Smith Chart Part 1

Sheet 1 of 13

Smith Chart Tutorial Part1 To begin with we start with the definition of VSWR, 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 VSWR:Define voltage standing wave ratio (VSWR)

Vmax V1 + V2 = Vmin V1 − V2 Voltage reflection coefficient Γ - Complex β

V1e+j. .l

β

V2e-j. .l

T.L

ZL

L Γ =

− j . β .l

V2 e V = 2 e − j 2 β .l + j .β .l V1 V1 e

At the load Γ (l = 0) ; Γ =

V2 ⇒ V1

but this may be complex number if there is an instantaneous phase change which we’ll call (φ) on reflection.

V2

Phase Diagram

V 2 V 2 j∅ = e ..... (a) L = 0 V1 V1

V1 θ

Sheet 2 of 13

At L > 0 Γ(L) = Γ( 0 )e − j .φ =

V2 V1

e j (φ − 2 βl )

For a lossless line V1 & V2 do not vary with L ∴ Γ is constant = Γ = Γ e j (φ − 2 β .l )

V2 V1

represented on Crank Diagram

Crank Diagram We use a crack diagram as a way of representing the reflection coefficient phasor.

V = V1 e + j .β .l + V2 e − j .β .l ∴

At L = 0

V V j φ − 2 β .l ) = 1 + 2 e − j 2 β .l = 1 + Γ e ( + j . β .l V1 V1 e

V = 1 + Γ e j.φ V1

AP

OP

P

|Γ| Between 0 & 1

φ O

L

A

At the origin of argand diagram.OP = magnitude of total voltage/incident voltage

P Vmin = 1-|Γ|

|Γ|

φ-2β.L

P’ C

O

B

A Vmax = 1+|Γ|

Sheet 3 of 13

As we saw previously the crack diagram with a circle drawn between points A & C is the beginnings of a Smith chart less the constant resistance and reactance circles/lines.

Vmax 1+ Γ VSWR - 1 = or Γ = VSWR + 1 Vmin 1- Γ

VSWR =

|V| C

C Vmax Length OP’ -crank diagram

Vmin

B

voltage v incident

B

0

L Lmin

At B φ - 2 β . l min = - π

φ = 2 β . l min - π =

4π . l min

λg

−π = φ

∴From standing wave pattern measure VSWR ⇒ | Γ | @ lmin ⇒ φ at load.

Sheet 4 of 13

SmithChart - Impedance (Z) or Admittance Y chart (1) Crank diagram + constant resistance & constant reactance circles. (2) Graphical solution to the equation

Z ( in ) Zo

=

1+ Γ 1− Γ

Γ = Γ e j ( φ − 2 β .l ) complex

(3) Smith Chart is a reflection coefficient diagram

θ = φ-2.β.L |Γ|

θ A

|Γ| = 1

A = |Γ| = 0

Smith Chart R=1 circle A is the matched point |Γ| =

0 Const r

no reflection

Const x Short circuit v=0,x=0 Open circuit x ⇒ ∞

O X = 0 ∴ pure resistance

A

F

R=0 pure reactance circle

Sheet 5 of 13

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 150 + j75Ω First normalise ie 150/50 = 3Ω ; 75/50 = 1.5Ω normalised impedance = 3+ j1.5Ω So the real part of the impedance will lie somewhere along the r = 3 constant resistance circle ie:R=3 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.

X=0.75 line

R=3 circle

3+j1.5

3

Sheet 6 of 13

Using the Smith Chart (1) Moving along the T.L = rotating around the Smith Chart.

FORWARDS

ZL

L FORWARDS (TO LOAD)

BACKWARDS BACKWARDS (FROM LOAD)

(2) Constant |Γ| or VSWR circles For a lossless line |Γ| & VSWR do not vary with L.

1+ Γ VSWR = 1- Γ

Vmin

S

Vmax =

Vmin =

Constant |Γ| or VSWR circles

|Γ| = 0 VSWR = 1

Z(max) = S (real VSWR) Zo Z(min) 1 = S Zo

Vmax |Γ| = 1 VSWR = ∞

Sheet 7 of 13

(3) Measure Lmin/λg ..... determines φ (at load).

|V| B

φ = 4π

lmin

λg

−π

angle of reflection coefficient L

Lmin FORWARD by Lmin/λg takes us the load.

Vmin

φ

B

Lmin/λg

ZL/Zo (4) Reading Z from chart also can get |Γ| & φ

x ZL φ

r

Sheet 8 of 13

(5)

Z/Zo

Z 1+ Γ = Zo 1 − Γ

|Γ| φ

1 Admittance = Y/Y0 =

1 φ -|Γ| Y/Yo

On a Smith chart point diametrically opposite

Note Y0 =

1 Zo

Y = G + j. β Conductance

Susceptance

On admittance chart r circles → g circles & x circles → b circles.

Note g =

G B and b = Yo Yo

Z Y gives Zo Yo

Sheet 9 of 13

(6) To transform an impedance along a T.L, rotate around the VSWR circle:-

Z(in)

ZL

Zo

Lmin BACKWARDS by Lmin/λg takes us to Zin.

l/λg

ZL/Zo

B Vmin Lmin/λg

Sheet 10 of 13

(7) Represent a series inductance on a smith chart.

Read values off the reactance scale 0.5

0.20

Therefore, assuming a frequency of say 1GHz the value of series inductance represented on the above Smith Chart is given by:-

Reactance (XL ) read from Smith chart = 0.5 - 0.2 Ω = 0.3Ω wrt 50Ω L =

N.XL 50 * 0.3 = = 2.38nH 2πf 2π * 1E 9

Similarly for a series capacitor

Sheet 11 of 13

(8) Represent a series capacitance on a smith chart.

Read values off the reactance scale 0.5

1.0

Therefore, assuming a frequency of say 1GHz the value of series capacitance represented on the above Smith Chart is given by:Reactance(XC ) read from Smith chart = 1.0 - 0.5 Ω = 0.5Ω w.r.t 50Ω C =

1 1 = = 6.36pF 2πf .N.XC 2π * 1E9 * 50 * 0.5

Where N is the normalising factor (usually 50 ohms)

To represent shunt reactance we need to plot admittance onto the Smith Chart. 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 180 degrees.

Sheet 12 of 13

(9) Represent a shunt inductance on a smith chart.

Read values off the admittance scale 0.8

0.2

Therefore, assuming a frequency of say 1GHz the value of shunt inductance represented on the above Smith Chart is given by:Admittance(YL ) read from Smith chart = (0.8 - 0.2) Ω = 0.6mhos w.r.t 50Ω L =

N 50 = = 13.26nH 2πf * YL 2π * 1E9 * 0.6

N = normaisation factor (usually 50 ohms)

Sheet 13 of 13

(10) Represent a shunt capacitance on a smith chart.

Read values off the admittance scale 1.0

0.2

Therefore, assuming a frequency of say 1GHz the value of shunt inductance represented on the above Smith Chart is given by:-

Admittance (YC ) read from Smith chart = (1.0 - 0.2) Ω = 0.8mhos w.r.t 50Ω C =

YC 0.8 = = 2.5pF 2πf * N 2π *1E 9 * 50

N = normaisation factor (usually 50 ohms)