Electromagnetics (ENGR 367) The Smith Chart: A Graphical Method for T-lines

Advantages of Knowing the Smith Chart Popularity: one of most widely used graphical methods for T-line analysis & design Popularity: one of most widely used graphical methods for T-line analysis & design –as an aid in a pencil & paper approach –for display by measurement and CAD tools Labor-saving: facilitates complex number calculations without losing too much accuracy Labor-saving: facilitates complex number calculations without losing too much accuracy Serves as a valuable tool for visualizing key aspects of T-line systems as they depend on frequency or position along the line Serves as a valuable tool for visualizing key aspects of T-line systems as they depend on frequency or position along the line

Requirements for Understanding the Smith Chart Have basic orientation to T-lines Have basic orientation to T-lines –Concepts (generator, load, matching, etc.) –Parameters (Z 0, Z in, ,, VSWR, Y=1/Z, etc.) Become familiar with some unique aspects Become familiar with some unique aspects –What curves on the graph represent –How to show Z or Y and move along the line Practice using the Smith Chart to find , VSWR, Z in, and do T-line matching Practice using the Smith Chart to find , VSWR, Z in, and do T-line matching

Outline of Lecture on Smith Chart Basic layout: shows what curves represent Basic layout: shows what curves represent Learn where to plot points: complex impedance (or admittance) Learn where to plot points: complex impedance (or admittance) Identify key reference points Identify key reference points Find out how to move along the line Find out how to move along the line Work examples of Smith Chart usage Work examples of Smith Chart usage Solve impedance Matching Problems Solve impedance Matching Problems Draw Conclusions Draw Conclusions

Basic Layout of the Smith Chart Within a circle of constant unit radius (  =1), graph explicitly shows a family of Within a circle of constant unit radius (  =1), graph explicitly shows a family of –Circles of constant resistance –Curves of constant reactance A polar coordinate graph of  =  e j  exists w/r/to the center of the chart that may be evaluated for   1 along with VSWR  1 on scales beneath the chart A polar coordinate graph of  =  e j  exists w/r/to the center of the chart that may be evaluated for   1 along with VSWR  1 on scales beneath the chart

Plotting Points on the Smith Chart Define Normalized Impedance Define Normalized Impedance Reflection Coefficient in terms of z L Reflection Coefficient in terms of z L Note: this keeps values w/in range on S.C.

Plotting Points on the Smith Chart Zero Reactance Line (Im{z} = x = 0): Zero Reactance Line (Im{z} = x = 0): –Also Horizontal Center Line of Pure Resistance Re{z} = r –Divides the Upper Half from the Lower Half Upper Half represents Im{z} > 0 (inductive reactance) [or alternatively: Im{y} > 0 (capactive susceptance)] Upper Half represents Im{z} > 0 (inductive reactance) [or alternatively: Im{y} > 0 (capactive susceptance)] Lower Half represents Im{z} < 0 (capacitive reactance) [or alternatively: Im{y} < 0 (inductive susceptance)] Lower Half represents Im{z} < 0 (capacitive reactance) [or alternatively: Im{y} < 0 (inductive susceptance)]

Essential (Key) Reference Points On The Smith Chart 1) Center (z = r = 1): matched condition –Also represents  = 0 and VSWR = 1 –Becomes the destination (objective) for impedance matching problems (HOME !) 2) Right Extreme (z = r =  ): open circuit –Also  = 1  0 ° and VSWR =  –An open circuited stub has Z in that begins here

Essential (Key) Reference Points On The Smith Chart 3) Left Extreme (z = r = 0): short circuit –Also  = 1  180 ° and VSWR =  –A short circuited stub has Z in that begins here Key Reference Points on the Smith Chart Key Reference Points on the Smith Chart * Center Point: matched condition (HOME) O Right Extreme: open circuit termination  Left Extreme: short circuit termination

Moving Toward the Generator (or Load) on the Smith Chart Use Circular Band at Outer Edge of Chart: has scales that indicate distance (in ) Use Circular Band at Outer Edge of Chart: has scales that indicate distance (in ) –To go toward the generator, move CW –To go toward the load, move CCW Normalized Input Impedance on the T-line Normalized Input Impedance on the T-line

Examples of Smith Chart Usage Example A (conditions same as in TLS3) Example A (conditions same as in TLS3) Given: Z 0 = Z L = 300 ,  =  l = 0.8 = 288°) Find: Z in, , VSWR Solution: take the following steps on the S.C.

Examples of Smith Chart Usage Solution: (Example A continued) Solution: (Example A continued)

Examples of Smith Chart Usage Example B (conditions same as in TLS3) Example B (conditions same as in TLS3) Given: Z L = R in  R in = 300   300  = 150  and Z 0 = 300 ,  =  l = 0.8 = 288° as before Find: Z in, , VSWR Solution: take these steps on the chart

Examples of Smith Chart Usage Solution: (Example B continued) Solution: (Example B continued)

Examples of Smith Chart Usage Example C (conditions same as in TLS3) Example C (conditions same as in TLS3) Given: Z L = (120 - j60) , Z 0 = 300  and  =  l = 0.8 = 288° as before Find: Z in, , VSWR Solution: follow these steps on the chart

Examples of Smith Chart Usage Solution: (Example C continued) Solution: (Example C continued)

Examples of Smith Chart Usage Example D (conditions same as in TLS3) Example D (conditions same as in TLS3) Given: Z L = - j300 , Z 0 = 300  and  =  l = 0.8 = 288° as before Find: Z in, , VSWR Solution: take these steps

Examples of Smith Chart Usage Solution: (Example D continued) Solution: (Example D continued)

Impedance Matching Problems Methods of T-line Matching (Load-to-line) Methods of T-line Matching (Load-to-line) –Single stub Series insertion Series insertion Parallel insertion Parallel insertion –Double stub tuner: usually a manuf’d device with movable shorts (act like trombone slides) –Quarter-wave transformer (QWT): fixed freq. –Tapered transformer (~ g long): for pulses and wideband applications

Matching Problem Using the Smith Chart: Example 1 Matching Problem Using the Smith Chart: Example 1 Single Stub Matching (P3-4-9, K&F, 5/e, p. 149) Single Stub Matching (P3-4-9, K&F, 5/e, p. 149) Given: a uniform Z 0 = 100  T-line terminated by a load Z L = (150 + j50)  Find: a) min. dist. d 1 from load to parallel shorted stub b) min. length d 2 of the shorted stub to match Solution: first examine the schematic diagram

Single Stub Matching Solution: take the following steps on S.C. Solution: take the following steps on S.C.

Single Stub Matching Solution: (continued) Solution: (continued)

Single Stub Matching Solution: (continued) Solution: (continued)

Single Stub Matching Design of the Single Stub Match has been completed since location and length of the shorted stub have been specified. Design of the Single Stub Match has been completed since location and length of the shorted stub have been specified. For further consideration For further consideration –What if an open circuited stub had been used? –Actual physical location and length of the stub depends on frequency and phase velocity since the wavelength = v p /f.

Matching Problem Using the Smith Chart: Example 2 QWT Matching (P3-4-10, K&F, 5/e, p. 151) QWT Matching (P3-4-10, K&F, 5/e, p. 151) Given: a 100  T-line terminated by a Z L = (300 + j200)  load Find: a) min. length d 1 to transform Z L to a pure resistance b) impedance of the QWT required for a match b) impedance of the QWT required for a match c) VSWR on each section of line c) VSWR on each section of line Solution: first examine the schematic diagram

QWT Matching Solution: take the following steps on S.C. Solution: take the following steps on S.C.

QWT Matching Solution: (continued) Solution: (continued)

QWT Matching Actual physical length of QWT depends on operating frequency and phase velocity since =v p /f Actual physical length of QWT depends on operating frequency and phase velocity since =v p /f For a microstrip QWT, Z( /4) is determined by its width a substrate of known  r and thickness (d), but must be within manufacturable limits For a microstrip QWT, Z( /4) is determined by its width a substrate of known  r and thickness (d), but must be within manufacturable limits As for Single Stub matching, QWT method works well for fixed frequency applications but its quality deteriorates for wideband applications As for Single Stub matching, QWT method works well for fixed frequency applications but its quality deteriorates for wideband applications

Matching with Double Stub Tuner Tuner design possible on the Smith Chart Tuner design possible on the Smith Chart Tuner has one more degree of freedom than the Single Stub, so more involved Tuner has one more degree of freedom than the Single Stub, so more involved In practice, tuning for match usually done experimentally with a pre-fabricated device In practice, tuning for match usually done experimentally with a pre-fabricated device

Variations of QWT Method for Wideband, Pulse & Digital Data Tx Description of Method Elements Real Estate (vs. QWT) Bandwidth (vs. QWT) Shorter than /4 lengths 2 /16 transformers Less (+) Less (-) Lumped Elements LC section Less (+) Less (-) Multiple /4 sections Match to inter- mediate imp’s. More (-) More (+) Tapered section 1 continuous piece More (--) More (++) Reference: Kraus & Fleisch, 5/e, pp (Bandwidth, Pulses & Transients).

Conclusions The Smitch Chart serves as a graphical method of T-line analysis and design to help visualize aspects of the system The Smitch Chart serves as a graphical method of T-line analysis and design to help visualize aspects of the system The Smith Chart represents complex normalized impedances (or admittances) on curves within a polar plot of  = 1 The Smith Chart represents complex normalized impedances (or admittances) on curves within a polar plot of  = 1

Conclusions Moving toward the generator (or load) on the Smith Chart corresponds to rotating on the constant  circle CW (or CCW) Moving toward the generator (or load) on the Smith Chart corresponds to rotating on the constant  circle CW (or CCW) The Smith Chart may be used to find , VSWR and Z in any position on a T-line ahead of a complex load impedance The Smith Chart may be used to find , VSWR and Z in any position on a T-line ahead of a complex load impedance

Conclusions Fixed frequency T-line impedance matching methods whose design may be illustrated on the Smith Chart include Fixed frequency T-line impedance matching methods whose design may be illustrated on the Smith Chart include –Single & Double Stub Tuner –Quarter-wave Transformer (QWT) For wideband applications, a tapered transformer provides more bandwidth at the expense of more space to implement For wideband applications, a tapered transformer provides more bandwidth at the expense of more space to implement

References and Other Resources Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999.