interactive matlab programs for impedance matching
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Interactive Matlab Programs for Impedance Matching Teaching in Microwave Engineering
Qiang Sui Liang Zhang Information Engineering School
Communication University of China Beijing, China
Hongtao Jia TaiYuan Television Station
TaiYuan Radio TV Broadcasting Station TaiYuan, China
Abstract—This paper presents four Graphics User Interface Matlab programs that students can use to explore the feature of different matching networks. Impedance matching is a typical and classical topic in microwave engineering teaching. The single-stub tuner and double-stub tuner are often used to match the complex load impedance to a transmission line. The single-section quarter-wave transformer and the multi-section quarter-wave Chebyshev transformer are used to match the real load impedance to a transmission line. The students can use the programs to verify their solutions easily.
Keywords-impedance matching;Matlab program;single-stub tuner;double-stub tuner;quarter-wave transformer.
I. INTRODUCTION Although students are already fluent in analysis of
impedance matching of synthesized method using lumped L and C, the “microwave engineering” based on distributed-circuit[1,2] is still a crabbed lesson. Many new concepts such as traveling wave, incident wave, reflected wave, reflection coefficient, standing wave, return loss, standing-wave ratio are difficult to understand for students. The Matlab[3,4] programs are introduced to help students understand the theory of the impedance matching. Students can use the programs to verify their solutions to homework problems as well.
This paper presents four Matlab programs which support the teaching of stub matching and quarter-wave transformer. The programs provide a computational laboratory to allow students to explore in greater depth the impedance matching theory. The programs permit a quick evaluation of the bandwidth of transformers, the stub-matching solutions, and also help students to develop an awareness of frequency dependence in their studies. This paper describes the function of the programs and shows how these can be done as an aid in solving impedance matching problems
II. STUB MATCHING As long as the load impedance has some nonzero real part,
a matching network can always be found. The matching technique that uses short-circuited transmission line (“stub”) is often used because such tuning circuit is convenient from a
microwave fabrication aspect. The single-stub tuner uses a single stub connected in parallel with the transmission line at a certain distance from the load. It is very easy to fabricate in microstrip or stripline form. The double-stub tuner uses two stubs connected in parallel with the transmission line in fixed positions. Such tuner is often fabricated in coaxial line form. Though the double-stub tuner cannot match all load impedances (“forbidden region” exists), it is still very important to introduce the matching principle to the students.
Students usually work with a paper smith chart, a pencil, a compass and their brain to determine the matching work. Both single and double stub matching program are based on analytic solution. The matching solution is also illustrated on the smith chart to help the students verify their solutions.
A. Single-stub shunt tuning In single-stub tuner, the two adjustable parameters are the
distance, D, from the load to the stub position, and the value of susceptance provided by the shunt stub. The basic idea is to select D so that the normalized admittance, yL, seen looking into the line at distance D from the load is of the form 1+jb.Then the normalized stub susceptance is chosen as –jb, resulting a matched condition. The interface of single-stub matching program (called SingleStubMatch.) is shown in Figure 1. The top left block shows the configuration of the single-stub tuner. The value of the load impedance, ZL, and the characteristic impedance, Z0, can be filled in conveniently. The right area illustrates the solutions on the smith chart. Fig.1 shows the calculated results when the load impedance ZL equals to 90-i80 ohm, and the characteristic impedance is 50 ohm. The first step is to get the normalized load impedance, zL=1.8-i1.6 and then convert ZL to the normalized admittance yL=0.31+i0.276. For the remaining steps the smith chart is considered as an admittance chart. Now notice that the VSWR circle intersects the 1+ib circle at point P1.Thus the length D is given by the yL position and the P1 position. The normalized admittance of point P1 is 1+i1.33. Thus, the solution requires the stub with a susceptance, -i1.33. The length of the shunt stub can be found on the smith chart by starting at yL=∞and moving along the outer edge of the chart toward the generator
This work was supported by the project of the State Administration of Radio Film and Television (Grand No. BG0109).
978-1-4244-4507-3/09/$25.00 ©2009 IEEE
to the –i1.33 point. The calculated D and L can be shown after pressing the D1 and L1 button.
Figure 1. SingleStubMatch interface
Obviously, there is another intersection of the VSWR circle and the 1+ib circle (denoted as P2). Another solution can be shown by clicking the D2 and L2 button. The SingleStubMatch is easy to use and the results can be got immediately when we change the value of the load impedance and the characteristic impedance of the transmission line.
B. Double-stub shunt tuning For the disadvantages of single-stub tuner [1], it is often
more convenient to have two stub lines attached at fixed positions and achieve matched condition by adjusting the lengths of the two stubs. The interface of double-stub matching program (called DoubleStubMatch) is shown in Figure 2. The layout of the interface is almost the same as that of SingleStubMatch. The first stub (named Stub1) can be an arbitrary distance, D1, from the load. The distance, D2, between the first stub and the second stub (named Stub2) is predetermined by the designer. Fig.2 shows a double-stub shunt tuner of matching the load impedance ZL=100+i50 ohm to Z0=50 ohm (the characteristic impedance of the line) with two stubs at 1/8λ spacing. The distance between Stub1 and the load, D1, is also 1/8λ. The solution of this problem is described as follows:
1. Normalizing ZL and finding the normalized admittance yL=0.4-i0.2 by drawing a constant VSWR circle. This normalization will be done automatically after entering the value of ZL and Z0 in the program.
2. Finding the normalized admittance at Stub1 position by moving distance D1 clockwise on the constant VSWR circle, which stops at point P1 (0.5+i0.5).
3. Drawing the auxiliary circle by moving every point on the g=1 circle 1/8λ toward the load (anticlockwise).
4. Adding the susceptance provided by Stub1 to 0.5+i0.5, the point P1 reaches P2 (0.5+i0.14) on the auxiliary circle. So the susceptance provided by Stub1 must be –i0.36, thus the length of Stub1 is determined (L1=0.194λ).
5. Moving point P2 along the constant VSWR circle 1/8λ to point P3 on the g=1 circle where the admittance is 1+i0.73.
6. Adding the susceptance provided by Stub2 to 1+i0.73, the point P3 reaches the center of the chart (the absolutely matching point). The susceptance provided by Stub2 must be –i0.73, thus the length of Stub2 is determined (L2=0.149λ).
The solutions can be immediately shown after pressing the button L1, L2 below “solution 1” and “solution 2” respectively.
Figure 2. DoubleStubMatch interface
DoubleStubMatch offers an additional function that can detect the “forbidden region” automatically. If the normalized admittance yL falls in the “forbidden region”, a reminding window will pop up when you press the button L1 or button L2.
III. QUARTER-WAVE TRANSFORMER The quarter-wave transformer is a simple and useful circuit
for matching any real load impedance to any line impedance. In the “microwave engineering” course, matching over a wider bandwidth is investigated using multi-section transformer. In the next two sections, the single and multi-section quarter-wave transformer programs are introduced.
A. Single-section quarter-wave matching transformer The single-section quarter-wave program (called
SingleSectionTransformer) interface is shown in Figure 3. The schematic diagram of single-section quarter-wave matching transformer is shown in the top left block. At the design frequency, fC, the length of the matching section is 4/Cλ , but at other frequencies the length is not 4/λ , so perfect match is no longer obtained. When Z0, ZL and the maximum reflection coefficient magnitude are decided, the relative bandwidth is calculated and the magnitude of the reflection coefficient versus normalized frequency f/fC is sketched in the right area of the interface after pressing the “Analyze” button.
Figure 3. SingleSectionTransformer Interface
For a constant maximum reflection coefficient magnitude, the bandwidth is different when the ratio of Z0 to ZL changes. The reflection coefficient magnitude versus normalized frequency for various mismatched loads is sketched as Figure 4.
Figure 4. Magnitude of reflection coefficient versus normalized frequency with different ratio ( Z0/ZL )
It is shown that the bandwidth increases for smaller load mismatches for a given maximum reflection coefficient magnitude that can be tolerated.
B. Multi-section transformer For engineering applications requiring more bandwidth
than a single quarter-wave section can provide, multi-section transformers can be used. This transformer consists of N equal-length (commensurate) sections of transmission lines. The total reflection coefficient Γ can be derived approximately from the theory of small reflections. Two of the most commonly used passband responses are the binomial (maximally flat) and the Chebyshev (equal ripple) response. For the same maximum reflection coefficient magnitude mΓ and the same sections N, the bandwidth of the Chebyshev transformer is substantially better than that of the binomial transformer.
The interface of the multi-section transformer program (called MultiSectionTransformer) is shown in Figure 5. This program is designed mainly for coaxial multi-section transformer adjusting. The configuration of the coaxial transformer is shown in the right bottom area. The diameter of the outer conductor of all sections keeps unchanged, thus the characteristic impedance of each section is determined by the
diameter of the inner conductor. All the input boxes displayed in the “Data Input” block should be filled in and checked before the button “Analyze” is pressed. The return loss or the VSWR versus the frequency is shown in the upper area of the interface when the button “Analyze” is pressed. The “Adjust Panel” block is just beside the “Data Input” block. The length and diameter of each inner conductor can be adjusted to optimize the return loss by the sliders with the specific slider step set by the “Sliderstep” toggle options. The process of adjustment helps students develop the awareness of the frequency dependence visually.
The optimized return loss of two-section and three-section transformer to match 50 ohm to 25 ohm in the frequency range of 470MHz - 862 MHz is shown in Fig. 5 and Figure 6, respectively. From the two figures, the “equal-ripple” response is obvious and the bandwith of three-section is much better than that of two-section.
Figure 5. Return loss versus frequency of two-section transformer
Figure 6. Return loss versus frequency of three-section transformer
Because there is a step change in the diameter of adjacent inner conductor, there exists the reactance associated with discontinuity [5]. This can be compensated for by making a small adjustment in the length of the matching section. In this program, the effect of the reactance is considered and the cascade [A] matrix [6] is chosen to calculate the total return loss of multi-section impedance transformer network. The network matrix of impedance transformer is closely related to
stepped-impedance filters. This program can inspire the students’ passion in computer aid design for microwave engineering.
IV. CONCLUSION This series of programs are intended as an aid to teaching
impedance matching at the introductory and intermediate level. These programs contain several impedance matching patterns for common matching problems and simple “graphic user interface”.
The stub-matching programs can give the more exactly solution than paper smith chart which also convenient for teacher to explain the principle.
The MultiSectionTransformer program introduces the possibility of refining design. It provides a quick computation of return loss or VSWR as a function of frequency, which is one of the key specifications. The adjusting on the interface is quite simple and the adjusting results can be used directly to the design of microwave transformer.
All the programs provided above are used to teach the basic principles of microwave matching. It will be better to use if the
four programs are integrated into one interface with a menu to choose matching pattern. It will be even better if the program is converted into an installing package and it can be used without the Matlab environment. If students progress beyond the intermediate level, a transition should be made to software which is oriented toward the design of real components. For example, the software of HFSS provides a realistic simulation environment for coaxial line and microstrip line components.
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