Starting from the conventional shape of the microstrip parallel coupled line BPF, a single band BPF can be simply designed. The amplitude ripple in the passband is selected to be 0.5 dB and the order of the filter is 3. This means that it is a three-pole filter. Designing such coupled line filter is achieved by transforming the cascaded networks by its equivalent RLC circuit using transmission line theory. The following design equations have to be considered [10] [11] .

β = ln [ coth ( G r 17.37 ) ] (1)

where G_r is the ripple in the passband.

γ = sinh ( β 2 n ) (2)

a k = sin [ ( 2 k − 1 ) π 2 n ] ,     k = 1 , 2 , 3 , ⋯ , n (3)

b k = γ 2 + sin 2 [ k π n ] ,     k = 1 , 2 , 3 , ⋯ , n (4)

g 0 = 1 , g 1 = 2 a 1 γ (5)

In our case, and at n = 3, β = 3.548, γ = 0.62643. So, a₁ = a₃ = 0.5, a₂ = a₄ = 1. Then b₁ = b₂ = 1.142, b₃ = 1.3924.

g k = 4 a k − 1 a k b k − 1 g k − 1 ,     k = 2 , 3 , 4 , ⋯ , n (6)

g m + 1 = 1 if n is odd (7)

g m + 1 = coth 2 [ β 4 ] if n is even (8)

From the mentioned equations, the normalized lowpass Chebyshev filter elements with ripple 0.5 dB are calculated and the values are as follow:

g 1 = g 3 = 1.5963 , g 2 = 1.0967 , g 4 = 1

In addition, the center frequency and the fractional bandwidth are calculated as follow:

w 0 = w 1 w 2 and fractional BW ( Δ ) = ( w 2 − w 1 ) / w 0

where w 0 = f c , w 1 = f c l , and w 2 = f c h .

In order to design the parallel coupled line filter, the following equations are used:

For the 1st coupling structure:

Z 0 J 1 = π Δ 2 g 1 (9)

For the intermediate structure:

Z 0 J n = π Δ 2 g n g n − 1 (10)

For the final coupling structure:

Z 0 J n + 1 = π Δ 2 g n g n + 1 (11)

where Z₀ is the characteristic impedance, and J is the inverter of admittance.

The characteristic impedance of the parallel coupled lines is divided into odd impedance (Z_0o) and even impedance (Z_0e), which can be calculated as:

( Z 0 o ) j , j + 1 = 1 Y 0 [ 1 − J j , j + 1 Y 0 + ( J j , j + 1 Y 0 ) 2 ] (12)

( Z 0 e ) j , j + 1 = 1 Y 0 [ 1 + J j , j + 1 Y 0 + ( J j , j + 1 Y 0 ) 2 ] (13)

According to the Ku-band satellite applications, the frequency band from 11.7 GHz to 12.2 GHz are assigned for downlink fixed satellite service (FSS) and frequencies from 12.2 GHz to 12.7 GHz are assigned for downlink broadcasting satellite services (BSS). In addition, the frequencies from 14 GHz to 14.5 GHz are assigned for uplink FSS [12] [13] .

In order to operate such presented filter at a single band, for example at 12.54 GHz downlink frequency, the calculated values of the odd and even impedances are given in Table 1.

The next step is to calculate the parallel coupled line dimensions for the four sections according to the calculated even and odd impedance values. These dimensions have been optimized using the Advanced Design System (ADS) electromagnetic momentum simulator, ver. 2019. The substrate material specifications have been defined correctly to the ADS simulator (ε_r = 10.2, h = 1.9 mm, and tanδ = 0.0023) and the optimized dimensions are given in Table 2.

The ADS structure for these parallel coupled line sections that constitute the downlink BPF is shown in Figure 1. The input/output coupled lines are connected

with the feed transmission lines, which have 50 Ω impedance. The transmission lines are constructed from the same substrate material (RT/Duroid 6010). Figure 2 depicts the S-parameters in dB for such BPF. Maximum return loss of 23.8 dB and insertion loss of 0.7 dB can be noticed at the downlink resonant frequency of 12.54 GHz.

On the same way, the calculated values of the odd and even impedances at 14.14 GHz uplink frequency are given in Table 3.

The optimized dimensions of the four sections of the parallel coupled line are given in Table 4.

The ADS structure for these parallel coupled line sections that constitute the uplink BPF is shown in Figure 3. Figure 4 depicts the S-parameters in dB for such BPF. Maximum return loss of 25.6 dB and insertion loss of 0.64 dB can be noticed at the uplink resonant frequency of 14.14 GHz.

Following the same procedure, a wide band BPF is proposed with a 3-dB cut off frequencies, f_CL = 12.4 GHz and f_CH = 14.8 GHz. The values of the odd and even impedances for the proposed wide band BPF have been calculated and presented in Table 5.

The optimized dimensions of the four sections of the proposed wide band parallel coupled line BPF are given in Table 6. The ADS structure for these parallel coupled line sections that constitute the proposed wide band BPF is shown in Figure 5.

A layout for such filter has been generated using ADS for preparing to the fabrication process. The proposed filter layout is shown in Figure 6(a). A prototype of the proposed wide band BPF is illustrated in Figure 6(b). The filter is fabricated on epoxy Rogers 6010 substrate with a dielectric constant of 10.2, thickness (h) of 1.9 mm, and loss tangent of 0.0023.

Figure 7 depicts both measured and simulated results of the proposed wide band BPF. The measured return loss of about 40 dB is achieved inside the ku-band and remains better than 10 dB across the entire band from 12.4 GHz to

14.8 GHz. The achieved measured insertion loss is 0.71 dB and 0.78 dB at 12.5 GHz and 14.6 GHz, respectively, and remains better than 3 dB across the mentioned band. A good agreement is found between simulated and measured results.

The simulated S-parameters are centered at 13.6 GHz while the measure results are shifted slightly and centered at 13.45 GHz. Any small difference between simulated and measured results may come from standard manufacture impacts and the copper surface roughness.

In this part, a dual band BPF can be generated by adding lumped capacitors to the coupled line sections. Figure 8 depicts a single section parallel coupled line that contains four ports. The input and output ports are terminated or connected to other cascaded sections while the other two ports are loaded by load impedances. In the case of loading port 2 by a series capacitor connected to ground, the value of this capacitor can be calculated for the dual operational frequencies by the following relation [14] :

Z L = 1 j w c , where w = 2 π f (14)

The initial values of the lumped capacitors have been calculated using equation (14) and optimized using ADS simulator in order to design a dual band BPF at 12.54 GHz and 14.14 GHz for downlink and uplink frequencies for Ku-band satellite applications. The optimization goals are directed toward improving the

selectivity and the stop band rejection ratio. The ADS structure for such dual band BPF is shown in Figure 9 and the optimized values for the 2-capacitors are C₁ = 2.1 pF and C₂ = 0.19 pF. Figure 10 depicts the S-parameters in dB for such dual band BPF. Maximum return loss of 17.567 dB and insertion loss of 0.197 dB can be noticed at the downlink resonant frequency of 12.5 GHz. Also, a return loss of 21.58 dB and insertion loss of 1.526 dB are achieved at the uplink resonant frequency of 14.17 GHz.

In this part, the lumped capacitors are replaced with the open circuited transmission lines. The lumped elements are not preferred due to their parasitics above 1 GHz and the limited available commercial values of these inductors and capacitors. Therefore, a dual band BPF has been proposed using distributed elements without degradation in the performance compared to lumped ones. Figure 11 shows the equivalent to the tuning capacitor which is connected shunt with the coupled line sections. This capacitor is replaced by open-circuited stub in shunt with the coupled sections.

The impedances of the two capacitors C₁ and C₂ are calculated and their equivalent Z_in1 and Z_in2 for the two open stubs transmission lines using the normal transmission line impedance equation [15] . The input impedance for an open-circuited transmission line equals –jZ₀cot(θ). For θ = 45˚, Z_in equals –jZ₀. This is equivalent to λ_g/8 transmission line.

The initial values of the two open stubs are calculated using this procedure, then they are optimized using ADS simulator to achieve acceptable performance compared to the lumped capacitors. The optimization limits are imposed in order to obtain realizable and acceptable dimensions for the open stubs and also the coupled line sections. The ADS structure for such dual band BPF using open stubs instead of lumped capacitors is shown in Figure 12. The frequency responses for that proposed dual band BPF are compared to that obtained by lumped capacitors and there is almost no degradation in the performance as illustrated in Figure 13.