L stub Z A = Z 0 Z R Z 0S. Single stub impedance matching
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1 Single stub impedance matching Impedance matching can be achieved by inserting another transmission line (stub) as shown in the diagram below Z A = Z 0 Z 0 Z R Z 0S d stub L stub Amanogawa, 006 Digital Maestro Series 9
2 There are two design parameters for single stub matching: Transmission Lines The location of the stub with reference to the load d stub The length of the stub line L stub Any load impedance can be matched to the line by using single stub technique. The drawback of this approach is that if the load is changed, the location of insertion may have to be moved. The transmission line realizing the stub is normally terminated by a short or by an open circuit. In many cases it is also convenient to select the same characteristic impedance used for the main line, although this is not necessary. The choice of open or shorted stub may depend in practice on a number of factors. A short circuited stub is less prone to leakage of electromagnetic radiation and is somewhat easier to realize. On the other hand, an open circuited stub may be more practical for certain types of transmission lines, for example microstrips where one would have to drill the insulating substrate to short circuit the two conductors of the line. Amanogawa, 006 Digital Maestro Series 93
3 Since the circuit is based on insertion of a parallel stub, it is more convenient to work with admittances, rather than impedances. Y A = Y 0 Y 0 = /Z 0 Y R = /Z R Y 0S d stub L stub Amanogawa, 006 Digital Maestro Series 94
4 For proper impedance match: ( d ) Y = Y + Y = Y = A stub stub 0 Z 0 Input admittance of the stub line Line admittance at location d stub before the stub is applied Y 0S Y stub + Y (d stub ) Y R = /Z R L stub d stub Amanogawa, 006 Digital Maestro Series 95
5 In order to complete the design, we have to find an appropriate location for the stub. Note that the input admittance of a stub is always imaginary (inductance if negative, or capacitance if positive) Y stub = jb stub A stub should be placed at a location where the line admittance has real part equal to Y 0 ( d ) = + ( d ) Y Y jb For matching, we need to have stub 0 stub B stub = B ( d ) stub Depending on the length of the transmission line, there may be a number of possible locations where a stub can be inserted for impedance matching. It is very convenient to analyze the possible solutions on a Smith chart. Amanogawa, 006 Digital Maestro Series 96
6 θ θ 3 z R First location suitable for stub insertion y(d stub ) Unitary conductance circle -0. Constant Γ(d) circle y R -3 Load location y(d stub ) - Second location suitable for stub insertion Amanogawa, 006 Digital Maestro Series 97
7 The red arrow on the example indicates the load admittance. This provides on the admittance chart the physical reference for the load location on the transmission line. Notice that in this case the load admittance falls outside the unitary conductance circle. If one moves from load to generator on the line, the corresponding chart location moves from the reference point, in clockwise motion, according to an angle θ (indicated by the light green arc) 4π θ= β d= d λ The value of the admittance rides on the red circle which corresponds to constant magnitude of the line reflection coefficient, Γ(d) = Γ R, imposed by the load. Every circle of constant Γ(d) intersects the circle Re { y } = (unitary normalized conductance), in correspondence of two points. Within the first revolution, the two intersections provide the locations closest to the load for possible stub insertion. Amanogawa, 006 Digital Maestro Series 98
8 The first solution corresponds to an admittance value with positive imaginary part, in the upper portion of the chart Line Admittance - Actual : Normalized : θ Stub Location : d stub = λ 4π Stub Admittance - Actual : Stub Length : ( d ) = + ( d ) Y Y j B y stub 0 stub ( d ) = + j b( d ) jb stub stub ( d ) stub Norma lized : jb( d ) stub λ Lstub = tan π Z ( ) 0sB dstub ( short) λ L ( ( )) stub = tan Z0 sb d stub π ( open) Amanogawa, 006 Digital Maestro Series 99
9 The second solution corresponds to an admittance value with negative imaginary part, in the lower portion of the chart Line Admittance - Actual : Normalized : θ Stub Location : d stub = λ 4π Stub Admittance - Actual : Stub Length : ( d ) = ( d ) Y Y j B y jb stub 0 stub ( d ) = j b( d ) stub stub ( d ) stub Norma lized : jb( d ) stub λ Lstub = tan π Z ( ) 0sB dstub ( short) λ L ( ( )) stub = tan Z0 sb d stub π ( open) Amanogawa, 006 Digital Maestro Series 00
10 If the normalized load admittance falls inside the unitary conductance circle (see next figure), the first possible stub location corresponds to a line admittance with negative imaginary part. The second possible location has line admittance with positive imaginary part. In this case, the formulae given above for first and second solution exchange place. If one moves further away from the load, other suitable locations for stub insertion are found by moving toward the generator, at distances multiple of half a wavelength from the original solutions. These locations correspond to the same points on the Smith chart. First set of locations Second set of locations λ = dstub + n λ = dstub + n Amanogawa, 006 Digital Maestro Series 0
11 Second location suitable for stub insertion Load location y(d stub ) 0. θ θ y R -0. z R -3 Unitary conductance circle Constant Γ(d) circle y(d stub ) - First location suitable for stub insertion Amanogawa, 006 Digital Maestro Series 0
12 Single stub matching problems can be solved on the Smith chart graphically, using a compass and a ruler. This is a step-by-step summary of the procedure: (a) Find the normalized load impedance and determine the corresponding location on the chart. (b) Draw the circle of constant magnitude of the reflection coefficient Γ for the given load. (c) Determine the normalized load admittance on the chart. This is obtained by rotating 80 on the constant Γ circle, from the load impedance point. From now on, all values read on the chart are normalized admittances. Amanogawa, 006 Digital Maestro Series 03
13 0.5 (a) Obtain the normalized load impedance z R =Z R /Z 0 and find its location on the Smith chart 3 0. z R y R 80 = λ /4-3 (b) Draw the constant Γ(d) circle -0.5 (c) Find the normalized load admittance knowing that y R = z(d=λ /4 ) From now on the chart represents admittances. - - Amanogawa, 006 Digital Maestro Series 04
14 (d) Move from load admittance toward generator by riding on the constant Γ circle, until the intersections with the unitary normalized conductance circle are found. These intersections correspond to possible locations for stub insertion. Commercial Smith charts provide graduations to determine the angles of rotation as well as the distances from the load in units of wavelength. (e) Read the line normalized admittance in correspondence of the stub insertion locations determined in (d). These values will always be of the form y y ( stub ) ( ) d = + jb top half of chart d = jb bottom half of chart stub Amanogawa, 006 Digital Maestro Series 05
15 First Solution (d) Move from load toward generator and stop at a location where the real part of the normalized line admittance is z R 3 First location suitable for stub insertion d stub =(θ /4π)λ (e) Read here the value of the normalized line admittance y(d stub ) = +jb θ Load location -0. y R Unitary conductance circle - Amanogawa, 006 Digital Maestro Series 06
16 Second Solution (d) Move from load toward generator and stop at a location where the real part of the normalized line 05 admittance is θ z R Unitary conductance circle Load location -0. y R (e) Read here the value of the normalized line admittance y(d stub ) = - jb - Second location suitable for stub insertion d stub =(θ /4π)λ Amanogawa, 006 Digital Maestro Series 07
17 (f) Select the input normalized admittance of the stubs, by taking the opposite of the corresponding imaginary part of the line admittance ( stub ) ( ) line: y d = + jb stub: y = jb stub stub line: y d = jb stub: y =+ jb stub (g) Use the chart to determine the length of the stub. The imaginary normalized admittance values are found on the circle of zero conductance on the chart. On a commercial Smith chart one can use a printed scale to read the stub length in terms of wavelength. We assume here that the stub line has characteristic impedance Z 0 as the main line. If the stub has characteristic impedance Z 0S Z 0 the values on the Smith chart must be renormalized as Y Z ± jb' =± jb =± jb Y Z 0 0s 0s 0 Amanogawa, 006 Digital Maestro Series 08
18 y = Short circuit (g) Arc to determine the length of a short circuited stub with normalized input admittance - jb -0.5 (f) Normalized input admittance of stub y stub = 0 - jb Amanogawa, 006 Digital Maestro Series 09
19 (g) Arc to determine the length of an open circuited stub with normalized input admittance - jb y = 0 Open circuit (f) Normalized input admittance of stub y stub = 0 - jb Amanogawa, 006 Digital Maestro Series 0
20 0.5 (f) Normalized input admittance of stub y stub = 0 + jb 0. (g) Arc to determine the length of a short circuited stub with normalized input admittance + jb 3 y = Short circuit Amanogawa, 006 Digital Maestro Series
21 0.5 (f) Normalized input admittance of stub y stub = 0 + jb 0. (g) Arc to determine the length of an open circuited stub with normalized input admittance + jb y = 0 Open circuit Amanogawa, 006 Digital Maestro Series
22 First Solution 0. matching condition y R After the stub is inserted, the admittance at the stub location is moved to the center of the Smith chart, which corresponds to normalized admittance and reflection coefficient 0 (exact matching condition). If you imagine to add gradually the negative imaginary admittance of the inserted stub, the total admittance would follow the yellow arrow, reaching the match point when the complete stub admittance is added. - Amanogawa, 006 Digital Maestro Series 3
23 First Solution Transmission Lines If the stub does not have the proper normalized input admittance, the matching condition is not reached 0.5 Effect of a stub with negative susceptance of insufficient magnitude 0. 3 Effect of a stub with positive susceptance y R Effect of a stub with negative susceptance of excessive magnitude - - Amanogawa, 006 Digital Maestro Series 4
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