AC
Lecture 7 Lecture 7 Smith Chart Hayt CH 11 Obj i D l Objective: Develop skill in using Smith kill i i S ih Chart for AC Tx line analysis ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The Smith Chart is a graphical tool used to model Smith Chart is a graphical tool used to model standing wave behaviour on transmission lines ( (this can be extended to other standing wave g systems such as waveguides). • The chart represents the complex plane as a The chart represents the complex plane as a series of interconnections between circles. The chart is designed to directly map the ratio of g y p voltage standing waves to current standing waves. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • • •
Real Axis Horiz Real Axis Horiz Imag. Axis Vert. Radius Mag. Г di
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The real axis values range from 0 to infinity with real axis values range from 0 to infinity with real 1 at the center of the chart. • The imaginary axis follows the outer perimeter of The imaginary axis follows the outer perimeter of the chart. With values ranging from 0 to infinity with positive 1 near the middle of the upper arc with positive 1 near the middle of the upper arc and negative 1 near the middle of the lower arc. • A short circuit (SC) lies on (0,0) while an open A short circuit (SC) lies on (0,0) while an open circuit (OC) lies on the infinity (opposite side of chart to the SC). ) ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart •
Short or Open Ct Short or Open Ct
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The position on the chart will be a function of position on the chart will be a function of the real and imaginary components of the load. load • The chart is a normalized function: • The load is normalized by the characteristic Th l d i li d b h h i i impedance of the transmission line. • For a 50 ohm line the normalized load impedance for ZL =100+j50 would be 2+j. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The normalized normalized impedance for some of the some of the “reference” loads are given loads are given in this figure.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The normalized load will provide a point on normalized load will provide a point on the complex plane. A circle, centered on the point (1 0) {the middle of the chart} that point (1,0) {the middle of the chart}, that crosses this point will have a radius proportional to the load reflection coefficient proportional to the load reflection coefficient.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Given Given a load of 100 a load of 100‐j200 j200 connected to a 50 connected to a 50 ohm line. Find the reflection coefficient using a Smith Chart a Smith Chart. • The normalized impedance is 2‐j4. The real circle RE=2 and the imaginary arc IM = 4 circle RE=2 and the imaginary arc IM =‐4 intersection will provide this location.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The real circle for r=2 is real circle for r=2 is shown on the right. • This will given the real This will given the real component of the normalized impedance normalized impedance.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The intersection of intersection of the r=2 and x=‐4 arc shown on the arc shown on the right. • This is the location This is the location of the normalized load impedance on load impedance on the complex plane. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Using Using the scale on the the scale on the bottom of the chart, the length of the line length of the line connecting the origin to the normalized load the normalized load impedance gives ГГ=0 0.825. 825 • The angle is ‐23 degrees ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Computing Computing the reflection coefficient directly the reflection coefficient directly gives: Г=0.82 with an angle of 23 degrees. • The chart will provide good accuracy within 2 The chart will provide good accuracy within 2 decimal places (graphical techniques trade off accuracy for speed). • The chart can also be used to compute input impedances as phase change is rotation along complex plane with same reflection coefficient magnitude. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The value of a standing wave is a function of value of a standing wave is a function of the relative phase between the forward and backward travelling waves. • For a loss‐less system, the amplitude of the backward travelling wave is constant and proportional to the load reflection coefficient. • This suggests that the value of the impedance at any point along a transmission line would lie on a circle on a smith chart. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • A A transmission line with a 100 ohm transmission line with a 100 ohm characteristic impedance is terminated by a 100‐j100 100 j100 load. load • The 30.0 cm long line is powered by a 600 MHz 10 volt (RMS) source The velocity of MHz 10 volt (RMS) source. The velocity of propagation along the line is 200 meters per microsecond Use a Smith Chart to compute microsecond. Use a Smith Chart to compute the impedance seen by the source. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The normalized load impedance is 1 normalized load impedance is 1 – j. The j The 0.3 0.3 0.90 line is wavelengths long. 2 6
• The load impedance is the start of the circular path (for input impedance we travel towards h (f i i d l d the source). We travel along the circle a di distance of 0.9 wavelengths (i.e. 0.5 + 0.4 f09 l h (i 0 5 0 4 wavelengths). ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The starting point of 1 starting point of 1 –jj corresponds to a corresponds to a position of 0.338 wavelengths on the outer rim of the chart. • Moving a distance of 0.9 wavelengths corresponds to 1.238 wavelengths (since every half wavelength is a full circle, the end position is 0.238 wavelengths). • This corresponds to the point 2.57 + j 0.42 or a load impedance of 257+j42. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Input Impedance Impedance intersection of impedance impedance circle and position on position on circle perimeter. perimeter ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • From From the TRLINE program the input the TRLINE program the input impedance is 253.5+j42.25 (close to the graphical evaluation) graphical evaluation). • The load is (0.25‐0.238)λ=0.012λ from the nearest maximum (of the VSWR) and 0 262λ nearest maximum (of the VSWR) and 0.262λ from the nearest minimum (of the VSWR). • You can locate the min. and max. of the VSWR Y l h i d f h VSWR relative to the load using a chart. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: •
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The lowest scale on left for reflection lowest scale on left for reflection coefficient. Uppermost left VSWR.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The right hand real axis intercepts are the right hand real axis intercepts are the VSWR values
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The Smith Chart can be used for both design Smith Chart can be used for both design and analysis. • Impedance matching requires a small (ideally Impedance matching requires a small (ideally zero) reflection coefficient . This means the curve on the chart will be around the center curve on the chart will be around the center. • The next example will illustrate this behaviour. • Given a band pass microwave filter stop‐band centered on 2 GHz. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: •
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: •
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The portion of the curve close to the center of portion of the curve close to the center of the chart have small reflection coefficients (closer to the center the smaller the (closer to the center, the smaller the coefficient). • Drawing a circle around a portion of the Drawing a circle around a portion of the center will define a stop band. • The transmission loss is a parameter related to Th i i l i l d the transmission coefficient. Т=Г+1 ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • Return Return loss: loss: 20 |1 Γ| • Transmission loss: • The transmission loss and return loss h i i l d l parameters are in common use. • The return loss is used for impedance matching (small reflection). • The transmission loss is used for filter design ( g (large transmission). ) ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Next Lecture Next Lecture • In the next lecture we will discuss Stub tuning In the next lecture we will discuss Stub tuning
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The Smith Chart is a graphical tool used to model Smith Chart is a graphical tool used to model standing wave behaviour on transmission lines ( (this can be extended to other standing wave g systems such as waveguides). • The chart represents the complex plane as a The chart represents the complex plane as a series of interconnections between circles. The chart is designed to directly map the ratio of g y p voltage standing waves to current standing waves. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • • •
Real Axis Horiz Real Axis Horiz Imag. Axis Vert. Radius Mag. Г di
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The real axis values range from 0 to infinity with real axis values range from 0 to infinity with real 1 at the center of the chart. • The imaginary axis follows the outer perimeter of The imaginary axis follows the outer perimeter of the chart. With values ranging from 0 to infinity with positive 1 near the middle of the upper arc with positive 1 near the middle of the upper arc and negative 1 near the middle of the lower arc. • A short circuit (SC) lies on (0,0) while an open A short circuit (SC) lies on (0,0) while an open circuit (OC) lies on the infinity (opposite side of chart to the SC). ) ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart •
Short or Open Ct Short or Open Ct
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The position on the chart will be a function of position on the chart will be a function of the real and imaginary components of the load. load • The chart is a normalized function: • The load is normalized by the characteristic Th l d i li d b h h i i impedance of the transmission line. • For a 50 ohm line the normalized load impedance for ZL =100+j50 would be 2+j. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The normalized normalized impedance for some of the some of the “reference” loads are given loads are given in this figure.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The normalized load will provide a point on normalized load will provide a point on the complex plane. A circle, centered on the point (1 0) {the middle of the chart} that point (1,0) {the middle of the chart}, that crosses this point will have a radius proportional to the load reflection coefficient proportional to the load reflection coefficient.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Given Given a load of 100 a load of 100‐j200 j200 connected to a 50 connected to a 50 ohm line. Find the reflection coefficient using a Smith Chart a Smith Chart. • The normalized impedance is 2‐j4. The real circle RE=2 and the imaginary arc IM = 4 circle RE=2 and the imaginary arc IM =‐4 intersection will provide this location.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The real circle for r=2 is real circle for r=2 is shown on the right. • This will given the real This will given the real component of the normalized impedance normalized impedance.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The intersection of intersection of the r=2 and x=‐4 arc shown on the arc shown on the right. • This is the location This is the location of the normalized load impedance on load impedance on the complex plane. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Using Using the scale on the the scale on the bottom of the chart, the length of the line length of the line connecting the origin to the normalized load the normalized load impedance gives ГГ=0 0.825. 825 • The angle is ‐23 degrees ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Computing Computing the reflection coefficient directly the reflection coefficient directly gives: Г=0.82 with an angle of 23 degrees. • The chart will provide good accuracy within 2 The chart will provide good accuracy within 2 decimal places (graphical techniques trade off accuracy for speed). • The chart can also be used to compute input impedances as phase change is rotation along complex plane with same reflection coefficient magnitude. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The value of a standing wave is a function of value of a standing wave is a function of the relative phase between the forward and backward travelling waves. • For a loss‐less system, the amplitude of the backward travelling wave is constant and proportional to the load reflection coefficient. • This suggests that the value of the impedance at any point along a transmission line would lie on a circle on a smith chart. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • A A transmission line with a 100 ohm transmission line with a 100 ohm characteristic impedance is terminated by a 100‐j100 100 j100 load. load • The 30.0 cm long line is powered by a 600 MHz 10 volt (RMS) source The velocity of MHz 10 volt (RMS) source. The velocity of propagation along the line is 200 meters per microsecond Use a Smith Chart to compute microsecond. Use a Smith Chart to compute the impedance seen by the source. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The normalized load impedance is 1 normalized load impedance is 1 – j. The j The 0.3 0.3 0.90 line is wavelengths long. 2 6
• The load impedance is the start of the circular path (for input impedance we travel towards h (f i i d l d the source). We travel along the circle a di distance of 0.9 wavelengths (i.e. 0.5 + 0.4 f09 l h (i 0 5 0 4 wavelengths). ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The starting point of 1 starting point of 1 –jj corresponds to a corresponds to a position of 0.338 wavelengths on the outer rim of the chart. • Moving a distance of 0.9 wavelengths corresponds to 1.238 wavelengths (since every half wavelength is a full circle, the end position is 0.238 wavelengths). • This corresponds to the point 2.57 + j 0.42 or a load impedance of 257+j42. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • Input Impedance Impedance intersection of impedance impedance circle and position on position on circle perimeter. perimeter ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • From From the TRLINE program the input the TRLINE program the input impedance is 253.5+j42.25 (close to the graphical evaluation) graphical evaluation). • The load is (0.25‐0.238)λ=0.012λ from the nearest maximum (of the VSWR) and 0 262λ nearest maximum (of the VSWR) and 0.262λ from the nearest minimum (of the VSWR). • You can locate the min. and max. of the VSWR Y l h i d f h VSWR relative to the load using a chart. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: •
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The lowest scale on left for reflection lowest scale on left for reflection coefficient. Uppermost left VSWR.
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The right hand real axis intercepts are the right hand real axis intercepts are the VSWR values
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • The The Smith Chart can be used for both design Smith Chart can be used for both design and analysis. • Impedance matching requires a small (ideally Impedance matching requires a small (ideally zero) reflection coefficient . This means the curve on the chart will be around the center curve on the chart will be around the center. • The next example will illustrate this behaviour. • Given a band pass microwave filter stop‐band centered on 2 GHz. ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: •
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: •
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Example: Smith Chart Example: • The The portion of the curve close to the center of portion of the curve close to the center of the chart have small reflection coefficients (closer to the center the smaller the (closer to the center, the smaller the coefficient). • Drawing a circle around a portion of the Drawing a circle around a portion of the center will define a stop band. • The transmission loss is a parameter related to Th i i l i l d the transmission coefficient. Т=Г+1 ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Smith Chart Smith Chart • Return Return loss: loss: 20 |1 Γ| • Transmission loss: • The transmission loss and return loss h i i l d l parameters are in common use. • The return loss is used for impedance matching (small reflection). • The transmission loss is used for filter design ( g (large transmission). ) ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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Next Lecture Next Lecture • In the next lecture we will discuss Stub tuning In the next lecture we will discuss Stub tuning
ECSE 352 AC Tx Line Notes (c) D. Davis 2011
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