1 Introduction

Power dividers are widely used in the wireless communication circuits and microwave applications, such as array antennas, phase shifters, modulators, power amplifiers, mixers and frequency multipliers for power division or combination [1,2,3]. The conventional Wilkinson power divider (WPD) occupies considerably large area and suffers from presence of unwanted harmonics in the frequency response [4,5,6]. So far, several methods have been presented for harmonic suppression and size reduction in microwave circuits [7,8,9,10].

In [11, 12] the electronic band gap structure, in [13, 14] the defected ground structure and in [15] lumped reactive elements (capacitance and inductance) are applied to miniaturize and suppress unwanted harmonics, which all of these structures are not desirable, because of etching process or additional lumped reactive components [1].

In [16,17,18,19,20,21] resonators are used to suppress unwanted harmonics. Unfortunately, applying resonators resulted in insertion loss increment in passband and more complex design.

In [22,23,24,25,26,27] open shunt stubs are applied to improve performance of power dividers. The open shunt stubs are the most common configuration of microstrip structure [23], which applied for various purposes, such as: impedance matching [24], harmonics suppression [25] and size reduction [27].

So far, in several approaches, open stubs are presented to design the power divider, but just a few works have utilized aperiodic stubs methods. Aperiodic stubs are applied in [23, 28] to design artificial transmission line (ATL) and LPF.

In some recent works [29, 30] coupled lines are applied to suppress unwanted harmonics, which in this method the insertion loss parameter is poor.

In the proposed paper, aperiodic stubs are applied to design a miniaturized and harmonics suppressed Wilkinson power divider (WPD) for the first time. The proposed divider has the ability of creating transmission zeros at desired frequencies by using aperiodic stubs. All dimensions of these aperiodic stubs are calculated analytically in the proposed paper. Also, a miniaturized size for the proposed divider is achieved by meandering of the aperiodic stubs lines. The proposed divider works at 700 MHz, which is a candidate band for fifth generation (5G) applications in many countries [31]. 5G networks have been considered as modern communication networks, which can supply high technology applications [32,33,34]. The performance analysis for small cell networks and transmission problems in wireless local area networks (WLANs) for 5G operating frequency bands are completely discussed in [35,36,37].

In the following sections, circuit analysis of the open stubs is presented in Sect. 2. The proposed 12 sections low pass filter (LPF) is based on aperiodic open stubs to suppress 12 unwanted harmonics (4th–15th), which is designed in Sect. 3. In Sect. 4, design process of the proposed Wilkinson power divider, consisting of two proposed LPFs and three open stubs at each port, is presented. Simulations and measurements results show that the proposed WPD has good performance.

2 Circuit analysis

The microstrip transmission lines are widely used in microwave circuits. Transmission line can be assumed as cascading of n-section aperiodic lines as depicted in Fig. 1. The electrical length of the transmission line (θ0T) must be equal to sum of n-aperiodic sections with electrical length of θ0n as shown in (1).

Fig. 1
figure 1

a Conventional and b aperiodic n-section cascading transmission lines

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$$\uptheta_{{0{\text{T}}}} =\uptheta_{01} +\uptheta_{02} +\uptheta_{03} + \cdots +\uptheta_{{0{\text{n}}}}$$
(1)

In low pass filter design, each section can be replaced by T-shaped structure, to reduce the size, as shown in Fig. 2. Each T-shaped structure includes two microstrip lines and an open circuit stub, with same characteristic impedance. T-shaped structure can be used as an ATL with smaller length compared to the conventional line.

Fig. 2
figure 2

a Conventional transmission line and b equivalent T-shaped circuit

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Therefore, in many works T-shaped structure are used to decrease the physical length of the conventional transmission line (TL). For example in Fig. 2, the conventional line has the same result with T-shape structure, but the electrical length of the T-shaped structure is shorter than conventional one, which is shown in Eqs. (2) and (3).

$$\uptheta_{{0{\text{n}}}} > 2\uptheta_{\text{n}}$$
(2)
$$\uptheta_{{0{\text{T}}}} > 2(\uptheta_{1} +\uptheta_{2} +\uptheta_{3} + \cdots +\uptheta_{\text{n}} )$$
(3)

The ABCD matrix for the conventional TL, which depicted in Fig. 2a, is given by (4) [24]:

$$\left[ {\begin{array}{*{20}c} {\text{A}} & {\text{B}} \\ {\text{C}} & {\text{D}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\text{Cos}\uptheta_{{0{\text{n}}}} } & {{\text{jZ}}_{0} \text{Sin}\uptheta_{{0{\text{n}}}} } \\ {{\text{j}}_{0} {\text{Y}}_{0} \text{Sin}\uptheta_{{0{\text{n}}}} } & {\text{Cos}\uptheta_{{0{\text{n}}}} } \\ \end{array} } \right]$$
(4)

Each T-shaped structure consists of three sections, two microstrip lines and one open circuit stub, as depicted in Fig. 2b. The ABCD matrix is demonstrated by M1 for microstrip lines and demonstrated by M2 for open stubs, as shown in (5) and (6) [24]:

$${\text{M}}_{1} = \left[ {\begin{array}{*{20}c} {{\text{Cos}}\uptheta_{\text{n}} } & {{\text{jZ}}_{0} {\text{Sin}}\uptheta_{\text{n}} } \\ {{\text{j}}_{0} {\text{Y}}_{0} {\text{Sin}}\uptheta_{\text{n}} } & {{\text{Cos}}\uptheta_{\text{n}} } \\ \end{array} } \right]$$
(5)
$${\text{M}}_{2} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ {{\text{j}}_{0} {\text{Y}}_{0} { \tan }\uptheta_{0} } & 1 \\ \end{array} } \right]$$
(6)

Therefore the ABCD matrix is calculated as follows for one T-shaped structure (MT), considering as cascading of three networks:

$${\text{M}}_{\text{T}} = {\text{M}}_{1} \times {\text{M}}_{2} \times {\text{M}}_{1}$$
(7)

Substituting (5), (6) in (7) resulted in (8):

$${\text{M}}_{\text{T}} = \left[ {\begin{array}{*{20}c} {{\text{Cos}}^{2}\uptheta_{\text{n}} - {\text{Sin}}^{2}\uptheta_{\text{n}} - .5{\text{Sin}}2\uptheta_{\text{n}} { \tan }\uptheta_{0} } & {{\text{jZ}}_{0} {\text{Sin}}2\uptheta_{\text{n}} - {\text{jZ}}_{0} {\text{Sin}}^{2}\uptheta_{\text{n}} { \tan }\uptheta_{0} } \\ {{\text{j}}_{0} {\text{Y}}_{0} \left( {{\text{Sin}}2\uptheta_{\text{n}} + {\text{Cos}}^{2}\uptheta_{\text{n}} { \tan }\uptheta_{0} } \right) } & {{\text{Cos}}^{2}\uptheta_{\text{n}} - {\text{Sin}}^{2}\uptheta_{\text{n}} - .5{\text{Sin}}2\uptheta_{\text{n}} { \tan }\uptheta_{0} } \\ \end{array} } \right]$$
(8)

The n-section cascading of T-shaped configuration is depicted in Fig. 3, which known as artificial transmission line and used as low pass filter to suppress n-desired harmonics [23]. Each T-shaped section can suppress a desired harmonic, corresponding to length of the open circuited stub.

Fig. 3
figure 3

Cascading of n-sections T-shaped configuration

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In this configuration, the electrical length and characteristic impedance of each T-shaped section is shown with θm and zm, respectively. Each T-shaped section creates a specific transmission zero at desired frequency, corresponding to the electrical length of the open circuited stub. The open circuited stub with electrical length of π/2 km can create a transmission zero at kmf0 frequency, where f0 is center frequency, m = 1, 2,…, n and km is assignable parameter, which specifies the stub length. If km is defined as multiplies of center frequency (f0), then the corresponding harmonic can be suppressed. The electrical length of each two microstrip lines in one T-shaped structure can be calculated as follows:

$$\uptheta_{\text{m}} = \tan^{ - 1} \left( {\frac{{\sqrt {\tan^{2} \left( {\frac{\pi }{{2k_{m} }}} \right) + 4 \sin^{2}\uptheta_{0m} } - \tan \left( {\pi /2k_{m} } \right)}}{{2\cos\uptheta_{0m} + 2}}} \right)$$
(9)

where in (9), θm is the electrical length of two microstrip lines in T-shaped structure and θ0m is the electrical length of the conventional transmission line equivalent T-shaped structure. The characteristic impedances of T-shaped structure can be obtained as:

$${\text{Z}}_{\text{m}} = {\text{Z}}\sqrt {\frac{{ \cos^{2}\uptheta_{m } \tan \left( {\frac{\pi }{{2k_{m} }}} \right) + \sin 2\uptheta_{m} }}{{\sin 2\uptheta_{m} - \sin^{2}\uptheta_{m} \tan \left( {\frac{\pi }{{2k_{m} }}} \right)}}}$$
(10)

All of the characteristic impedance were considered equal (Z1 = Z2 = ··· = Zn) in the proposed structure, to minimize the discontinuities and decrease the complexity of the analysis. To find the unknown design parameters, the equations are defined as bellow:

$${\text{F}}_{11} =\uptheta_{0} - \mathop \sum \limits_{m = 1}^{n} (\uptheta_{0m} )$$
(11)

and

$${\text{Fi}}_{1} = \left( {{\text{Z}}_{1} /{\text{Z}}_{0} } \right) - \left( {{\text{Z}}_{\text{i}} /{\text{Z}}_{0} } \right)^{2}$$
(12)

where in (12), i = 2, 3,…, n. So, the constructed equations involve n-equations. The desired electrical length at the operating frequency of f0 is demonstrated by θ0 in (11) and θ01, θ02,…, θ0n are the n-unknown variables. To find θ01, θ02,…, θ0n variables, unknown parameters of Z1/Z0, Z2/Z0,… Zn/Z0, in (12) can be written as θ01, θ02,…, θ0n, using (9) and (10). As results θ01, θ02,…, θ0n are determined. After this step, the values of θ1, θ2,…, θn and Z1 can be calculated using (9) and (10). The Newton’s method is used to find the answer of this equation with several variables. The solution to the problem of F = [0 0 … 0]T is formulated as an iterative process

$${\text{X}}_{{{\text{i}} + 1}} = {\text{X}}_{\text{i}} - {\text{J}}^{ - 1} {\text{F}}$$
(13)

where X in (13) is equal to [θ01 θ02 … θ0n]T. The Jacobian matrix is defined in (14)–(16), with the initial guess of X° = [1° 1°… 1°]T can be calculated as follows: Using (10)–(16) allows one to synthesize the value of unknown parameters. More detail about analysis and methods of finding unknown parameter are completely described in [23].

$${\text{J}} = \left[ {\begin{array}{*{20}c} { - \,1} & { - \,1} & { - \,1} & { - \,1} & \ldots & { - \,1} \\ {\frac{{\partial \left( {Z_{1} /Z_{0} } \right)^{2} }}{{\partial\uptheta_{1} }}\frac{{\partial\uptheta_{1} }}{{\partial\uptheta_{01} }}} & { - \frac{{\partial \left( {Z_{2} /Z_{0} } \right)^{2} }}{{\partial\uptheta_{2} }}\frac{{\partial\uptheta_{2} }}{{\partial\uptheta_{02} }}} & 0 & 0 & \ldots & 0 \\ \vdots & 0 & { - \frac{{\partial \left( {Z_{3} /Z_{0} } \right)^{2} }}{{\partial\uptheta_{3} }}\frac{{\partial\uptheta_{3} }}{{\partial\uptheta_{03} }}} & 0 & \ldots & 0 \\ \vdots & 0 & 0 & { - \frac{{\partial \left( {Z_{4} /Z_{0} } \right)^{2} }}{{\partial\uptheta_{4} }}\frac{{\partial\uptheta_{4} }}{{\partial\uptheta_{04} }}} & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & 0 \\ {\frac{{\partial \left( {Z_{1} /Z_{0} } \right)^{2} }}{{\partial\uptheta_{1} }}\frac{{\partial\uptheta_{1} }}{{\partial\uptheta_{01} }}} & 0 & 0 & 0 & \ldots & { - \frac{{\partial \left( {Z_{n} /Z_{0} } \right)^{2} }}{{\partial\uptheta_{n} }}\frac{{\partial\uptheta_{n} }}{{\partial\uptheta_{0n} }}} \\ \end{array} } \right]$$
(14)
$$\frac{{\partial \left( {Z_{m} /Z_{0} } \right)^{2} }}{{\partial\uptheta_{m} }} = \frac{{2\tan \left( {\frac{\pi }{{2k_{m} }}} \right)\left[ { \tan^{2}\uptheta_{m} + \tan \left( {\frac{\pi }{{2k_{m} }}} \right) - \tan\uptheta_{m} - 1} \right]}}{{\left( {1 + \tan^{2}\uptheta_{m} } \right)\left[ {\sin 2\uptheta_{m} - \sin^{2}\uptheta_{m} \tan \left( {\frac{\pi }{{2k_{m} }}} \right)} \right]^{2} }}$$
(15)
$$\frac{{\partial\uptheta_{m} }}{{\partial\uptheta_{0m} }} = \frac{1}{{\left( {1 + \tan^{2}\uptheta_{m} } \right)}}\frac{{\sin\uptheta_{0m} \left[ {4\cos\uptheta_{0m} - \tan \left( {\frac{\pi }{{2k_{m} }}} \right)\sqrt {\tan^{2} \left( {\frac{\pi }{{2k_{m} }}} \right) + 4 \sin^{2}\uptheta_{0m} } + \tan^{2} \left( {\frac{\pi }{{2k_{m} }}} \right) + 4 } \right]}}{{\left[ {2\cos\uptheta_{m} + 2} \right]^{2} \sqrt {\tan^{2} \left( {\frac{\pi }{{2k_{m} }}} \right) + 4 \sin^{2}\uptheta_{0m} } }}$$
(16)

3 Design procedure

The proposed Wilkinson power divider consists of two LPFs in branch lines and three open stubs at each port. At first, a conventional power divider at 700 MHz is designed. Then, a primitive divider is designed with three open stubs at each port. This divider only suppresses second and third harmonics. On the other hand, to have more harmonic suppression, the LPFs should be designed. To design the LPFs with harmonic suppression, 12 stubs are added in the LPF structure. At the end, the designed LPFs are added in the primitive divider. The circuit design flowchart of the proposed divider is shown in Fig. 4, to show the design procedures.

Fig. 4
figure 4

The circuit design flowchart of the proposed divider

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3.1 Design of proposed LPF based on aperiodic stub

In power divider design process, applied low pass filters have a critical role. The proposed divider works at 700 MHz frequency and suppressed fourteen harmonics. The second and third unwanted harmonics are eliminated by applied open stubs at each ports of divider and other twelve harmonics are suppressed by proposed LPF. Therefore the proposed LPF should be designed to suppress 4th–15th harmonics (2.8–10.5 GHz). The first transmission zero of the designed LPF should be located at 2.8 GHz, so to have proper design the cut off frequency of filter should be at 2.5 GHz. The proposed LPF is created by cascading of 12 sections of T-shaped structure with characteristic impedance of 142 Ω. Because the center frequency is 700 MHz, this filter must create 12 transmission zeros at multiples of the fundamental frequency, they could be located at:

$$2.8 + \left( {{\text{m}} - 1} \right) \times 0.7\,{\text{GHz}}$$
(17)

where m = 1, 2, 3,…, 12. Table 1 shows the calculated and extracted values from simulations for twelve-section LPF, with 142 Ω lines. The results show little difference between calculation values and simulations results, which confirm the validity of design process.

Table 1 Analysis and simulation values of twelve-sections LPF with 142 Ω lines
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The Layout of the designed LPF with twelve T-shaped sections is shown in Fig. 5. The designed LPF is simulated by the advanced design system (ADS) software with RO4003 substrate (ɛr = 3.365, thickness = 20 mil, loss tangent of 0.0033).

Fig. 5
figure 5

Layout of the LPF with twelve T-shaped sections

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The dimensions of the designed filter with mentioned substrate is 17.7 mm × 26 mm (0.353 λg × 0.24 λg). Unfortunately, this filter occupies large area.

The EM simulation results of the LPF with twelve T-shaped sections, is depicted in Fig. 6. As seen in this figure, the designed LPF creates twelve transmission zeros at desired frequencies with high attenuation levels, which features good harmonics suppression 4th–15th for the proposed divider.

Fig. 6
figure 6

EM simulation results of the LPF with twelve T-shaped sections

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As previously mentioned, the LPF with twelve T-shaped sections occupies large area; Therefore, to have both compact size and harmonics suppression properties, the proposed LPF is presented using bended stub lines as shown in Fig. 7.

Fig. 7
figure 7

Layout of the proposed LPF

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In the proposed LPF, twelve T-shaped sections are bended, resulted in size reduction. The overall size of proposed LPF is only 3.6 mm × 30.6 mm (0.415 λg × 0.048 λg), which resulted in more than 75% size reduction, compared to the LPF with twelve T-shaped sections. The Electromagnetic (EM) simulation results of the designed LPF with twelve T-shaped sections, is shown in Fig. 8. The performance of the proposed filter is very good, which creates twelve transmission zeros at desired frequencies with high attenuation levels.

Fig. 8
figure 8

Electromagnetic simulation results of the proposed LPF

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3.2 Design process of the proposed Wilkinson power divider

The layout of the conventional WPD is depicted in Fig. 9. The size of the conventional divider at 700 MHz with mentioned substrate (RO4003 with: ɛr = 3.365, thickness = 20 mil, loss tangent of 0.0033) is 32.8 mm × 39.5 mm.

Fig. 9
figure 9

Conventional Wilkinson divider at 700 MHz

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The layout of the Wilkinson divider using twelve T-shaped sections LPF is shown in Fig. 10. Three open stubs at three ports of the divider are applied to suppress lower harmonics (2nd and 3rd). Also, two LPFs which were introduced in the past section are applied as branch lines to suppress 4th to 15th harmonics.

Fig. 10
figure 10

Designed WPD using twelve T-shaped sections LPF

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The EM simulation results of this WPD are depicted in Fig. 11. The results show that the divider works correctly at 700 MHz and suppress the 2nd–15th harmonics with high levels of attenuation.

Fig. 11
figure 11

EM simulation results of the designed WPD using twelve T-shaped sections LPF at 700 MHz

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Unfortunately, this divider occupies large area. To reduce the large size of the presented divider, the applied twelve sections LPFs are replaced by proposed bended stubs LPFs, which described in the past section.

The Layout and fabricated photograph of the designed WPD are demonstrated in Figs. 12a, b, respectively. The size of the proposed divider is only 11.7 mm × 30.6 mm, which reduces the size of the divider more than 73% compared to the conventional WPD at 0.7 GHz. The electromagnetic (EM) simulation and measurement results of the designed WPD are depicted, in Fig. 13.

Fig. 12
figure 12

a Layout and b fabricated photograph of the proposed WPD

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Fig. 13
figure 13

EM simulation and measurement results of the proposed WPD

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The results show that, the proposed divider successfully suppresses the 2nd–15th harmonics with high level of attenuations. A comparison between proposed power divider and other related works are summarized in Table 2. As result shows, the proposed power divider not only extremely reduces the circuit size but also has very good harmonics suppression compared to the reported works.

Table 2 Performance comparison of the proposed WPD and other previous related works
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4 Conclusion

A compact LPF and a miniaturized, harmonics suppressed 0.7 GHz divider using open stubs are proposed in this paper. Three open stubs are used to suppress second and third harmonics at three ports of the divider and two novel 12-section low pass filters, based on aperiodic stubs, are applied to suppress 4th to 15th harmonics. The size of the proposed filter is very compact and results in a miniaturized power divider. The proposed divider not only has perfect harmonics suppression, but also extremely decreases the circuit size.