Impedance and admittance smith chart pdf
Determine the complex point representing the given reflection coefficient Γ(d) on the chart. Find normalized resistance circle r = This vector represents the reflection coefficient -0.
IMPEDANCE AND ADMITTANCE SMITH CHART PDF SERIES
The chart provides directly the magnitude and the phase angle of Γ(d) Example: Find Γ(d), given ( ) 0 Z d = 5 + j100 Ω with Z = 50 Ω Amanogawa, Digital Maestro Series 173ġ0 1. The intersection of the two curves indicates the reflection coefficient in the complex plane. Find the arc of constant normalized reactance x 4. Normalize the impedance Given Z(d) Find Γ(d) ( d) Z R X zn ( d) = + j = r+ j x Z Z Z Find the circle of constant normalized resistance r 3. x = 0.5 Im(Γ ) x = 1 x ± x = 0 x = Re(Γ ) x = -1 Amanogawa, Digital Maestro Series 171Ĩ Basic Smith Chart techniques for loss-less transmission lines Given Z(d) Find Γ(d) Given Γ(d) Find Z(d) Given Γ R and Z R Find Γ(d) and Z(d) Given Γ(d) and Z(d) Find Γ R and Z R Find d max and d min (maximum and minimum locations for the voltage standing wave pattern) Find the Voltage Standing Wave Ratio (VSWR) Given Z(d) Find Y(d) Given Y(d) Find Z(d) Amanogawa, Digital Maestro Series 17ĩ 1. r = 0 Im(Γ ) r = 1 Re(Γ ) r = 5 r = 0.5 r Amanogawa, Digital Maestro Series 170ħ The result for the imaginary part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized reactance x are found on a circle with 1 1 1, x x Center = Radius = As the normalized reactance x varies from - to, we obtain a family of arcs contained inside the domain of the reflection coefficient Γ 1. In order to obtain universal curves, we introduce the concept of normalized impedance z n ( d) ( ) 1+ Γ( d) 1 ( d) = Z d Z = Γ 0 Amanogawa, Digital Maestro Series 166ģ The normalized impedance is represented on the Smith chart by using families of curves that identify the normalized resistance r (real part) and the normalized reactance x (imaginary part) ( ) Re( ) Im( ) z d = z + j z = r+ jx n n n Let s represent the reflection coefficient in terms of its coordinates Now we can write ( d) Re( ) jim( ) Γ = Γ + Γ ( ) j ( ) ( ) j ( ) 1 Re Im r + jx = + Γ + Γ 1 Re Γ Im Γ ( ) ( ) j ( ) ( ) 1 Re Γ Im Γ + Im Γ = ( 1 Re Γ ) + Im ( Γ) Amanogawa, Digital Maestro Series 167Ĥ The real part gives r = ( ) ( ) 1 Re Γ Im Γ ( 1 Re( Γ )) + Im ( Γ) Add a quantity equal to zero = 0 r ( ) 1 1 Re Γ 1 + Re Γ 1 + rim ( Γ ) + Im ( Γ ) + = 0 1+ r 1+ r ( ( ) ) ( ) ( ( ) ) ( ( ) ) 1 1 r Re Γ 1 + Re Γ ( 1 + r) Im ( Γ ) = 1+ r 1+ r r r 1 1+ Re Γ Re Γ + + ( 1+ r) Im ( Γ ) = 1+ r 1+ r ( r) ( ) ( ) ( 1 + r) r ( ) ( ) 1 Im ( ) Re Γ + Γ = 1+ r 1+ r Equation of a circle Amanogawa, Digital Maestro Series 168ĥ The imaginary part gives x x = Im ( Γ) ( 1 Re( Γ )) + Im ( Γ) ( ( )) ( ) x ( ) 1 Re Γ + Im Γ Im Γ = Γ + Γ Γ + = x x x ( 1 Re( )) Im ( ) Im ( ) ( ( )) 1 Re Im ( ) Im ( ) ( ( ) ) Re 1 Im( ) 1 1 Γ + Γ Γ + = x x x 1 1 Γ + Γ = x x Multiply by x and add a quantity equal to zero = 0 Equation of a circle Amanogawa, Digital Maestro Series 169Ħ The result for the real part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized resistance r are found on a circle with r 1,0 1+ r 1+ r Center = Radius = As the normalized resistance r varies from 0 to, we obtain a family of circles completely contained inside the domain of the reflection coefficient Γ 1. It is obvious that the result would be applicable only to lines with exactly characteristic impedance Z 0. To do so, we start from the general definition of line impedance (which is equally applicable to a load impedance when d=0) Zd ( ) ( ) ( ) V d = I d Z 0 1 +Γ 1 Γ ( d) ( d) This provides the complex function Zd ( ) f = Γ Γ that we want to graph.
Amanogawa, Digital Maestro Series 165Ģ The goal of the Smith chart is to identify all possible impedances on the domain of existence of the reflection coefficient.
In the case of a general lossy line, the reflection coefficient might have magnitude larger than one, due to the complex characteristic impedance, requiring an extended Smith chart.
This is also the domain of the Smith chart. Im(Γ ) 1 Re(Γ ) The domain of definition of the reflection coefficient for a loss-less line is a circle of unitary radius in the complex plane.
From a mathematical point of view, the Smith chart is a 4-D representation of all possible complex impedances with respect to coordinates defined by the complex reflection coefficient. The chart provides a clever way to visualize complex functions and it continues to endure popularity, decades after its original conception. 1 Smith Chart The Smith chart is one of the most useful graphical tools for high frequency circuit applications.