An Introduction S-PARAMETERS S-parameters are a useful method for representing a circuit as a “black box” S-parameters are a useful method for representing a circuit as a “black box”
The external behaviour of this black box can be predicted without any regard for the contents of the black box. S-parameters are a useful method for representing a circuit as a “black box”
The external behaviour of this black box can be predicted without any regard for the contents of the black box.
This black box could contain anything:
a resistor, a transmission line or an integrated circuit. A “black box” or network may have any number of ports.
This diagram shows a simple network with just 2 ports. A “black box” or network may have any number of ports.
This diagram shows a simple network with just 2 ports.
Note :
A port is a terminal pair of lines. S-parameters are measured by sending a single frequency signal into the network or “black box” and detecting what waves exit from each port.
Power, voltage and current can be considered to be in the form of waves travelling in both directions. S-parameters are measured by sending a single frequency signal into the network or “black box” and detecting what waves exit from each port.
Power, voltage and current can be considered to be in the form of waves travelling in both directions.
For a wave incident on Port 1, some part of this signal reflects back out of that port and some portion of the signal exits other ports. I have seen S-parameters described as S 11, S21, etc. Can you explain?
First lets look at S 11 .
S11 refers to the signal reflected at Port 1 for the signal incident at Port 1. I have seen S-parameters described as S 11, S21, etc. Can you explain?
First lets look at S 11 .
S11 refers to the signal reflected at Port 1 for the signal incident at Port 1.
Scattering parameter S 11 is the ratio of the two waves b1/a1. I have seen S-parameters described as S 11, S21, etc. Can you explain?
Now lets look at S 21 .
S21 refers to the signal exiting at Port 2 for the signal incident at Port 1.
Scattering parameter S 21 is the ratio of the two waves b2/a1. I have seen S-parameters described as S 11, S21, etc. Can you explain?
Now lets look at S 21 .
S21 refers to the signal exiting at Port 2 for the signal incident at Port 1.
Scattering parameter S 21 is the ratio of the two waves b2/a1. I have seen S-parameters described as S 11, S21, etc. Can you explain?
Now lets look at S 21 .
S21 refers to the signal exiting at Port 2 for the signal incident at Port 1.
Scattering parameter S 21 is the ratio of the two waves b2/a1.
I have seen S-parameters described as S 11, S21, etc. Can you explain?
A linear network can be characterised by a set of simultaneous equations describing the exiting waves from each port in terms of incident waves.
S11 = b1 / a1
S12 = b1 / a2
S21 = b2 / a1
S22 = b2 / a2
Note again how the subscript follows the parameters in the ratio (S 11 =b1/a1, etc...)
S-parameters are complex (i.e. they have magnitude and angle) because both the magnitude and phase of the input signal are changed by the network.
(This is why they are sometimes referred to as complex scattering parameters). These four S-parameters actually contain eight separate numbers: the real and imaginary parts (or the modulus and the phase angle) of each of the four complex scattering parameters. Quite often we refer to the magnitude only as it is of the most interest.
How much gain (or loss) you get is usually more important than how much the signal has been phase shifted. What do S-parameters depend on?
S-parameters depend upon the network and the characteristic impedances of the source and load used to measure it, and the frequency measured at.
i.e.
if the network is changed, the S-parameters change.
if the frequency is changed, the S-parameters change.
if the load impedance is changed, the S-parameters change.
if the source impedance is changed, the S-parameters change. What do S-parameters depend on?
S-parameters depend upon the network and the characteristic impedances of the source and load used to measure it, and the frequency measured at.
i.e.
if the network is changed, the S-parameters change.
if the frequency is changed, the S-parameters change.
if the load impedance is changed, the S-parametersIn the Si9000e change. S-parameters are quoted with source and load if the source impedance is changed, the S-parametersimpedances chang ofe. 50 Ohms A little math…
This is the matrix algebraic representation of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry about the leading diagonal. A little math…
This is the matrix algebraic representation of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry about the leading diagonal.
In the case of a 2-port network, that means that S 21 = S 12 and interchanging the input and output ports does not change the transmission properties. A little math…
This is the matrix algebraic representation of 2 port S-parameters:
Some matrices are symmetrical. A symmetrical matrix has symmetry about the leading diagonal.
In the case of a symmetrical 2-port network, that means that S 21 = S 12 and interchanging the input and output ports does not change the transmission properties.
A transmission line is an example of a symmetrical 2-port network. A little math…
Parameters along the leading diagonal,
S11 & S 22 , of the S-matrix are referred to as reflection coefficients because they refer to the reflection occurring at one port only. A little math…
Parameters along the leading diagonal,
S11 & S 22 , of the S-matrix are referred to as reflection coefficients because they refer to the reflection occurring at one port only.
Off-diagonal S-parameters, S 12 , S 21 , are referred to as transmission coefficients because they refer to what happens from one port to another. Larger networks:
A Network may have any number of ports. Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n 2 coefficients (S-parameters), each one representing a possible input-output path. Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n 2 coefficients (S-parameters), each one representing a possible input-output path.
The number of rows and columns in an S-parameters matrix is equal to the number of ports. Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n 2 coefficients (S-parameters), each one representing a possible input-output path.
The number of rows and columns in an S-parameters matrix is equal to the number of ports.
For the S-parameter subscripts “ij”, “j” is the port that is excited (the input port) and “i” is the output port. Larger networks:
A Network may have any number of ports.
The S-matrix for an n-port network contains n 2 coefficients (S-parameters), each one representing a possible input-output path.
The number of rows and columns in an S-parameters matrix is equal to the number of ports.
For the S-parameter subscripts “ij”, “j” is the port that is excited (the input port) and “i” is the output port.
Sum up…
• S-parameters are a powerful way to describe an electrical network • S-parameters change with frequency / load impedance / source impedance / network
•S11 is the reflection coefficient st •S21 describes the forward transmission coefficient (responding port 1 !) • S-parameters have both magnitude and phase information • Sometimes the gain (or loss) is more important than the phase shift and the phase information may be ignored • S-parameters may describe large and complex networks
• If you would like to learn more please see next slide: Further reading:
Agilent papers http://www.sss-mag.com/pdf/an-95-1.pdf http://www.sss-mag.com/pdf/AN154.pdf
National Instruments paper http://zone.ni.com/devzone/nidzgloss.nsf/webmain/D2C4FA88321195FE8625686B00542 EDB?OpenDocument
Other links: http://www.sss-mag.com http://www.microwaves101.com/index.cfm http://www.reed-electronics.com/tmworld/article/CA187307.html http://en.wikipedia.org/wiki/S-parameters
Online lecture OLL-140 Intro to S-parameters - Eric Bogatin Online lecture OLL-141 S 11 & Smith charts - Eric Bogatin www.bethesignal.com Description TRANSMISSION LINES Terminology and Conventions
Sinusoidal Source
V(t)= Vo cos (ωt + φ)
j(ωt +φ) jφ jωt V()t = Re {Voe }= Re {Voe e }
j = −1
jφ Voe is a complex phasor
36 Phasors
Im Vo jφ Voe = Vo cos ()()φ + jV o sin φ Vo sin (φ) ω φ
Vo cos (φ) Re • In these notes, all sources are sine waves • Circuits are described by complex phasors • The time varying answer is found by multiplying phasors by e jω t and taking the real part
37 TEM Transmission Line Theory
Charge on the inner conductor:
∆q = Cl∆xV
where C l is the capacitance per unit length Azimuthal magnetic flux: ∆Φ = Ll∆xI
where L l is the inductance per unit length 38 Electrical Model of a Transmission Line
L l ∆ x i i + ∆ i v v + ∆ v C l ∆ x
Voltage drop along the inductor: di v − ()v + ∆v = L ∆x l dt Current flowing through the capacitor: dv i + ∆i = i − C ∆x l dt
39 Transmission Line Waves
Limit as ∆x->0 ∂v ∂i ∂B = −Ll ∇ × E = − ∂x ∂t ∂t ∂i ∂v ∂D = −Cl ∇ × H = ∂x ∂t ∂t Solutions are traveling waves x x v()x,t = v+ t − + v− t + vel vel v+ x v− x ()x,ti = t − − t + Zo vel Zo vel v+ indicates a wave traveling in the +x direction v- indicates a wave traveling in the -x direction 40 Phase Velocity and Characteristic Impedance
vel is the phase velocity of the wave 1 vel = LlCl For a transverse electromagnetic wave (TEM), the phase velocity is only a property of the material the wave travels through 1 1 = LlCl µε
The characteristic impedance Z o
Ll Zo = Cl has units of Ohms and is a function of the material AND the geometry 41 Pulses on a Transmission Line
R v+ L
Pulse travels down the transmission line as a forward going wave only (v +). However, when the pulse reaches the load resistor: v v+ + v− = R L = i v+ v− − Zo Zo so a reverse wave v - and i - must be created to satisfy the boundary 42 condition imposed by the load resistor Reflection Coefficient
The reverse wave can be thought of as the incident wave reflected from the load v− R − Z = L o = Γ Reflection coefficient + v R L + Zo Three special cases:
RL = ∞ (open) Γ = +1
RL = 0 (short) Γ = -1
RL = Z o Γ = 0
A transmission line terminated with a resistor equal in value to the characteristic impedance of the transmission line looks the same to 43 the source as an infinitely long transmission line Sinusoidal Waves
Experiment: Send a SINGLE frequency ( ω) sine wave into a transmission line and measure how the line responds
v+ = V + cos ()ωt − βx = Re {V +e− jβxe jωt }
wave number ω 2πf 2π = vel phase velocity β = = β vel λ
By using a single frequency sine wave we can now define complex impedances such as: di v = L V = jωLI Zind = jωL dt dv 1 i = C Z = dt I = jωCV cap jωC 44 Standing Waves
x Z o ZL d x = 0
At x=0 Z − Z V − = ΓV + = L o Z + Z Along the transmission line: L o
V = V +e− jβx + ΓV +e+ jβx V = V + ()()1− Γ e− jβx + 2V +Γ cos βx traveling wave standing wave 45 Voltage Standing Wave Ratio (VSWR)
Large voltage 1.5 Large current
1
+ V V (1+ Γ ) (1+ Γ ) 0.5 max = = + Vmin V ()1− Γ ()1 − Γ 0 = VSWR
0.5
1 1 Γ = 2 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Position
The VSWR is always greater than 1 46 Voltage Standing Wave Ratio (VSWR)
V V + (1+ Γ ) (1+ Γ ) max = = + Vmin V ()1− Γ ()1 − Γ = VSWR
1 Γ = 2
Incident wave Reflected wave Standing wave The VSWR is always greater than 1 47 Reflection Coefficient Along a Transmission Line
x Z o ZL d x = 0 towards load Γ towards generator ZL − Zo G ΓL = ZL + Zo
+ − jβx + + jβx V = V e + ΓLV e Wave has to travel down and back V V +e+ jβ(− )d Γ = forward = Γ = Γ e− 2j βd G V L + − jβ(− )d L reverse gen V e 48 Impedance and Reflection
Im {Γ} ΓL There is a one-to-one correspondence between ΓG and ZL
θ = −2βd Re {Γ} ZG − Zo ΓG = ZG + Zo
1 + ΓG ZG = Zo 1− ΓG ΓG 1+ Γ e− 2j βd Z = Z L G o − 2j βd 1− ΓLe 49 Impedance and Reflection: Open Circuits
For an open circuit Z L= ∞ so ΓL = +1 Impedance at the generator: − jZ Z = o G tan ()βd For βd<<1 − jZ o 1 ZG ≈ = looks capacitive βd jωCld For βd = π/2 or d= λ/4
ZG = 0 An open circuit at the load looks like a short circuit at the generator if the generator is a quarter wavelength away from the load 50 Impedance and Reflection: Short Circuits
For a short circuit Z L= 0 so ΓL = -1 Impedance at the generator:
ZG = jZ o tan (βd)
For βd<<1
ZG ≈ jZ oβd = jωLld looks inductive
For βd = π/2 or d= λ/4
ZG → ∞ A short circuit at the load looks like an open circuit at the generator if the generator is a quarter wavelength away from the load 51 Incident and Reflected Power
x Z Z Ps o L d x = − d x = 0 Voltage and Current at the generator (x=-d) + + jβd + − jβd VG = V(− )d = V e + ΓLV e + + V + jβd V − jβd IG = (I − )d = e − ΓL e Zo Zo The rate of energy flowing through the plane at x=-d 1 P = Re {V I *} 2 G G +2 +2 1 V 1 2 V P = − ΓL reflected power forward power 2 Zo 2 Zo 52 Incident and Reflected Power
• Power does not flow! Energy flows. – The forward and reflected traveling waves are power orthogonal • Cross terms cancel – The net rate of energy transfer is equal to the difference in power of the individual waves • To maximize the power transferred to the load we want:
ΓL = 0
which implies: ZL = Zo
When Z L = Z o, the load is matched to the transmission line
53 Load Matching
What if the load cannot be made equal to Zo for some other reasons? Then, we need to build a matching network so that the source effectively sees a match load.
Ps Z0 M ZL
Γ = 0
Typically we only want to use lossless devices such as capacitors, inductors, transmission lines, in our matching network so that we do not dissipate any power in the network and deliver all the available power to the load.
54 Normalized Impedance
It will be easier if we normalize the load impedance to the characteristic impedance of the transmission line attached to the load. Z z = = r + jx Zo 1+ Γ z = 1− Γ Since the impedance is a complex number, the reflection coefficient will be a complex number Γ = u + jv
1− u 2 − v2 2v r = x = 2 2 ()1− u 2 + v2 ()1− u + v 55 dB and dBm
A dB is defined as a POWER ratio. For example: Prev ΓdB = 10 log Pfor = 10 log Γ 2 = 20 log ()Γ
A dBm is defined as log unit of power referenced to 1mW:
P P = 10 log dBm 1mW
56 Description Z AND S PARAMETERS Two Port Z Parameters
We have only discussed reflection so far. What about transmission? Consider a device that has two ports:
I I1 2
V1 V2
The device can be characterized by a 2x2 matrix:
V1 = Z11 I1 + Z12 I2 V2 = Z21 I1 + Z22 I2 [V ]= Z [I ][ ] 58 Scattering (S) Parameters
Since the voltage and current at each port (i) can be broken down into forward and reverse waves: + − Vi = Vi + Vi + − ZoIi = Vi − Vi We can characterize the circuit with forward and reverse waves: − + + V1 = S11 V1 + S12 V2 − + + V2 = S21 V1 + S22 V2 [V − ]= []S [V + ] 59 Z and S Parameters
Similar to the reflection coefficient, there is a one-to-one correspondence between the impedance matrix and the scattering matrix:
−1 [S]= ([Z]+ Zo [1]) ([Z]− Zo [1])
−1 [][][]Z = Zo ()1 + S ()[][]1 − S
60 Normalized Scattering (S) Parameters
The S matrix defined previously is called the un-normalized scattering matrix. For convenience, define normalized waves: + Vi ai = 2Z oi − Vi bi = 2Z oi
Where Z oi is the characteristic impedance of the transmission line connecting port (i)
2 |a i| is the forward power into port (i) 2 |b i| is the reverse power from port (i) 61 Normalized Scattering (S) Parameters
The normalized scattering matrix is:
b1 = s11 a1 + s12 a 2 b2 = s21 a1 + s22 a 2 [b ]= s [a ][ ]
Where:
Z o j s = S j,i Z j,i oi If the characteristic impedance on both ports is the same then the normalized and un-normalized S parameters are the same. 62 Normalized S parameters are the most commonly used. Normalized S Parameters
The s parameters can be drawn pictorially
a1 b2 s21
s11 s22 s12 b1 a2
s11 and s 22 can be thought of as reflection coefficients
s21 and s 12 can be thought of as transmission coefficients s parameters are complex numbers where the angle corresponds to a
phase shift between the forward and reverse waves 63 Examples of S parameters
τ 0 e− jωτ []s = 1 2 − jωτ Zo e 0
Transmission Line
−1 0 1 2 []s = 0 −1 Short
1 2 G 0 0 []s = G 0 64 Amplifier Examples of S parameters
1 2 0 0 1 []s = 1 0 0 0 1 0
1 0 0 []s = Z o 1 0 2 Isolator 65 Lorentz Reciprocity If the device is made out of linear isotropic materials (resistors, capacitors, inductors, metal, etc..) then: [s]T = [s]
or
s i,j = s j,i for i ≠ j
This is equivalent to saying that the transmitting pattern of an antenna is the same as the receiving pattern
reciprocal devices: transmission line short directional coupler non-reciprocal devices: amplifier isolator circulator 66 Lossless Devices
The s matrix of a lossless device is unitary: T [s* ][][]s = 1 2 1 = ∑ s j,i for all j i 2 1 = ∑ s j,i for all i j Lossless devices: transmission line short circulator Non-lossless devices: amplifier isolator 67 Network Analyzers Network analyzers measure S parameters as a function of frequency At a single frequency, network analyzers send out forward waves a1 and a2 and measure the phase and amplitude of the reflected waves b1 and b2 with respect to the forward waves. b b s = 1 s = 2 11 a 21 a 1 a 2 =0 1 a 2 =0 a 1 a2 b b s = 1 s = 2 12 a 22 a b1 b 2 a1 =0 2 a1 =0 2
68 Network Analyzer Calibration To measure the pure S parameters of a device, we need to eliminate the effects of cables, connectors, etc. attaching the device to the network analyzer Connector Y Connector X yx 21
y21 s21 x21
y11 y22 s11 s22 x11 x22
y12 s12 x12
yx 12
We want to know the S parameters at these reference planes
We measure the S parameters at these reference planes 69 Network Analyzer Calibration
• There are 10 unknowns in the connectors • We need 10 independent measurements to eliminate these unknowns – Develop calibration standards – Place the standards in place of the Device Under Test (DUT) and measure the S- parameters of the standards and the connectors – Because the S parameters of the calibration standards are known (theoretically), the S parameters of the connectors can be determined and can be mathematically eliminated once the DUT is placed back in the measuring fixtures.
70 Network Analyzer Calibration • Since we measure four S parameters for each calibration standard, we need at least three independent standards. • One possible set is: 0 1 Thru []s = 1 0 −1 0 Short []s = 0 −1
0 e− jωτ Delay* []s = − jωτ *ωτ ~90degrees e 0 τ 71 Phase Delay A pure sine wave can be written as: j(ωt −βz) V = Voe The phase shift due to a length of cable is: θ = βd ω = d vph
= ωτ ph The phase delay of a device is defined as: arg (S ) τ = − 21 ph ω
72 Phase Delay
• For a non-dispersive cable, the phase delay is the same for all frequencies. • In general, the phase delay will be a function of frequency. • It is possible for the phase velocity to take on any value - even greater than the velocity of light – Waveguides – Waves hitting the shore at an angle
73 Group Delay • A pure sine wave has no information content – There is nothing changing in a pure sine wave – Information is equivalent to something changing • To send information there must be some modulation of the sine wave at the source The modulation can be de-composed into different frequency components m V = V cos ()ωt + V []cos ()()()ω + ∆ω t + cos ()ω − ∆ω t o o 2
V = Vo (1+ m cos (∆ωt))cos (ωt)
74 Group Delay The waves emanating from the source will look like
V = Vo cos (ωt − βz) m + V cos ()()()ω + ∆ω t − β + ∆β z o 2 m + V cos ()()()ω − ∆ω t − β − ∆β z o 2
Which can be re-written as:
V = Vo (1+ m cos (∆ωt − ∆βz))cos (ωt − βz)
75 Group Delay The information travels at a velocity v = 1 ⇒ 1 gr ∆β ∂β ∆ω ∂ω
The group delay is defined as: d τgr = vgr ∂β = d ∂ω ∂()arg ()S = − 21 ∂ω
76 Phase Delay and Group Delay
Phase Delay: arg (S ) τ = − 21 ph ω
Group Delay: ∂(arg (S )) τ = − 21 gr ∂ω
77 Description SMITH CHART Smith Chart
• Impedances, voltages, currents, etc. all repeat every half wavelength • The magnitude of the reflection coefficient, the standing wave ratio (SWR) do not change, so they characterize the voltage & current patterns on the line • If the load impedance is normalized by the characteristic impedance of the line, the voltages, currents, impedances, etc. all still have the same properties, but the results can be generalized to
any line with the same normalized impedances 79 Smith Chart
• The Smith Chart is a clever tool for analyzing transmission lines • The outside of the chart shows location on the line in wavelengths • The combination of intersecting circles inside the chart allow us to locate the normalized impedance and then to find the impedance anywhere on the line
80 Smith Chart
• Thus, the first step in analyzing a transmission line is to locate the normalized load impedance on the chart • Next, a circle is drawn that represents the reflection coefficient or SWR. The center of the circle is the center of the chart. The circle passes through the normalized load impedance • Any point on the line is found on this circle. Rotate clockwise to move toward the generator (away from the load) • The distance moved on the line is indicated on the
outside of the chart in wavelengths 81 Smith Charts The impedance as a function of reflection coefficient can be re- written in the form: 2 2 r 2 1 1− u − v u − + v2 = r = 2 ()1− u 2 + v2 1+ r ()1+ r
2v 2 x = 2 1 1 2 2 ()u −1 + v − = ()1− u + v x x 2
These are equations for circles on the (u,v) plane
82 Smith Chart – Real Circles 1 Im {Γ}
0.5
r=0 r=1/3 r=1 r=2.5 1 0.5 0 0.5 1 Re {Γ}
0.5
1 83 Smith Chart – Imaginary Circles Im 1 {Γ}
x=1/3 x=1 x=2.5 0.5
1 0.5 0 0.5 1 Re {Γ}
x=-1/3 x=-1 x=-2.5 0.5
1 84 Smith Chart
85 Smith Chart Example 1
Given:
ΓL = 5.0 ∠45 °
Zo = 50 Ω
What is ZL?
ZL = 50 Ω( 35.1 + 35.1j ) = 67 5. Ω + 67j 5. Ω
86 Smith Chart Example 2
Given:
ZL = 15 Ω − 25j Ω
Zo = 50 Ω
What is ΓL? 15 Ω − 25j Ω z = L 50 Ω = 3.0 − 5.0j
ΓL = .0 618 ∠ −124 °
87 Smith Chart Example 3
Given: ZL = 50 Ω + 50j Ω
Zo = 50 Ω τ = 78.6 nS
Zin = ?
What is Zin at 50 MHz? 50 Ω + 50j Ω z = L 50 Ω = 0.1 + 0.1j
ΓL = .0 445 ∠64 ° 2ωτ = 244 ° − 2j βd − 2j ωτ Γin = ΓLe = ΓLe 2ωτ = 244 ° Γin = .0 445 ∠180 °
ZL = 50 Ω( 38.0 + 0.0j )= 19 Ω 88 Admittance
A matching network is going to be a combination of elements connected in series AND parallel. Z1 Impedance is well suited when working with Z2 series configurations. For example:
V = ZI ZL = Z1 + Z2
Impedance is NOT well suited when working with parallel configurations. Z1 Z2
Z1Z2 ZL = Z1 + Z2
For parallel loads it is better to work with admittance. Y1 Y2 1 I = YV Y1 = YL = Y1 + Y2 Z1 89 Normalized Admittance Y y = = YZ o = g + jb Yo 1− Γ y = 1+ Γ 2 1− u 2 − v2 g 1 u + + v2 = g = 2 ()1+ u 2 + v2 1+ g ()1+ g
− 2v 2 b = 2 1 1 2 2 ()u +1 + v + = ()1+ u + v b b2
These are equations for circles on the (u,v) plane
90 Admittance Smith Chart
1 1 Im {Γ} Im {Γ} b=-1 b=-1/3 b=-2.5 0.5 0.5
g=2.5 g=1 g=1/3 g=0 1 0.5 0 0.5Re {Γ} 1 1 0.5 0 0.5Re {Γ 1}
b=2.5 b=1/3
0.5 0.5 b=1
1 1
91 Impedance and Admittance Smith Charts • For a matching network that contains elements connected in series and parallel, we will need two types of Smith charts – impedance Smith chart – admittance Smith Chart • The admittance Smith chart is the impedance Smith chart rotated 180 degrees. – We could use one Smith chart and flip the reflection coefficient vector 180 degrees when switching between a series configuration to a
parallel configuration. 92 Admittance Smith Chart Example 1
Given: y = 1+ 1j
What is Γ?
• Procedure: Plot y • Plot 1+j1 on chart Flip 180 • vector = .0 445 ∠64 ° degrees Read Γ • Flip vector 180 degrees Γ = .0 445 ∠ −116 °
93 Admittance Smith Chart Example 2
Given: Γ = 5.0 ∠ + 45 ° Zo = 50 Ω What is Y?
• Procedure: Plot Γ • Plot Γ • Flip vector by 180 degrees • Read coordinate Read y Flip 180 degrees y = 38.0 − 36.0j 1 Y = ()38.0 − 36.0j 50 Ω Y = ()6.7 − 2.7j 10x −3 mhos
94 Constant Imaginary Smith Chart Impedance Lines
Impedance Z=R+jX =100+j50 Normalized z=2+j for
Zo=50 Constant Real Impedance Circles
95 Smith Chart •Impedance divided by line impedance (50 Ohms) – Z1 = 100 + j50 – Z2 = 75 -j100 – Z3 = j200 – Z4 = 150 – Z5 = infinity (an open circuit) – Z6 = 0 (a short circuit) – Z7 = 50 – Z8 = 184 -j900
•Then, normalize and plot. The points are plotted as follows: – z1 = 2 + j – z2 = 1.5 -j2 – z3 = j4 – z4 = 3 – z5 = infinity – z6 = 0 – z7 = 1 – z8 = 3.68 -j18S 96 Toward Constant Generator Reflection Coefficient Circle
Away From Scale in Generator Wavelengths
Full Circle is One Half Wavelength Since Everything Repeats
97 Smith Chart Example
• First, locate the normalized impedance on the chart for Z L = 50 + j100 • Then draw the circle through the point • The circle gives us the reflection coefficient (the radius of the circle) which can be read from the scale at the bottom of most charts • Also note that exactly opposite to the normalized load is its admittance. Thus, the chart can also be used to find the admittance. We use this fact in stub matching
98 Matching Example
Ps Z0 = 50 Ω M 100 Ω
Γ = 0
Match 100 Ω load to a 50 Ω system at 100MHz
A 100 Ω resistor in parallel would do the trick but ½ of the power would be dissipated in the matching network. We want to use only lossless elements such as inductors and capacitors so we don’t dissipate any power in the matching network
99 Matching Example
We need to go from z=2+j0 to z=1+j0 on the Smith chart We won’t get any closer by adding series impedance so we will need to add something in parallel. We need to flip over to the admittance chart
Impedance Chart
100 Matching Example
y=0.5+j0 Before we add the admittance, add a mirror of the r=1 circle as a guide.
Admittance Chart
101 Matching Example
y=0.5+j0 Before we add the admittance, add a mirror of the r=1 circle as a guide Now add positive imaginary admittance.
Admittance Chart
102 Matching Example
y=0.5+j0 Before we add the admittance, add a mirror of the r=1 circle as a guide Now add positive imaginary admittance jb = j0.5 jb = j0.5 5.0j = 2j π()100 MHz C 50 Ω C = 16 pF
Admittance 16 pF 100 Ω Chart
103 Matching Example
We will now add series impedance Flip to the impedance Smith Chart We land at on the r=1 circle at x=-1
Impedance Chart
104 Matching Example
Add positive imaginary admittance to get to z=1+j0
jx = 0.1j ()()0.1j 50 Ω = 2j π 100 MHz L L = 80 nH 80 nH
16 pF
100 Ω
Impedance Chart
105 Matching Example
This solution would have also worked 32 pF
160 nH 100 Ω
Impedance Chart
106 Matching Bandwidth
0 80 nH
5 16 pF 10 100 Ω
15 100 MHz
20
25 Reflection Coefficient (dB) 30
35 50 MHz
40 50 60 70 80 90 100 110 120 130 140 150 Frequency (MHz)
Because the inductor and capacitor impedances change with frequency, the match works over a narrow frequency range Impedance Chart
107