Voltage reflection coefficient Γ. L e V V. = e. At the load Γ (l = 0) ; Γ = V V

Transcription

1 of 3 Smith hart Tutorial Part To begin with we start with the definition of SWR, which is the ratio of the reflected voltage over the incident voltage. The Reflection coefficient Γ is simply the complex (ie has phase) version of SWR:- Define voltage standing wave ratio (SWR) max min oltage reflection coefficient Γ - omplex e +j. β.l 2 e -j. β.l T. Z Γ 2e e j. β. l + j. β. l 2 e j2β. l At the load Γ (l 0) ; Γ 2 but this may be complex number if there is an instantaneous phase change which we ll call () on reflection j e... (a) 0 Phase Diagram θ

2 2 of 3 At > 0 Γ () Γ e j. (0) 2 e ( 2βl) j For a lossless line & 2 Γ Γ e do not vary with Γ ( 2β. l) j is constant 2 represented on rank Diagram rank Diagram We use a crack diagram as a way of representing the reflection coefficient phasor. j β l e + e +.. j. β. l 2 2 j + e + + j. β. l e β. l j( β. l) Γ e 2 2 At 0 OP + Γe j. AP P Γ Between 0 & O A At the origin of argand diagram.op magnitude of total voltage/incident voltage P min - Γ Γ P -2β. O B A max + Γ

3 3 of 3 As we saw previously the crack diagram with a circle drawn between points A & is the beginnings of a Smith chart less the constant resistance and reactance circles/lines. SWR max min + - Γ Γ or Γ SWR - SWR + max ength OP -crank diagram B B min voltage incident v 0 min At B - 2 β. l - π min 2 β. l - π min 4 π. l λ g min π From standing wave pattern measure SWR l min at load.

4 4 of 3 Smithhart - Impedance (Z) or Admittance Y chart () rank diagram + constant resistance & constant reactance circles. (2) Graphical solution to the equation Z + Γ Γ ( in) j ( 2β. l ) Z o Γ Γ e complex (3) Smith hart is a reflection coefficient diagram Γ θ -2.β. A θ A Γ 0 Γ Smith hart Γ 0 no reflection A is the matched point Short circuit v0,x0 R circle onst r onst x Open circuit x X 0 pure resistance O A F R0 pure reactance circle

5 5 of 3 Impedance is plotted on the smith chart by first normalising to the characteristic impedance of the system (usually 50 ohms). In a 50 ohm system the centre of the smith chart is a pure 50 ohms. For example say we wanted to plot an impedance of 50 + j75ω First normalise ie 50/50 3Ω ; 75/50.5Ω normalised impedance 3+ j.5ω So the real part of the impedance will lie somewhere along the r 3 constant resistance circle ie:- R3 circle 3 Next we follow the constant reactance line at 0.75 to find the intersection of the r 3 circle to get to our impedance point. X0.75 line R3 circle 3+j.5 3

6 6 of 3 Using the Smith hart () Moving along the T. rotating around the Smith hart. FORWARDS Z FORWARDS (TO OAD) BAKWARDS (FROM OAD) BAKWARDS (2) onstant Γ or SWR circles For a lossless line Γ & SWR do not vary with. SWR + - S Γ Γ max Z(max) Zo min Z(min) Zo S (real SWR) S min onstant Γ or SWR circles Γ 0 SWR max Γ SWR

7 7 of 3 (3) Measure min/λ g... determines (at load). B 4π lmin π λ g angle of reflection coefficient min FORWARD by min/λ g takes us the load. min B min/λ g (4) Reading Z from chart also can get Γ & Z/Zo x Z r

8 8 of 3 (5) Z + Γ Z o Γ Γ Z/Zo Admittance Y/Y 0 - Γ Y/Yo On a Smith chart point diametrically opposite Z Z o gives Y Y o Note Y 0 Z o Y G + j. β onductance Susceptance On admittance chart r circles x circles g circles & b circles. Note g G Y o and b B Y o

9 9 of 3 (6) To transform an impedance along a T., rotate around the SWR circle:- Z(in) Zo Z min BAKWARDS by min/λ g takes us to Zin. l/λ g Z/Zo min B min/λ g

10 0 of 3 (7) Represent a series inductance on a smith chart. Read values off the reactance scale Therefore, assuming a frequency of say GHz the value of series inductance represented on the above Smith hart is given by:- Reactance (X ) read from Smith chart Ω 0.3Ω wrt 50Ω N.X 2πf 50 * π * E 2.38nH Similarly for a series capacitor

11 of 3 (8) Represent a series capacitance on a smith chart. Read values off the reactance scale Therefore, assuming a frequency of say GHz the value of series capacitance represented on the above Smith hart is given by:- Reactance(X ) read fromsmithchart Ω 0.5Ω w.r.t 50Ω 2π f.n.x 2π * E 9 * 50 * pF WhereNis the normalising factor (usually 50 ohms) To represent shunt reactance we need to plot admittance onto the Smith hart. It is easiest to use a Smith chart with both impedance (usually in black) lines and admittance lines (usually in red) on the same chart. Or you can rotate the Smith chart 80 degrees.

12 2 of 3 (9) Represent a shunt inductance on a smith chart. Read values off the admittance scale Therefore, assuming a frequency of say GHz the value of shunt inductance represented on the above Smith hart is given by:- Admittance(Y ) read fromsmith chart ( ) Ω 0.6mhos w.r.t 50Ω N 2πf * Y π * E * nH N normaisation factor (usually 50 ohms)

13 3 of 3 (0) Represent a shunt capacitance on a smith chart. Read values off the admittance scale Therefore, assuming a frequency of say GHz the value of shunt inductance represented on the above Smith hart is given by:- Admittance (Y ) read from Smith chart (.0-0.2) Ω 0.8mhos w.r.t 50Ω Y 2πf * N π*E *50 2.5pF N normaisation factor (usually 50 ohms)