Outline of This Thesis

This thesis has four chapters and organized as follows. Chapter 2 introduces characteristics and functions, i.e., the scattering matrix and circuit parameters of a conventional rat race coupler. The second part of Chapter 2 derives the formulae for synthesizing the ring hybrids. Some solutions are presented and the corresponding circuit bandwidths are discussed. In addition, measured results of an experimental circuit are compared with simulation data for validation of the theory.

In Chapter 3, an application to circuit miniaturization is presented. To reduce the circuit size, stepped-impedance sections are utilized to replace the four arms of a rat race coupler. The technique is based on the transmission line theory and the derivation is given in details. Furthermore, comparison between the rat races with and without stepped-impedance substitutions is also discussed. The measurement results of an experimental circuit are used to validate the simulation results based on the proposed

INTRODUCTION 4

theory.

Chapter 4 draws the conclusion of this thesis and proposes some suggestions of the future works.

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 5

CHAPTER 2

Synthesis of a Generalized Rat Race Coupler

This chapter will first describe basic characteristics of a rat race coupler. Then, derivation for generalized synthesis of a rat race is presented. The design graphs of circuit parameters, including the characteristic impedances and electric lengths, are displayed. Finally, tradeoffs between circuit size and bandwidth are discussed. With the graphs and tables shown in this chapter, a rat race coupler can be readily designed and synthesized.

2.1 CHARACTERISTICS OF A RAT RACE COUPLER

The rat race coupler, or 180^o hybrid junction, is a four-port network with an input port, an isolated port, and two output ports with in- and out-of-phase signals. Figure 2.1 shows the circuit schematic of a conventional rat race ring coupler, consisting of an internal ring with impedance

2 Z

0 and four feed lines with identical port impedance Z₀. A signal applied to port 1 will be evenly split into two in-phase components at ports 2 and 4 while port 3 is isolated. If the input is taken at port 2, it will be equally split into two components at ports 1 and 3 with 180 ^o phase difference, and port 4 is isolated. A rat race ring coupler can also be used as a power combiner.

When input signal is applied to ports 2 and 4, a half of the sum of the inputs will be formed at port 1, while a half of the difference will be formed at port 3. Therefore, ports 1 and 3 are referred as the sum and difference ports respectively. Thus, the scattering matrix for an ideal 3dB rat race ring coupler has the following form [19]:

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 6

Figure 2.1 Circuit schematic of the conventional rat race coupler.

2.2 GENERALIZED SYNTHESIS

A generalized rat race ring coupler is derived in this section. Both the characteristic impedances and electric lengths of the four arms can be designed by utilizing the provided formulae. In addition, tradeoffs between circuit size and bandwidth are demonstrated and discussed.

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 7

2.2.1 Formulation

Figure 2.2 shows the layout of the rat race under investigation of port designation.

The reference port admittance is normalized to unity. The parameters Yi and

θ

i (i = 1, 2, and 3) represent the characteristic admittances and electric lengths of the sections, respectively. Even- and odd-mode analysis can be used due to the circuit symmetry.

Based on even-odd analysis, the four-port network can be reduced to a two-port shown in Figure 2.3. Let jY_a and jY_b be the input admittances seen at ports 1 and 2 looking into the sections loaded with YL. The ABCD matrix in Figure 2.3 can be

readily derived as

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 8

θ

₂

θ

2

θ

3

θ

₃

θ

₁

θ

1

Y

2

Y

₁

1 4

3 2

Y

₂

Y

3

Y

1

P

Q

Y

₃

Figure 2.2 Schematic of the rat race coupler under analysis

1 2

Y

3

θ

₃

θ Y

1

1

Y

2

θ

₂

jY

a

jY

b

Y

L

Y

_L

Figure 2.3 Reduced circuit for even- and odd-mode analysis

When Y_L = 0, namely, the even-mode,

Y

_a =

Y

₁tan

θ

₁ and

Y

_b =

Y

₃ tan

θ

₃. If the excitation is taken at port 1, the reflection and transmission coefficients can be derived as

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 9 obtained. Therefore, the two coefficients can be derived in a similar fashion. Let

2

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 10

with –cot

θ

1 and –cot

θ

3, respectively. Since the coupler is reciprocal and symmetric about the PQ plane, only six entries of its 4×4 S-parameter matrix, i.e., Sm1 (m = 1, 2, 3 and 4), S₂₂ and S₃₂ have to be derived. When excitation is taken at port 2, it can be validated that the reflection and transmission coefficients are the results in (2.3) and (2.5) by interchanging the indices 1 and 3.

Next, the following inter-port properties are used to formulate the conditions for solving the circuit parameters:

1) Isolation: S31 = (Te – To)/2 = 0 ⇒ E1 = H1 and E2 = H2. It leads to Y1(tan

θ

1 + cot

θ

1)

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 11

When excitation is applied to port 2, the same result in (2.6) can be obtained while deriving S₄₂ = 0.

2) Input matching: S₁₁ = (Γe + Γo)/2 = 0 ⇒ E3 = –H₃ and E₄ = –H₄. The same conditions are obtained when S₂₂ = 0 is used.

3) Outputs: S21 = (Te + To)/2 =

α S

41 =

α

(Γe – Γo)/2. It results in αe^jφ1(E3 + jE4), where |α| denotes the magnitude of S21/S₄₁ while

φ

1 is the phase difference

between the outputs, ports 2 and 4. Thus, two possible results, (2.7) and (2.8), can be obtained.

1

2

sin 2

sin

θ

θ

=

α^R (2.7a) π

φ₁

= 2

n (2.7b)

1

2

sin 2

sin

θ

θ

= −

α^R (2.8a)

( )

^π

φ₁

= n 2 + 1

(2.8b)

However,

θ

1 or

θ

2 in (2.8a) must be larger than 90^o to satisfy the equation. To

minimize the circumference of the coupler, (2.7a) and (2.7b) are used instead of (2.8a) and (2.8b) herein. Similarly, (2.7a) and (2.9) can be obtained when deriving

S

₁₂/S₃₂.

φ

2 is the phase difference between ports 1 and 3 while excitation is taken at port 2, and the magnitude of S12/S32 is the same as S21/S41, i.e. |α|.

( )

^π

φ₂

= n 2 + 1

(2.9)

Based on the results of properties 2) and 3), we have E₄ = X₄ = 0. With the aid of (2.6), the following conditions can then be obtained:

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 12

2 1 2

1

cot 2 cot 2

1 1

R R θ θ

Y

= + − (2.10)

where R = Y₂/Y₁. In (2.6b), the solution of the minus sign is just that of the plus sign with interchange of

θ

1 and

θ

3 (See also Figure. 2.2). Therefore, choosing the plus sign will not lose any solution since Y₁ = Y₃, as shown in (2.6a). Furthermore, for minimizing the ring size, n = 1 is used herein. Based on (2.7a), (2.7b), and (2.9), |

α

| can be controlled by

θ

1,

θ

2, and R while the phase difference between two output ports is either 0^o or 180^o. As shown in Figure 2.4, if R is fixed and |

α

|’s range is designed to be from -10 dB to 10 dB, every

θ

1 value results in one specific

θ

2 curve. The dashed lines in Figure 2.4 represent the maxima of

θ

1, consisting of real solutions of

θ

2 for |

α

| from -10 dB to 10 dB. Namely, |

α

|’s range can not be realized from -10 dB to 10 dB if

θ

1 value is larger than the dashed line. Therefore, R and

θ

1 must be chosen properly to meet the requirement of |

α

|. Note also that the point A in Figure 2.4(b) denotes the conventional rat race coupler with

θ

1 = 45^o,

θ

2 = 90^o, and |

α

| = 0 dB.

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 13

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 14

For a 3-dB rat race coupler, |

α

| equals 1. Since 3-dB rat race couplers have been widely used in many applications, we focus on the rat race couplers with |

α

| = 1 below.

When |

α

| is 0 dB, (2.7a) can be written as (2.11). Figure 2.5(a) plots the solution

θ

2

versus

θ

1 for various R values based on (2.11). One can see that for any

θ

1 there are two

θ

2 solutions, and vice versa. Note that the solution curves in Figure 2.5(a) are bisymmetric about

θ

1 = 45^o and

θ

2 = 90^o. For circuit size miniaturization, only the

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 15

which is identical to that given in [16], where all the solutions to the ring hybrid design are in one curve of Figure 2.5(b). One can validate that (2.11) and (2.10) are equivalent to (1a) and (1b) of [17]. The condition that

θ

1 and

θ

2 have to obey in (2) of [17] is an inequality. Here, it has an explicit form in (2.11) and is plotted in Figure 2.5(a). Based on (2.10), Figure 2.5(b) plots the Z₁ = Y₁^–1 solutions against

θ

1. For each enforcing cot2

θ

1 = 1 in (2.10) or by evaluating (2.14) using the L’Hospital’s rule.

When Z1 is a small number, say 0.2, the

θ

1 value will be close to that given in (2.13) since each curve has a large slope when Z₁ = 0. The Z₁ solution curves for R ≥ 1 have various upper bounds which are also functions of R and can be derived from (2.11):

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 16

θ

1 = sin^–1(R^–1sin

θ

2)/2 ≤ sin^–1(R^–1)/2 (2.14)

For example, when R = 5 and 1.25, the upper bounds are

θ

1 = 5.77^o and 26.57^o, respectively. When R ≤ 1, the upper

θ

1 bounds can be calculated from the corresponding lower bound in (2.14) since the Y1–1

curves are symmetric about

θ

1 = 45^o.

Based on (2.6b) and (2.11), the total circumference of the ring can be expressed in terms of

θ

1 as

Φ(

θ

1) = 4

θ

1 + 2sin^–1(Rsin2

θ

1) + 180^o (2.15)

Figure 2.5(c) plots the total length l of the hybrid ring normalized with respect to 1.5λ, or

Φ = Φ(

θ

1)/540 (2.16)

for the given R values. Note that as compared with the traditional 1.5λ-ring, the normalized area is square of Φ.

One can design the circuit starting from a given size reduction, e.g., Φ = 0.7, and Figure 2.5(c) shows that there are many possible R values. Alternatively, the design can start from a given

θ

1, say

θ

1 = 30^o > 22.5^o, a smaller R value will lead to a better area reduction. Note that when R = 1, as in [16], the best theoretical size reduction is l

= λ (normalized area = 4/9) under the limit of Y1–1

= 0 where

θ

1 = 22.5^o. If Y2 is different from Y₁, i.e., R ≠ 1, a hybrid ring with l < λ can be obtained. It is also

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 17

possible to start the design from a given Y1–1

in Figure 2.5(b). Once the

θ

1 and R values are chosen,

θ

2 can be determined by invoking the solution curves in Figure 2.5(a).

0 30 60 90

0 60 120 180

(degree)

1

10 20 40 50 70 80

30 90 150

R = 1

0.6 0.4 0.2

5.00 2.50 1.67 1.25

0.2 0.6 0.4 0.8

1.25 1.67 2.50 5.00

0.8

θ θ

²

(degree) ^{R =} 1

(a)

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 18

Φ = Normalized total length l /1.5 λ

0.9

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 19

of the sections evaluated at the operation frequency.

2.2.2 Simulation and measurement

Figure 2.6 compares simulation and measured responses of a rat race coupler, built on a substrate with

ε

_r = 2.2 and thickness = 0.508 mm, with l = 0.97λ at f^o = 2.5 GHz. The ring has a mean radius of 13.47 mm and a normalized area of (0.97/1.5)² = 41.82%. Important circuit parameters include

θ

1 = 9.4^o,

θ

2 = 65.8^o,

θ

3 = 99.4^o, and R = 2.83. The simulation is done by the IE3D [18]. The magnitude responses are in Figure 2.6(a), and the relative phases in Figure 2.6(b). At fo, the measured |S11|, |S21|, |S31| (isolation) and |S41| are −21.4 dB, −3.37 dB, −29.56 dB and −3.36 dB, respectively.

The best measured |S₁₁| is −33.9 dB at 2.47 GHz. The measured results show good agreement with the simulation. Figure 2.6(c) shows the photo of the measured circuit.

Simulation Measured 0

-50 -40 -30 -10

-20

3 3.5 2.5

2 1.5

Frequency (GHz)

41312111

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

|S |

11 21

31

|S | |S | |S |41

|S |₄₁

|S |31

(a)

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 20

Frequency (GHz)

1.5 2 2.5 3 3.5

-70

-160

-250

-340

270 360

180

90

0

(degree ) -

41

S

21

S S

12

S

32

- (degree )

0

Measured Simulation

(b)

(c)

Figure 2.6 Performances of the experimental rat race coupler. (a) Magnitude responses. (b) Phase responses. (c) Photo of the circuit. Y1–1 = 62.15 Ω (W1 = 1.09 mm), Y₂^–1 = 21.96 Ω (W2 = 4.64 mm),

θ

1 = 9.4^o,

θ

2 = 65.8^o,

θ

3 = 99.4^o.

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 21

Figure 2.7 plots the simulation bandwidths of the new rat race rings. Although the entire circuit has four stepped-impedance junctions, only the circuits with R = 5 and 0.2 need slight trimming for tuning the |S₃₁| dips at f_o = 2.5 GHz. A ring with R = 0.2 is used for test the circuit bandwidth. The parameters are

θ

1 = 33.8^o,

θ

2 = 10.7^o,

θ

3

= 123.8^o, Y1–1 = 20.48 Ω and Y2–1 = 102.39 Ω. The bandwidths measured by |S11|

= –15 dB, |S₃₁| = –20 dB, |S₁₂/S₃₂| = ±0.5 dB, |S21/S₄₁| = ±0.5 dB, ∠S21 – ∠S41 = ±5^o and ∠S12 – ∠S32 = 180^o ± 5^o are 6.8%, 28.9%, 24.8%, 8.8%, 8.8% and 8.5%, respectively. The bandwidth by |S11| = –15 dB has the smallest value, so that it is used as a basis in Figure 2.7 for demonstration. When

θ

1 ≤ 45^o, for a given R value, a larger

θ

1 has a larger bandwidth, except for R = 1 and

θ

1 ≥ 35^o. For example, when R = 1.25, the bandwidth changes from 5% to 35% when

θ

1 is varied from 20^o to 25^o. One can see that when R or 1/R is larger, the circuit possesses smaller bandwidth. It is interesting to note that a uniform hybrid ring (R = 1) can have a bandwidth from about 5% to 40% by choosing a proper

θ

1.

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 22

0.1 0

0.2 0.2

0.3 0.4 0.5 0.6

Fractional bandwidth

^0.4

0.6 0.8

1.25 1

1.67 2.5

R=5

5 10 15 20 25 30 35 40 45

0

θ

₁

(degree)

Figure 2.7 Bandwidths of the new rat race couplers. Bandwidth is defined by the frequencies where |S11| = –15 dB. Substrate:

ε

r = 2.2 and thickness = 0.508 mm.

To show more details on the tradeoffs between bandwidth and normalized circumference (Φ in (2.16)), Table 2.1 summarizes the data shown in Figure 2.5(c) and Figure 2.7. For the extreme cases of R = 0.2 and R = 5, some line widths are too small to be accepted for simulation so that not all solutions in Figure 2.5(c) are given in Figure 2.7 and Table 2.1. In Table 2.1, ΦL and ΦH denote the lower and upper limits of Φ in our simulation, and ΔL and ΔH are their bandwidths, respectively. It is noted that ΦL and ΦH will change when the substrate or the design frequency is changed.

SYNTHESIS OF A GENERALIZED RAT RACE COUPLER 23

TABLE 2.1TRADEOFFS BETWEEN BANDWIDTH AND NORMALIZED CIRCUMFERENCE OF THE RAT RACE COUPLER

Φ (See (2.16)) Δ (%,|S11|=–15DB)

R

ΦL ΦH ΔL ΔH

0.2 0.62 0.65 6.80 12.2

0.4 0.66 0.75 9.50 24.2

0.6 0.68 0.80 9.20 34.5

0.8 0.68 0.86 7.70 51.5

1.0 0.67 1.00 2.70 39.0

1.25 0.68 0.85 5.40 36.0

1.67 0.67 0.79 6.10 22.6

2.5 0.65 0.73 4.50 12.6

5.0 0.63 0.65 2.90 4.50

CIRCUIT MINIATURIZATION WITH STEPPED-IMPEDANCE SECTIONS 24

CHAPTER 3

Circuit Miniaturization with Stepped-Impedance Sections

Circuit area occupation is always a critical issue in the microwave frequency band, especially at low frequencies. Therefore, many approaches are proposed to save circuit size, as discussed in Chapter 1. This chapter presents an effective way to miniaturize the size of rat race couplers using stepped-impedance sections. Finally, effectiveness of the miniaturization on the circuit bandwidths is also discussed.

3.1 CIRCUIT MINIATURIZATION

3.1.1 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

References

[1] R. K. Settaluri, G. Sundberg, A. Weisshaar and V. K. Tripathi, “Compact folded

[1] R. K. Settaluri, G. Sundberg, A. Weisshaar and V. K. Tripathi, “Compact folded

在文檔中 環形耦合器之廣義合成及其電路微小化之應用 (頁 11-0)