Stepped-Impedance Section

The technique in [13] is employed here to further miniaturize the hybrid ring in Figure 2.6.

Z ,

_{i i}

θ Z ,

_{L L}

θ Z ,

_{L L}

θ

Z , 2

H H

θ

Figure 3.1 Substitution of a uniform section by a stepped-impedance section for circuit miniaturization.

As mentioned in [13], a uniform transmission line is equivalent to a section of

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 25

stepped-impedance transmission line as long as they have the same ABCD matrix at the designed frequency. Figure 3.1 shows a uniform transmission line section with characteristic impedance Zi and electric length

θ

i, which will be substituted for a stepped-impedance section on the right. The substitution has two sections with characteristic impedance ZL and electric length

θ

L at both ends and a section with characteristic impedance Z_H and electric length 2

θ

H in between. The ABCD matrix of the stepped-impedance section can be derived as

L

The next step is to equate (3.1) to the ABCD matrix of the uniform transmission line in Figure 3.1 shown in (3.3). Due to circuit symmetry and reciprocity, the conditions A = D and AD – BC = 1 can be guaranteed. Therefore, there are only two conditions available in (3.1).

D

i

A

= =cos

θ

(3.3a)

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 26

By equating (3.1a) to (3.3a), we have

( ) ( ) ( )

sides of (3.4) have to be maximized. It results in

⎟⎠

By utilizing (3.5) and (3.6), the four arms of the ring in Figure 2.6 can be replaced by stepped-impedance sections. The 2

θ

1 arm and each

θ

2 arm are substituted for one stepped-impedance section, respectively. However, the 2

θ

3 arm is replaced by two identical stepped-impedance sections.

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 27

Figure 3.2 shows design graphs of the substitutions for Y1-section (62.15Ω, 18.8^o), Y₂-section (21.96Ω, 65.8^o), and Y₃-section (62.15Ω, 198.8^o) of Figure 2.6(c), respectively. Z_L and Z_H are normalized to Z₀. As shown in Figure 3.2, it can be observed that the larger the value r is, the smaller the section

θ

0 is. Therefore, to minimize circuit area, r must be as large as possible. Nevertheless, large r results in large ZH, the characteristic impedance of the high impedance section. The line width resolution of our PCB (printed circuit board) process is about 0.15 mm which is 150 Ω for a substrate with

ε

r = 2.2 and thickness 0.508 mm. Moreover, large r value also leads to small Z_L which may not be practical. Thus, practical r value has an upper limit. The range of r is set to be from 1 to 10 in Figure 3.2. Table 3.1 displays the circuit parameters of the stepped-impedance substitutions. Note that when the substrate dielectric constant or the design frequency is increased, the ring area becomes smaller and hence improves the size reduction factor.

3.5

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 28 stepped-impedance section of (a) the Y1-section, (b) the Y2-section, and (c) the

Y

₃-section.

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 29

TABLE 3.1

θ

_L,

Z

_L AND

Z

_H OF THE STEPPED-IMPEDANCE SECTIONS FOR SUBSTITUTING THE ARMS OF THE RAT RACE COUPLER IN FIGURE 2.6

Section

θ

L =

θ

H

Z

L (Ω) WL (mm) ZH (Ω) WH (mm)

Y

₁ 3.3^o 25.4 3.89 149.0 0.15

Y

2 9.8^o 6.6 18.03 55.0 1.35

Y

3 13.0^o 13.5 8.24 134.6 0.21

3.1.2 Simulation and measurement

Figure 3.3(a) and 3.3(b) show the performances of a fabricated circuit designed at 1.03 GHz. The values of

θ

L (=

θ

H), ZL and ZH for the four arms are in Table 3.1. The length of the Y₁-section of the circuit in Figure 2.6 is 2

θ

₁ = 18.8^o, and the total length of the stepped-impedance section is only 4

θ

L = 13.2^o. Similarly, Y2-section (65.8^o) and

Y

3-section (198.8^o) are replaced by their substitutes of total lengths 39.2^o and 104^o, respectively. Thus the stepped-impedance sections contribute an area reduction factor of (195.6^o/349.2^o)² = 31.4%. The total circumference is 0.54λ and the normalized circuit area is only 13.12%. The size reduction is much better than that of the 7λ/6-ring in [13] and believed to be the best miniaturization of planar rat race couplers in open literature. The measured |S₁₁|, |S₂₁|, |S₃₁| and |S₄₁| are –22.5 dB, –3.34 dB, –25 dB and –3.56 dB, respectively. The best isolation (|S31|) is –35.5 dB at 1.01 GHz. The measured responses are in good agreement with the simulation data. Fig.

3.3(c) shows the photo of the experimental rat race coupler. When the frequency of the design in Fig. 3.3 is increased to 2.5 GHz, the total stepped-impedance peripheral becomes 0.78λ and the normalized circuit area is increased to 27.2%, since the ring

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 30

area limits the line width of the low-impedance sections.

|S |, | S |, | S |, | S | (dB)

11213141

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 31

(c)

Figure 3.3 Performances of the 0.54λ-ring coupler. (a) Magnitude responses. (b) Relative phase responses. (c) Photo of the experimental rat race coupler. Geometric parameters are in Table 3.1.

TABLE 3.2 BANDWIDTHS OF THE CONVENTIONAL 1.5λ-RING, THE 0.97λ-RAT RACE IN

FIGURE 2.6, AND THE 0.78λ- AND 0.54λ-CIRCUITS IN FIGURE 3.3

|S11| = –15 dB (Input matching)

|S31| = –20 dB (Isolation)

∠S₄₁ – ∠S₂₁ =

±5^o

∠S₁₂ – ∠S₃₂ = 180^o ± 5^o Circuit

(2.5 GHz)

Sim. Mea. Sim. Mea. Sim. Mea. Sim. Mea.

1.5λ-ring 39.5% - 31.3% - 16.1% - 15.3% -

0.97λ-ring 3.9% 4.0% 11.9% 9.4% 5.7% 2.9% 10.8% 2.9%

0.78λ-ring 3.2% - 5.2% - 12.4% - 4.4% -

0.54λ-ring (1 GHz)

4.6% 3.9% 6.3% 5.9% 16.7% 15.5% 7.2% 9.1%

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 32

Table 3.2 compares the bandwidths of the conventional 1.5λ ring, the 0.97λ rat race in Figure 2.6, and the 0.54λ and 0.78λ circuits in Figure 3.3. The leading three circuits are designed at 2.5 GHz. The circuit in Figure 2.6 offers smaller bandwidths than the traditional rat race coupler. In particular, the bandwidths measured by |S11|

= –15 dB and |S₃₁| = –20 dB of the circuit in Figure 2.6 are about respectively one tenth and one third of those of the 1.5-λ ring. The simulation bandwidths measured by

∠S41 – ∠S21 = ±5^o and ∠S12 – ∠S32 = 180^o ± 5^o of the 0.78λ circuit and that in Figure 2.6 are between 4.4% and 12.4%. A comparison of the data of 0.54λ and 0.78λ in Figure 3.3 shows that ring miniaturized by the stepped-impedance approach at a lower frequency has larger simulation bandwidths.

CONCLUSION 33

CHAPTER 4

Conclusion

In this thesis, generalized synthesis for a rat race ring coupler is derived and demonstrated. Tradeoffs between peripheral and bandwidth are also clarified.

Moreover, to reduce the circuit size, stepped-impedance sections are adopted to replace the arms of a rat race. Bandwidths of the rat race couplers mentioned in this paper are compared. Finally, some suggestions for the advanced research are provided.

4.1 SUMMARY OF THIS THESIS

In the first part of chapter 2, the functions and characteristics of a rat race coupler are briefly introduced. Then, generalized synthesis for rat race couplers is performed.

Design equations are provided for calculating the electric lengths and the characteristic impedances of the four arms. There are two degrees of freedom in choosing the geometric parameters for synthesis of the rat race couplers. The upper and lower bounds of the solutions are given in analytical expressions. A 0.97λ-ring operating at 2.5 GHz is then fabricated and measured. Finally, operation bandwidths of the synthesized rat race couplers are simulated and discussed. In general, bandwidth decreases when circumference is reduced. Moreover, larger impedance ratio of the arms leads to better size reduction but a smaller bandwidth.

The arms of a rat race are substituted for stepped-impedance sections to achieve

CONCLUSION 34

circuit miniaturization in Chapter 3. By utilizing the technique, the size of 0.97λ-ring is further reduced at 1 GHz. The realized circuit occupies only 13.12% of the area of a conventional 1.5λ-ring and its performances are compared with the 0.97λ-ring and the conventional rat race. In general, the size reduction leads to a decreased circuit bandwidth.

4.2 SUGGESTIONS FOR FUTURE STUDIES

Generalized synthesis of a rat race coupler and its application to circuit miniaturization are presented in this thesis. At least three related topics can be developed for further research. First, as mentioned in Chapter 2, the power ratio of two output ports can be controlled by the impedance ratio (R) and electric lengths (

θ

1

and

θ

2). Nevertheless, the range of the power ratio is restricted to circuit parameters.

If a wide range of |

α

| (|S21/S41|, |S12/S32|) is required, the impedance ratio or electric length may be impractical. Additional elements such as shunt stubs can probably solve the problem.

Second, stepped-impedance sections can be utilized for other applications such as rat race couplers with dual-band function. Since stepped-impedance sections provide more degrees of freedom in circuit design, the required conditions for two operation frequencies may be satisfied.

Third, designing a wide bandwidth rat race coupler is also an interesting topic. In general, the fractional bandwidth of a rat race coupler realized in microstrip substrate is less than 60%. It’s a challenge to synthesize a microstrip rat race with a bandwidth

CONCLUSION 35

more than 60%. Cascading similar elements together generally produces a wider bandwidth than a single element. Consequently, cascading two or more rat race rings can be a possible approach for wide-band application.

REFERENCES 36

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REFERENCES 38

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