ECE 453 – Jose Schutt‐Aine 1

Jose E. Schutt-AineElectrical & Computer Engineering

University of Illinoisjesa@Illinois.edu

ECE 453Wireless Communication Systems

The Smith Chart

ECE 453 – Jose Schutt‐Aine 2

TL Equations

tantan

R oo

o R

Z jZ lZ l ZZ jZ l

2( ) j lRl e

1 ( )( )1 ( )o

zZ z Zz

( )( )( )

o

o

Z z ZzZ z Z

Impedance Transformation 

Reflection Coefficient Transformation 

Reflection Coefficient – to Impedance 

Impedance to Reflection Coefficient 

2( ) 1 j z j z

Ro

VI z e eZ

2( ) 1j z j zRV z V e e

Voltage Current

ECE 453 – Jose Schutt‐Aine 3

Derivation of the Smith Chart

o1 ( z )Z( z ) Z1 ( z )

n1 ( z ) 1Z ( z )1 ( z ) 1

The relationship between impedance and reflection coefficient is given by:

where Zo is the characteristic impedance of the system.  The normalized impedance is

r ij nZ r jx

The reflection coefficient and the normalized impedance have the form:

and

ECE 453 – Jose Schutt‐Aine 4

( ( =

(r i r i

2 2r i

1 ) j 1 ) j1 )

( (=(

2 2r i r i r i

2 2r i

1 j 1 ) j 1 )r jx1 )

Therefore

Separating real and imaginary components,

r i

r i

1 jr jx1 j

Derivation of the Smith Chart

(

2 2r i2 2

r i

1r1 )

Isolating the real part from both sides

ECE 453 – Jose Schutt‐Aine 5

2 2 2r r i r ir 1 2 1

2 2r i r( r 1 ) ( r 1 ) 2r 1 r

2 2 rr i

2r 1 r1 r 1 r

2 22 2 rr i 2 2

2r r 1 r r1 r (1 r ) 1 r (1 r )

22

r i 2

r 11 r (1 r )

Multiplying through by the denominator,

Completing the square

Derivation of the Smith Chart

or

ECE 453 – Jose Schutt‐Aine 6

r ,01 r

11 r

This is the equation of a circle centered at 

and of radius 

22

r i 2

r 11 r (1 r )

Derivation of the Smith Chart

Equating the imaginary parts gives

i2 2

r i

2x(1 )

2 2r r i ix 1 2 2

2 2r r i ix 2x x 2 x or

ECE 453 – Jose Schutt‐Aine 7

11,x

1x

This is the equation of a circle centered at 

of radius 

2 2 ir r i 2 2

2 1 12 1 1 1x x x

2

2r i 2

1 11x x

Derivation of the Smith Chart

ECE 453 – Jose Schutt‐Aine 8

n

n

Z 1 r 1 jxZ 1 r 1 jx

1/ 22 2

2 2

( r 1 ) x 1( r 1) x

n1 ( z )Z1 ( z )

n

1 1 ( z )yZ 1 ( z )

The reflection coefficient is given by

We also have

The Smith Chart

Thus, going from normalized impedance to normalized admittance corresponds to a 180 degree shift.

ECE 453 – Jose Schutt‐Aine 9

The Smith Chart

3 ways to move on the Smith chartConstant SWR circle  displacement along TLConstant resistance (conductance) circleaddition of reactance (susceptance)Constant reactance (susceptance) arc  addition of resistance (conductance)

ECE 453 – Jose Schutt‐Aine 10

Results of several different experiments are plotted on a Smith chart. Each experimentmeasured the input reflection coefficient from a low frequency (denoted by a circle) toa high frequency (denoted by a square) of a one‐port. Determine the load that wasmeasured. The loads that were measured were one of those shown on the table below.You should make no assumptions about how low the low frequency was, nor abouthow high the high frequency was. For each of the measurements below indicate theload using the load identifier above (e.g., i, ii, etc.) There may be more than one correctanswer.

(a) What type of load gives rise to the refection coefficient indicated by curve A?(b) What type of load gives rise to the refection coefficient indicated by curve B?(c) What type of load gives rise to the refection coefficient indicated by curve C?(d) What type of load gives rise to the refection coefficient indicated by curve D?(e) What type of load gives rise to the refection coefficient indicated by curve E?(f) What type of load gives rise to the refection coefficient indicated by curve F?

Smith Chart Example

ECE 453 – Jose Schutt‐Aine 11

Smith Chart Example

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Load Descriptioni An inductorii A capacitoriii A reactive load at the end of a transmission lineiv A resistive load at the end of a transmission linev A parallel connection of an inductor, a resistor, and a capacitor going through resonance and with a transmission

line offsetvi A series connection of a resistor, an inductor and a capacitor going through resonance and with a transmission line

offsetvii A series resistor and inductorviii A shunt connection of a resistor and a capacitor

Smith Chart Example

ECE 453 – Jose Schutt‐Aine 13

Load Description

A An inductor – or a reactive load at the end of a transmission line (i andiii)

B A resistive load at the end of a transmission line (iv)

C A capacitor (ii)

D A shunt connection of a resistor and a capacitor (viii)

E A series resistor and inductor (vii)

F A series connection of a resistor, an inductor and a capacitor goingthrough resonance and with a transmission line offset (vi)

Smith Chart Example

Solutions

ECE 453 – Jose Schutt‐Aine 14

Develop a two‐element matching network to match a source with an impedance of RS = 25  to a load RL = 200 

Smith Chart Example

ECE 453 – Jose Schutt‐Aine 15

Solution (from source to load)1. Normalize to 50 .  Then zG = rG = 0.5 and zL = rL = 4.02. Enter Smith chart at 0.5+j03. Identify circle r = 0.54. Normalized admittance at load is 0.255. Identify r = 0.25 circle6. Find center of r = 0.25 circle  (0,0.2)7. Rotate center by 180 degrees8. From new center draw circle of radius 0.25/(1+0.25) 

which intersects r = 0.5 circle at 0.5+j1.323 = zC.9. Rotate zC by 180 degrees. By construction, it will 

land on g = 0.25 circle at yC = 0.25‐j0.66110. Move along g = 0.25 circle until intersection with 

horizontal axis.

Smith Chart Example

ECE 453 – Jose Schutt‐Aine 16

Smith Chart Example

ECE 453 – Jose Schutt‐Aine 17

S C Gx x x 1.323 0 1.323

S S oX x Z 1.323 50 66.1

P L Cb b b 0 ( 0.661) 0.661

PP

o

b 0.661B 132 mSZ 50

P PX 1 / B 75.6

Smith Chart ExampleChange in normalized reactance from source to point C

Reactance value

Change in normalized susceptance from point C to load

Reactance value (in parallel)

Susceptance value

ECE 453 – Jose Schutt‐Aine 18

Smith Chart Example

SX 66.1

PX 75.6