12/15/2016 ECE 4380/5390 Spring 2013 Instructor: Dr. Raymond Rumpf Office: A-337 E-Mail: [email protected] Topic #6 Smith Charts Smith Charts 1 Outline • Construction of the Smith Chart • Admittance and impedance • Circuit theory • Determining VSWR and • Impedance transformation • Impedance matching Smith Charts 2 1 12/15/2016 Construction of the Smith Chart 2 12/15/2016 Polar Plot of Reflection Coefficient The Smith chart is based on a polar plot of the voltage reflection coefficient . The outer boundary corresponds to || = 1. The reflection coefficient in any passive system must be|| ≤ 1. j e radius on Smith chart angle measured CCW from right side of chart Smith Charts 5 Normalized Impedance All impedances are normalized. This is usually done with respect to the characteristic impedance of the transmission line Z0. Z z Z0 Smith Charts 6 3 12/15/2016 Reflection Coefficient form Normalized Impedance We can write the reflection coefficient in terms of normalized impedances. Z Z L 0 ZZ Z Z z 1 L 000 L Z Z ZZLL0 L 0 z1 ZZ00 Smith Charts 7 Derivation of Smith Chart: Solve for Load Impedance Solving the previous equation for load impedance, we get z 1 L zL 1 zz11 LL 1 zzLL 1 zL zzLL1 1 zL 11 1 z L 1 Smith Charts 8 4 12/15/2016 Derivation of Smith Chart: Real and imaginary parts The load impedance and reflection coefficient can be written in terms of real and imaginary parts. zrjxLL L r j i Substituting these into the load impedance equation yields 1 z L 1 1rij rjxLL 1rij 1ri j rjxLL 1ri j Smith Charts 9 Derivation of Smith Chart: Solve for rL and xL We solve or previous equation for rL and xL by setting the real and imaginary parts equal. 1 j rjx ri LL 22 1ri j 1ri 11rijj ri r L 2 2 11 jj ri ri 1ri 11 jj 1 1 2 rr riiri 2 2 1ri 122jj jj ririirii 2 2 2 1ri i xL 2 1222 j 2 ri i 2 1 2 ri 1ri 1222 rij i 2222 11ri ri Smith Charts 10 5 12/15/2016 Derivation of Smith Chart: Rearrange equation for rL We rearrange the equation for rL so that it has the form of a circle. 122 r ri L 2 2 1ri 22 22 2 2 1ri rrLL2 r L1 1ri ri 0 rL rrLL11 r L 1 2 2 22 2 2 1 r i rrrLLL2 1 10ri ri rrLL r L rrrLLL111 2 2 2 22r i 1 2 210rr i rrLL2 rrLL11 rrrLLL ri 22 rL 1 rr11 210rr 22 r 22 r LL Lr Lr r Li i L 2 22 22 rrr1 21110rr r r LLL2 Lr L r L i L ri 22 r 1 L rrLL11 22rLr rL 1 2 0 2 rir 1 r 1 r 1 L L L 2 ri 2 can be factored r 1 L rL 1 Smith Charts 11 Derivation of Smith Chart: Rearrange equation for xL We rearrange the equation for xL so that it has the form of a circle. 2 x i L 2 2 1ri 2 2 2i 1ri xL 2 2 10 2 ri i xL swap terms can be factored 2 2 11 ri10 2 xxLL Smith Charts 12 6 12/15/2016 Derivation of Smith Chart: Two families of circles Constant Resistance Circles Constant Reactance Circles 22 22 rL 2 1 2 11 ri ri1 rrLL11 xLLx These have centers at These have centers at rL 1 ri 0 ri1 rL 1 xL Radii Radii 1 1 1 rL xL Smith Charts 13 Derivation of Smith Chart: Putting it all together Lines of constant Lines of constant Lines of constant resistance inductive reactance reflection coefficient Superposition ++= Lines of constant capacitive reactance We ignore what is outside the || = 1 circle. We don’t draw the constant || circles. This is the Smith chart! Smith Charts 14 7 12/15/2016 Alternate Way of Visualizing the Smith Chart Lines of constant resistance Lines of constant reactance Reactance Regions open L circuit short circuit C Smith Charts 15 3D Smith Chart The 3D Smith Chart unifies passive and active circuit design. 2D 3D EE3321 ‐‐ Final Lecture 8 12/15/2016 Summary of Smith Chart Smith Charts 17 Impedance and Admittance on the Smith Chart 9 12/15/2016 Admittance Coordinates We could have derived the Smith chart in terms of admittance. You can make an admittance Smith chart by rotating the standard Smith chart by 180. Smith Charts 19 Impedance/Admittance Conversion The Smith chart is just a plot of complex numbers. These could be admittance as well as impedance. To determine admittance from impedance (or the other way around)… 1. Plot the impedance point on the Smith chart. 2. Draw a circle centered on the Smith chart that passes through the point (i.e. constant VSWR). 3. Draw a line from the impedance point, through the center, and to the other side of the circle. 4. The intersection at the other side is the admittance. impedance admittance Smith Charts 20 10 12/15/2016 Visualizing Impedance/Admittance Conversion Smith Charts 21 Example #1 – Step 1 Plot the impedance on the chart zj0.2 0.4 Smith Charts 22 11 12/15/2016 Example #1 – Step 2 Draw a constant VSWR circle zj0.2 0.4 Smith Charts 23 Example #1 – Step 3 Draw line through center of chart zj0.2 0.4 Smith Charts 24 12 12/15/2016 Example #1 – Step 4 Read off admittance zj0.2 0.4 yj1.0 2.0 Smith Charts 25 Example #2 – Step 1 Plot the impedance on the chart zj0.5 0.3 Smith Charts 26 13 12/15/2016 Example #1 – Step 2 Draw a constant VSWR circle zj0.5 0.3 Smith Charts 27 Example #2 – Step 3 Draw line through center of chart zj0.5 0.3 Smith Charts 28 14 12/15/2016 Example #2 – Step 4 Read off admittance zj0.5 0.3 yj1.5 0.9 Smith Charts 29 Circuit Theory 15 12/15/2016 Adding a Series Capacitor Suppose we have an initial impedance of z = 0.5 + j0.7 And we add a series capacitor of zC = -j1.0. Since we do not change the resistance, we walk toward capacitance (CCW) around the constant resistance circle. The “angular” distance covers X=-j1.0 around the constant R circle. Smith Charts 31 Adding a Series Inductor Suppose we have an initial impedance of z = 0.8 - j1.0 And we add a series inductor of zL = j1.8. Since we do not change the resistance, we walk toward Inductance (CW) around the constant resistance circle. The “angular” distance covers X=j1.8 around the constant R circle. Smith Charts 32 16 12/15/2016 Adding a Shunt Capacitor Here we use the Smith chart rotated by 180° for admittance. In terms of admittance, capacitance is a positive quantity, but the positive direction is now downward on the Smith chart. Smith Charts 33 Adding a Shunt Inductor We again use the Smith chart rotated by 180° for admittance. In terms of admittance, inductance is a negative quantity, but the negative direction is now upward on the Smith chart. Smith Charts 34 17 12/15/2016 Summary Series C Shunt C Series L Shunt L Smith Charts 35 Example #1 – Circuit Analysis Smith Charts 36 18 12/15/2016 Example #2 – Circuit Analysis 3.3157 nH Z ? 50 1 in 1.9894 pF ZRjL in jC || 1 RjL jC 1 RjL f 2.4 GHz jC RjL Z0 50 12 LC j RC 50j 1.5080 1010 3.3157 10 9 2 1 1.5080 1010 3.3157 10 9 1.9894 10 12j 1.5080 10 10 50 1.9894 10 12 20j 40 50 20j 40 0.0769 j 0.6154 0.62 82.9 50 20j 40 1 0.0769j 0.6154 VSWR 4.2654 1 0.0769j 0.6154 Smith Charts 37 Example #2 – Circuit Analysis Normalize Impedances j1.0 1 zin ? 1 j1.5 50 r 1.0 50 j2 2.4 1099 3.3157 10 Lj1.0 50 1 912 j2 2.4 10 1.9894 10 1 C 50 j1.5 Smith Charts 38 19 12/15/2016 Example #2 – Circuit Analysis Plot load impedance j1.0 1 1 zin ? j1.5 z 1 Smith Charts 39 Example #2 – Circuit Analysis Add series inductor j1.0 1 1 zin ? j1.5 zj1 Smith Charts 40 20 12/15/2016 Example #2 – Circuit Analysis Convert to admittance 1 j1.0 1 j1.5 yin ? yj0.5 0.5 Smith Charts 41 Example #2 – Circuit Analysis Add shunt capacitance 1 j1.0 1 j1.5 yin ? yj0.5 1.0 Smith Charts 42 21 12/15/2016 Example #2 – Circuit Analysis Convert to impedance j1.0 1 1 zin ? j1.5 zj0.4 0.8 We now know zin zin z 0.4 j 0.8 Smith Charts 43 Example #2 – Circuit Analysis Denormalize the impedance j1.0 1 1 zin ? j1.5 zj0.4 0.8 ZZzin 0 in 20 j 40 Smith Charts 44 22 12/15/2016 Example #2 – Circuit Analysis All together 3.3157 nH 50 1.9894 pF Zjin 20 40 Smith Charts 45 Determining VSWR and 23 12/15/2016 The Horizontal Bar on the Smith Chart VSWR Reflectance 2 Reflection Coefficient Smith Charts 47 Determining VSWR 1. Plot the normalized load impedance on the Smith chart.