1. Introduction

The demand for high-integration, miniature wireless communication systems has boosted the interest in low-cost, high-performance integrated passive devices (IPDs) during the last few years. The semiconductor-based IPD grows as an attractive technology capable of realizing many types of passive components using a thick conductor on a low-loss substrate. The power divider with low insertion loss and compact size is one of the most important devices in communication systems^{[1, 2]}. Owing to its advantages in size, cost and realizable system on chip, lumped-element power dividers have been widely designed and reported^[3-9].

The lumped-element network has been chosen as the transmission-line counterpart to realize an integrated power divider. Four types of discrete lumped-element Wilkinson power dividers were proposed and tested^[3]. The available relative bandwidth of the lumped power divider indicates its potential for IPD application. An active inductor was used in the T-type $LC$ network as a shunt inductor to achieve low insertion loss on a loss-silicon substrate^[4]; however, this method sacrifices DC power and linearity. The transmission line was replaced by $\pi$-type $LC$-network equivalents in Refs. [5, 6]. Low insertion loss was achieved by Si IPD technology^[5] and superconductor technology^[6], respectively. T-type $LC$-network equivalents have also been suggested to realize a lumped-element power divider using SI-GaAs-based IPD technology^{[7, 8]}. A wide-band and miniature power divider was realized with GaAs technology^[9], in which the transmission line was implemented by bridged T-coils.

Owing to low conductor and substrate loss, the high quality factor is one of the most attractive advantages in IPD. However, it increases the production and packaging costs for multichip connection. In contrast, passive devices can easily be integrated with other functional circuits in a single chip using ordinary semiconductor processes. However, the thinner conductor in an ordinary process has a large ohmic loss resulting in the lower quality factor^[10]. The wide-band power divider using ordinary GaAs technology^[9] exhibits higher insertion loss compared with relative reports^{[5, 7, 8]} that use IPD technology. In this paper, the quality factor of the inductor is discussed and calculated for its impact on insertion loss. According to the discussion on the quality factor, it is possible to achieve low insertion loss with ordinary GaAs technology. As depicted in Section 2.2, a quality factor of more than 20 can provide available loss of less than 0.5 dB. The inductor physical dimensions are chosen and optimized to obtain an adequate quality factor and an appropriate size. Then, a power divider using ordinary GaAs technology is analyzed, designed and fabricated. The measured results show that the power divider with ordinary GaAs technology not only exhibits low insertion loss as small as IPD technology, but also provides smaller size and easiness to integrate in a system.

2. Circuit design and analysis

2.1 Lumped Wilkinson power divider

The conventional Wilkinson power divider is shown in Fig. 1. One-quarter wavelength transmission line with characteristic impedance $\sqrt 2 Z_0$ and ballast resistor with impedance 2$Z_{0}$ are the fundamental elements of the Wilkinson power divider, where $Z_{0}$ is the terminal impedance. The transmission line can be equivalent to the $\pi$-type $LC$ network, as shown in Fig. 2. Then, the lumped Wilkinson power divider can be realized by replacing the transmission line with a $\pi$-type $LC$ network, as shown in Fig. 3.

The $ABCD$ matrix of transmission line with characteristic impedance $Z_{\rm C}$ and length $\ell$ in Fig. 2 can be expressed as:

$\begin{equation} [ABCD]_{\rm TL} = \begin{bmatrix} \cos \beta \ell & {\rm j}Z_{\rm C} \sin \beta \ell \\ {\rm j}Y_{\rm C} \sin \beta \ell &\cos \beta \ell \\ \end{bmatrix}. \end{equation}$

(1)

In Eq. (1), $Y_{\rm C}$ is the characteristic admittance, $\beta$ is the propagation constant and $\beta \ell$ is the electric length of the transmission line. $\beta \ell =90^\circ$ when the transmission line is one-quarter wavelength, in which the corresponding frequency is $\omega_{0}$.

The $ABCD$ matrix of the $\pi$-type $LC$ network in Fig. 2 is as follows:

$\begin{equation} [ABCD]_{\rm LC} = \begin{bmatrix} 1-\omega ^2LC&{\rm j}\omega L \\ {\rm j}2\omega C-{\rm j}\omega ^3LC^2& 1-\omega ^2LC \\ \end{bmatrix}. \end{equation}$

(2)

In Eq. (2), $L$ is the inductance of the series inductor and $C$ is the capacitance of the shunt capacitor in the $LC$ network. When $\beta \ell =90^\circ$ and $[ABCD]_{\rm TL} =[ABCD]_{\rm LC}$, the following equations can be achieved: $\omega _0 =\frac{1}{\sqrt {LC}}$ and $Z_{\rm C} =\sqrt {\frac{L}{C}}$, where $\omega _0$ is the center frequency and $\ell$ is one-quarter wavelength at frequency $\omega_{0}$.

2.2 Design consideration of quality factor

Table 1 shows the characteristics of different technologies. In this design, the ordinary GaAs technology is chosen for integrated lumped power divider design. As shown in Table 1, the ordinary GaAs technology has a thinner conductor thickness so that the quality factor is smaller than in special IPD technology. The limited quality factor of the inductor has to be considered, especially for insertion-loss sensitive devices, such as power dividers.

A non-ideal chip inductor can be simplified as an inductor parallel with a resistor $R_{\rm p}$, as shown in Fig. 3. Then, the insertion loss can be calculated from the $ABCD$ matrix. The $\Pi$-type $LC$ network is used for 50 to 100 $\Omega$ transformation, which is complex for insertion loss calculation. Two $\pi$-type $LC$ networks are connected with the same terminal impedance $Z_{0} =$ 50 $\Omega$, as shown in Fig. 4. The insertion loss of the power divider can easily be calculated from dual-$\pi$-type $LC$ networks.

The parasitic resistance $R_{\rm p}$ can be simply obtained from quality factor $Q$ and inductance $L$, i.e., $R_{\rm p}=\omega LQ$. The $ABCD$ matrix of the $\pi$-type $LC$ network considering $R_{\rm p}$ can be given as:

$\begin{equation} [ABCD]_1 = \begin{bmatrix} 1-\dfrac{\omega ^2LCR_{\rm p} }{{\rm j}\omega L+R_{\rm p}} & \dfrac{{\rm j}\omega LR_{\rm p} }{{\rm j}\omega L+R_{\rm p}} \\ {\rm j}2\omega C-\dfrac{{\rm j}\omega ^3C^2LR_{\rm p}}{{\rm j}\omega L+R_{\rm p}} & 1-\dfrac{\omega ^2LCR_{\rm p}}{{\rm j}\omega L+R_{\rm p}} \\ \end{bmatrix}. \end{equation}$

(3)

The insertion loss IL$_{\pi}$ can be obtained from the elements of matrix $[ABCD]_{1}$:

$\begin{equation} \begin{split} {\rm IL}_\pi = {}& \{2/[A^2+BC+(AB+BD)/2Z_0 \\& +2Z_0 (AC+DC)+BC+D^2]\}^{1/2}. \end{split} \end{equation}$

(4)

Then, the insertion loss IL of the power divider can be given as: IL $=$ IL$_{\pi}$. The relationship between the insertion loss and the quality factor can be achieved when $f_{0}$ is fixed. As shown in Fig. 5, the insertion loss varies with the frequency for different quality factors. In this design, the center frequency is set at 5.85 GHz. The insertion loss is less than 0.5 dB when $Q$ $\geqslant$ 20, as shown in Fig. 5. The minimum quality factor $Q_{\rm min}$ $=$ 20 in the operation frequency range can be obtained by the dimension being optimized in ordinary GaAs technology. The electromagnetic (EM) simulator Agilent Momentum is used for EM simulation and optimization. The parameters of the two-turn rotund inductor in the power divider are optimized as follows: width $=$ 20 $\mu$m, diameter $=$ 170 $\mu$m and space $=$ 10 $\mu$m. The simulated average quality factor is bigger than 23 from 5 to 7 GHz, which can guarantee an available insertion loss.

3. Measured results

The power divider is fabricated using Win 0.5 $\mu$m pHEMT technology. Printed-circuit board Rogers 4003 with relative permittivity of 3.55, loss tangent of 0.002 and thickness of 0.508 mm was applied as a test board to measure the performance of the fabricated GaAs power divider, as shown in Fig. 6. The chip photo of the circuit with an area of 0.9 $\times$ 0.85 mm$^{2}$ is shown in Fig. 7. The measured results and simulated results are depicted in Fig. 8. The microstrip and coaxial RF connector loss have been deducted from the measured results. As shown in Fig. 8, the measured return loss and isolation are better than 15 dB and 20 dB at the operation bandwidth of 5.15 to 6.15 GHz, respectively. The power divider features an average insertion loss of 0.5 dB and a minimum insertion loss of 0.42 dB at 5.5 GHz after deducting PCB influence. Table 2 shows the performance of the power divider fabricated using ordinary GaAs technology in this paper compared with some previous works. The measured results show excellent RF performances with low cost and compact size.

4. Conclusion

A compact IPD power divider with low insertion loss was constructed with ordinary GaAs technology. The non ideal chip inductor was analyzed for its impact on insertion loss. Results indicate that the integrated lumped power divider is appropriate for the system in package (SiP) or multi-chip module applications.