stopband filtering characteristic depends upon the characteristic impedance of the open-ended transmission line. This type of bandstop filter provides wide bandstop characteristics. The design procedure starts with a ladder-type bandpass prototype filter by using equation (14) [Reference Hong and Lancaster24]:

(14)$$\Omega = \Omega _C \times \alpha \times \tan \left({\displaystyle{\pi \over 2} \times \displaystyle{\,f \over {\,f_0}}} \right) = 0.59$$

where $\alpha = \cot [ {( {\pi /2} ) ( {1-FBW/2} ) } ] = 1.63$, Ω and Ω_C are normalized frequency variable and cut-off frequency of prototype bandpass filter, and f ₀ and f are the mid-band frequency and frequency variables of the bandstop filter. FBW of the bandstop filter is defined by using equation (15):

(15)$$FBW = \displaystyle{{\,f_2-f_1} \over {\,f_0}} = 0.6, \;\quad f_0 = \displaystyle{{\,f_1 + f_2} \over 2} = 15.3\,{\rm GHz}$$

f ₁ and f ₂ are the lower and higher cut-off frequencies of the bandstop filter. These types of bandstop characteristics periodically exist, that is an odd multiple of frequency f ₀. At this frequency, open circuit stub length is an odd multiple of λ_g/4 where λ_g denotes the guided wavelength at frequency f ₀ so they shorted with the 50 Ω line (port-1 and port-2) provides bandstop characteristic. To design nth order optimum bandstop filter by using the optimum transfer function technique shown in equation (16) at design frequency (6.75 GHz) [Reference Hong and Lancaster24]:

(16)$$\vert {S_{21}( f ) } \vert ^2 = \displaystyle{1 \over {1 + \varepsilon ^2 \times F_n^2 ( f ) }} = 0.25$$

where ɛ denotes the ripple constant in passband and F_n is the filtering function which can be calculated by using equation (17):

(17)$$\eqalign{F_n( f ) = & T_n\left({\displaystyle{t \over {t_c}}} \right)T_{n-1}\left({\displaystyle{{t\sqrt {1-t_c^2 } } \over {t_c\sqrt {1-t^2} }}} \right)-U_n\left({\displaystyle{t \over {t_c}}} \right)U_{n-1}\left({\displaystyle{{t\sqrt {1-t_c^2 } } \over {t_c\sqrt {1-t^2} }}} \right)\cr = & 2.25\;( {{\rm at}\;{\rm design}\;{\rm frequency} = 6.75\,{\rm GHz}} ) } $$

where t is Richard's transform variable:

(18)$$t = j\tan \left({\displaystyle{\pi \over 2} \times \displaystyle{\,f \over {\,f_0}}} \right) = 0.879j\;{\rm and}\;t_c = j\tan \left({\displaystyle{\pi \over 4} \times ( 2-FBW) } \right) = 0.86j$$

T_n(x) and U_n(x) are Chebyshev's functions of order n and are calculated by using equations (19) and (20) [Reference Hong and Lancaster24]:

(19)$$T_n( x) = \cos ( {n \times {\cos }^{{-}1}x} ) $$

(20)$$U_n( x) = \sin ( {n \times {\cos }^{{-}1}x} ) $$

The characteristic impedance of each open-ended section can be determined by using equations (21)–(23):

(21)$$Z_{\,port \hbox{-} 1} = Z_{\,port \hbox{-} 2} = Z_0 = 50\Omega $$

(22)$$Z_i = \displaystyle{{Z_0} \over {g_i}}$$

so that Z ₁ = Z ₅ = 97.30 Ω, Z ₂ = Z ₄ = 53.32 Ω, and Z ₃ = 50.14 Ω:

(23)$$Z_{\,j, j + 1} = \displaystyle{{Z_0} \over {J_{\,j, j + 1}}}$$

so that Z _{1, 2} = Z _{4, 5} = 66.66 Ω and Z _{2, 3} = Z _{3, 4} = 69.95 Ω.

For bandstop lower cut-off frequency f ₁ = 10.6 GHz and higher cut-off frequency f ₂ = 20 GHz, then f ₀ = 15.4 GHz and FBW = 60%. The determined characteristic impedance value of the open-ended line to produce bandstop characteristics from 10.6 to 20 GHz is given in Table 2 [Reference Hong and Lancaster24].

Table 2. Element values for optimum bandstop filter for ripple less than 0.5 dB