1 Introduction

Nowadays, microwave and radio-frequency devices are widely used in various applications [1,2,3,4,5]. The design of miniaturized devices is an important demand for modern wireless networks. Dividers are important components in the recent RF circuits and systems such as frequency multipliers, mixer and power amplifiers [6]. Recently, design a miniaturized power divider with harmonics rejections is an important challenge. So far, different methods are introduced, to reduce the circuit size of power dividers [7,8,9,10]. Using non-uniform transmission lines (TLs) instead of usual uniform TLs in [7], leads to decrease the overall size of the reported divider up to 52% compared to the conventional divider. In [8], coupled line sections are used to decrease the size of the power divider. Using microstrip EBG cell [9] and defected ground structure (DGS) [10], are also another methods to achieve size reduction, which are difficult to implement and not suitable methods to reduce the circuit size [11].

Compact resonators [12,13,14,15,16], low-pas filters [17,18,19,20,21,22] and band-pass filters [23,24,25] can be used in the dividers structure for size reduction and harmonics suppression purposes. Unfortunately, this method increases the insertion loss in the pass-band.

Stub-loaded resonators in the reported structure in [26], resulted in 20 dB suppression for the 2nd and 3rd harmonics, besides size reduction. Using a coupling structure instead of the traditional quarter-wavelength lines in [27] resulted in a wide harmonics rejection band.

In some recent works, artificial intelligence and neuro-based approaches are used to design optimum devices [28,29,30].

In this work, a miniaturized WPD with unwanted harmonics elimination is proposed. Composite transmission lines and resonator cells are used instead of bulky quarter wavelength transmission lines, which occupied large areas in the conventional divider. Furthermore using this technique, not only eliminates unwanted harmonics with good attenuation level, but also reduces 52% of the occupied area compared to the conventional one at the main frequency of 1.5 GHz. Section II of the paper will present theoretical design procedures. Simulation and experimental results are expressed in section III. The conclusion of the paper is presented in section IV.

2 Design Procedure

In this design, an equal Wilkinson power divider is proposed based on the unequal divider structure. Figure 1a demonstrates, the conventional equal WPD. In the conventional structure, two long quarter-wavelength microstrip branches (√2Z01) and a lumped resistor (2Z01) exist. Figure 2b, depicts the structure of the conventional unequal divider. If the power division ratio (k) in the unequal divider is 1, it becomes an equal one. The unequal divider consists of two microstrip lines between the input port and an isolation resistor and two lines between output ports and isolation elements. The characteristic impedance values of each line and isolation resistor value are discussed in [6].

Fig. 1
figure 1

Wilkinson power divider a equal conventional, b unequal conventional, c proposed one

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Fig. 2
figure 2

The block diagram of the primitive power divider

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The block diagram schematic of the proposed WPD is demonstrated in Fig. 1c. The proposed divider consists of four short transmission lines, two inductors, two resonator cells, a capacitor and a resistor to isolate output ports. The design procedure and analysis of the proposed divider, has three parts. At first, the primitive structure (part I) is designed and explained, then long λ/4 branches are replaced by composite lines (part II). At the end (part III) resonator cells are used instead of output microstrip lines.

2.1 The Primitive Divider Structure (Part I)

The conventional unequal divider as shown in Fig. 1b, which occupied large areas, because of large input microstrip transmission lines. According to the reported method in [31] the primitive power divider is designed. The block diagram of the primitive divider is illustrated in Fig. 2. Characteristic impedances of the primitive divider could be defined as follows:

$${\text{Z}}_{02} = {\text{Z}}_{03} = 2{\text{Z}}_{01}$$
(1)
$$_{{Z^{\prime\prime}_{04} = Z^{\prime\prime}_{05} = \sqrt 2 Z_{01} }}$$
(2)
$${\text{R}}_{{1}} = {\text{ 4Z}}_{{0{1}}}$$
(3)
$${\text{C}}_{1} = \frac{1}{{4{\text{Z}}_{01} \omega \tan \theta }}$$
(4)

where ω is the operation angular frequency. The C1 and R1 are capacitor and resistor, which are used to output ports isolation.

2.2 Composite Transmission Lines (Part II)

In order to have more size reduction, output applied transmission lines (shown in Fig. 2), which occupied large areas are replaced by equal composite transmission lines with compact size as shown in Fig. 3.

Fig. 3
figure 3

a Conventional transmission line and its, b equivalent composite transmission lines

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According to the ABCD matrix discussed in [6], the transmission matrix of each section in Fig. 3 can be expressed as follows:

$${\text{[ABCD1] = }}\left[ {\begin{array}{*{20}c} 0 & {{\text{jz}}} \\ {\frac{{\text{j}}}{{\text{z}}}} & 0 \\ \end{array} } \right]$$
(5)
$${\text{[ABCD2] = }}\left[ {\begin{array}{*{20}c} {\cos \theta^{\prime}} & {{\text{jz}}^{^{\prime}} \sin \theta^{\prime}} \\ {\frac{j}{{{\text{z}}^{^{\prime}} }}\sin \theta^{\prime}} & {\cos \theta^{\prime}} \\ \end{array} } \right]$$
(6)
$${\text{[ABCD3] = }}\left[ {\begin{array}{*{20}c} 1 & {{\text{j}}\omega {\text{L}}^{^{\prime}} } \\ 0 & 1 \\ \end{array} } \right]$$
(7)

Equating, the [ABCD] matrices of the composite transmission lines in Fig. 3b with the conventional transmission line, in Fig. 3a, resulted as:

$$\left[ {{\text{ABCD}}_{{1}} } \right] \, = \, \left[ {{\text{ABCD}}_{{2}} } \right] \times \left[ {{\text{ABCD}}_{{3}} } \right] \times \left[ {{\text{ABCD}}_{{2}} } \right]$$
(8)
$$\left[ {\begin{array}{*{20}c} {\cos \theta^{\prime}} & {jz^{\prime}\sin \theta^{\prime}} \\ {\frac{j}{{z^{\prime}}}\sin \theta^{\prime}} & {\cos \theta^{\prime}} \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}c} 1 & {j\omega L^{\prime}} \\ 0 & 1 \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}c} {\cos \theta^{\prime}} & {jz^{\prime}\sin \theta^{\prime}} \\ {\frac{j}{{z^{\prime}}}\sin \theta^{\prime}} & {\cos \theta^{\prime}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & {jz} \\ \frac{j}{z} & 0 \\ \end{array} } \right]$$
(9)
$$\left[ {\begin{array}{*{20}c} {\cos^{2} \theta^{\prime}\_\frac{j}{{z^{\prime}}}\sin \theta^{\prime}\cos \theta^{\prime}\_\sin^{2} \theta^{\prime}} & {2jz^{\prime}\sin \theta^{\prime}\cos \theta^{\prime} + j\omega L^{\prime}\cos^{2} \theta^{\prime}} \\ {\frac{1}{2}\sin \theta^{\prime}\cos \theta^{\prime}\_\frac{{j\omega L^{\prime}}}{{z^{{\prime}{2}} }}\sin^{2} \theta^{\prime} + \frac{j}{{z^{\prime}}}\sin \theta^{\prime}\cos \theta^{\prime}} & {j\sin^{2} \theta^{\prime}\_\frac{{\omega L^{\prime}}}{{z^{\prime}}}\sin \theta^{\prime}\cos \theta^{\prime} + \cos^{2} \theta^{\prime}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & {jz} \\ \frac{j}{z} & 0 \\ \end{array} } \right]$$
(10)

After some algebraic calculations, the characteristic impedance \(({\text{Z}}^{^{\prime}} )\), electrical length \(\left( {\theta^{\prime}} \right)\) of the composite transmission line and lumped inductance \(\left( {{\text{L}}^{^{\prime}} } \right)\) have been obtained as below:

$$\omega {\text{L}}^{^{\prime}} = \frac{{2{\text{z}}^{^{\prime}} }}{{\tan 2\theta^{\prime}}}$$
(11)
$${\text{z}}^{^{\prime}} = {\text{z}}\tan \theta^{^{\prime}}$$
(12)

Figure 4, shows the structure of power divider using composite transmission lines. The characteristic impedance of Z04, Z05 and lumped elements values L01, L02 are calculated based on (11–12). In this design θ02 and θ03 are considered equal.

$${\text{Z}}_{04} = {\text{Z}}_{05} = \sqrt 2 {\text{Z}}_{01} \tan \theta_{02}$$
(13)
$${\text{L}}_{01} = {\text{L}}_{02} = \frac{{2{\text{Z}}_{05} }}{{\omega \tan 2\theta_{02} }}$$
(14)
Fig. 4
figure 4

Structure of the power divider using composite transmission lines

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2.3 Proposed Resonator Cells (Part III)

In order to eliminate unwanted harmonics, the designed resonator is proposed as depicted in Fig. 5a. The scattering parameters of this cell are depicted in Fig. 5b.

Fig. 5
figure 5

The proposed resonator cell a layout, b frequency response

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The size dimensions of proposed resonator cell based on RO4003 substrate with 20 mil thickness and dielectric constant of 3.38, are shown in Fig. 5, which listed as follows: L1 = 6.4, L2 = 6.2, L3 = 0.2, L4 = 3.5, L5 = 0.3, W1 = 2.4, W2 = 3.3, W3 = 0.7 and W4 = 0.2 (all in mm). The all gaps between microstrip lines in proposed resonator are 0.1 mm.

3 Experimental Results and Simulation

Figure 6a shows the structure of the proposed WPD. The layout of the square shaped and circular shaped of the conventional WPDs, are demonstrated in Fig. 6b, c respectively. According to (1)–(4) and (13)–(14) with little optimization, the electrical lengths and characteristic impedances of the proposed divider are obtained as follows: Z01 = 50 Ω, Z02 = 100 Ω, Z04 = 29 Ω, L01 = L02 = 6.1 nH, C1 = 3.6 pF, R1 = 200 Ω, θ01 = 9.25° and θ02 = 22.5°.

Fig. 6
figure 6

Wilkinson power divider a layout of the proposed, b layout of the square-shaped and c circular-shaped of the conventional

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The proposed divider is fabricated on RO4003 substrate (H = 20 mil, ℰr ℰ= 3.38 and tanα = 0.0022). The measurement and electromagnetic (EM) simulation curves of the designed WPD are depicted in Fig. 7a, b. As seen in Fig. 7a, the obtained values of S12 and S13 are − 3.15 dB. The input return loss is more than 20 dB at operating frequency and harmonics suppression is up to 10 GHz with a more than 20 dB suppression level. As seen in Fig. 7b, more than 20 dB output return losses and better than 30 dB output ports isolation are obtained at 1.5 GHz. The overall circuit size of the proposed WPD is only 0.15 × 0.05 λg, which shows more than 52% size reduction compared to the typical WPD.

Fig. 7
figure 7

Simulated and measurement results of designed WPD a input return loss and Insertion loss, b isolation and output return losses

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A performance comparison between some related WPDs and the proposed divider are listed in Table 1. The proposed WPD has a small area compared with other reported works and also has very good harmonics suppression performance.

Table 1 Performance comparison between proposed WPD and other works
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4 Conclusion

A compact Wilkinson power divider using lumped component elements and resonator cells for harmonics suppression has been described. The proposed divider has more than 52% size reduction compared to the conventional WPD. More than 20 dB return losses at all input and output ports and better than 30 dB output ports isolation are obtained at 1.5 GHz for the fabricated device.