Introduction to Frequency Domain Analysis (3 Classes)

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Introduction to Frequency Domain Analysis (3 Classes) Many thanks to Steve Hall, Intel for the use of his slides Reference Reading: Posar Ch 4.5 http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf. Outline. Motivation: Why Use Frequency Domain Analysis 2-Port Network Analysis Theory

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Introduction to Frequency

Domain Analysis (3 Classes)

Many thanks to Steve Hall, Intel for the use of his slides

http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf

Outline
• Motivation: Why Use Frequency Domain Analysis
• 2-Port Network Analysis Theory
• Scattering Matrix
• Transmission (ABCD) Matrix
• Mason’s Rule
• Cascading S-Matrices and Voltage Transfer Function
• Differential (4-port) Scattering Matrix
Motivation: Why Frequency Domain Analysis?
• Time Domain signals on T-lines lines are hard to analyze
• Many properties, which can dominate performance, are frequency dependent, and difficult to directly observe in the time domain
• Skin effect, Dielectric losses, dispersion, resonance
• Frequency Domain Analysis allows discrete characterization of a linear network at each frequency
• Characterization at a single frequency is much easier
• Frequency Analysis is beneficial for Three reasons
• Ease and accuracy of measurement at high frequencies
• Simplified mathematics
• Allows separation of electrical phenomena (loss, resonance … etc)
Key Concepts

Here are the key concepts that you should retain from this class

• The input impedance & the input reflection coefficient of a transmission line is dependent on:
• Termination and characteristic impedance
• Delay
• Frequency
• S-Parameters are used to extract electrical parameters
• Transmission line parameters (R,L,C,G, TD and Zo) can be extracted from S parameters
• Vias, connectors, socket s-parameters can be used to create equivalent circuits=
• The behavior of S-parameters can be used to gain intuition of signal integrity problems
Review – Important Concepts
• The impedance looking into a terminated transmission line changes with frequency and line length
• The input reflection coefficient looking into a terminated transmission line also changes with frequency and line length
• If the input reflection of a transmission line is known, then the line length can be determined by observing the periodicity of the reflection
• The peak of the input reflection can be used to determine line and load impedance values
Two Port Network Theory
• Network theory is based on the property that a linear system can be completely characterized by parameters measured ONLY at the input & output ports without regard to the content of the system
• Networks can have any number of ports, however, consideration of a 2-port network is sufficient to explain the theory
• A 2-port network has 1 input and 1 output port.
• The ports can be characterized with many parameters, each parameter has a specific advantage
• Each Parameter set is related to 4 variables
• 2 independent variables for excitation
• 2 dependent variables for response
Network characterized with Port Impedance
• Measuring the port impedance is network is the most simplistic and intuitive method of characterizing a network

I

I

I

I

1

1

2

2

port

2

2

-

-

port

+

+

+

+

Port 2

V

V

V

V

Port 1

1

1

2

2

Network

Network

-

-

-

-

Case 1: Inject current I1 into port 1 and measure the open circuit voltage at port 2 and calculate the resultant impedance from port 1 to port 2

Case 2: Inject current I1 into port 1 and measure the voltage at port 1

and calculate the resultant input impedance

Impedance Matrix
• A set of linear equations can be written to describe the network in terms of its port impedances

Where:

If the impedance matrix is known, the response of the system can be predicted for any input

Or

Open Circuit Voltage measured at Port i

Current Injected at Port j

Zii the impedance looking into port i

Zij the impedance between port i and j

Impedance Matrix: Example #2

Calculate the impedance matrix for the following circuit:

R2

R1

R3

Port 2

Port 1

Impedance Matrix: Example #2

Step 1: Calculate the input impedance

R2

R1

+

-

R3

I1

V1

Step 2: Calculate the impedance across the network

R1

R2

+

-

R3

I1

V2

Impedance Matrix: Example #2

Step 3: Calculate the Impedance matrix

Assume: R1 = R2 = 30 ohms

R3=150 ohms

Measuring the impedance matrix

0.1nH

Port 1

T-line

0.1nH

Port 2

0.3pF

0.3pF

Zo=50 ohms, length=5 in

Question:

• What obstacles are expected when measuring the impedance matrix of the following transmission line structure assuming that the micro-probes have the following parasitics?
• Lprobe=0.1nH
• Cprobe=0.3pF

Assume F=5 GHz

Measuring the impedance matrix

0.1nH

Port 1

T-line

0.1nH

Port 2

0.3pF

0.3pF

Zo=50 ohms, length=5 in

• Open circuit voltages are very hard to measure at high frequencies because they generally do not exist for small dimensions
• Open circuit  capacitance = impedance at high frequencies
• Probe and via impedance not insignificant

Without Probe Capacitance

0.1nH

T-line

Zo = 50

Port 1

Port 2

Port 2

Z21 = 50 ohms

With Probe Capacitance @ 5 GHz

Zo = 50

Port 2

Port 1

106 ohms

106 ohms

Z21 = 63 ohms

• The impedance matrix is very intuitive
• Relates all ports to an impedance
• Easy to calculate

• Requires open circuit voltage measurements
• Difficult to measure
• Open circuit reflections cause measurement noise
• Open circuit capacitance not trivial at high frequencies

Note: The Admittance Matrix is very similar, however, it is characterized

with short circuit currents instead of open circuit voltages

Scattering Matrix (S-parameters)
• Measuring the “power” at each port across a well characterized impedance circumvents the problems measuring high frequency “opens” & “shorts”
• The scattering matrix, or (S-parameters), characterizes the network by observing transmitted & reflected power waves

a2

a1

2

2

-

-

port

port

Port 1

Port 2

R

R

Network

Network

b2

b1

ai represents the square root of the power wave injected into port i

bj represents the power wave coming out of port j

Scattering Matrix
• A set of linear equations can be written to describe the network in terms of injected and transmitted power waves

Where:

Sii = the ratio of the reflected power to the injected power at port i

Sij= the ratio of the power measured at port j to the power injected at port i

Making sense of S-Parameters – Return Loss
• When there is no reflection from the load, or the line length is zero, S11 = Reflection coefficient

R=50

Zo

R=Zo

Z=-l

Z=0

S11 is measure of the power returned to the source,

and is called the “Return Loss”

Making sense of S-Parameters – Return Loss
• When there is a reflection from the load, S11will be composed of multiple reflections due to the standing waves

Zo

RL

Z=0

Z=-l

• If the network is driven with a 50 ohm source, then S11 is calculated using the input impedance instead of Zo

50 ohms

S11 of a transmission line

will exhibit periodic effects

due to the standing waves

Example #3 – Interpreting the return loss
• Based on the S11 plot shown below, calculate both the impedance and dielectric constant

R=50

Zo

R=50

L=5 inches

0.45

0.4

0.35

0.3

S11, Magnitude

0.25

0.2

0.15

0.1

0.05

0

1.0

1.5

2.0

2.5

3..0

3.5

4.0

4.5

5.0

Frequency, GHz

Example – Interpreting the return loss

0.45

1.76GHz

2.94GHz

0.4

Peak=0.384

0.35

0.3

S11, Magnitude

0.25

0.2

0.15

0.1

0.05

0

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Frequency, GHz

• Step 2: Calculate Er using the velocity
• Step 1: Calculate the time delay of the t-line using the peaks
Example – Interpreting the return loss
• Step 3: Calculate the input impedance to the transmission line based on the peak S11 at 1.76GHz

Note: The phase of the reflection should be either +1 or -1 at 1.76 GHz because it is aligned with the incident

• Step 4: Calculate the characteristic impedance based on the input impedance for x=-5 inches

Er=1.0 and Zo=75 ohms

Making sense of S-Parameters – Insertion Loss
• When power is injected into Port 1 with source impedance Z0 and measured at Port 2 with measurement load impedance Z0, the power ratio reduces to a voltage ratio

a2=0

a1

2

2

-

-

port

port

V1

Zo

Zo

V2

Network

Network

b2

b1

S21 is measure of the power transmitted from

port 1 to port 2, and is called the “Insertion Loss”

Loss free networks
• For a loss free network, the total power exiting the N ports must equal the total incident power
• If there is no loss in the network, the total power leaving the network must be accounted for in the power reflected from the incident port and the power transmitted through network
• Since s-parameters are the square root of power ratios, the following is true for loss-free networks
• If the above relationship does not equal 1, then there is loss in the network, and the difference is proportional to the power dissipated by the network
Insertion loss example

Question:

• What percentage of the total power is dissipated by the transmission line?
• Estimate the magnitude of Zo (bound it)
Insertion loss example
• What percentage of the total power is dissipated by the transmission line ?
• What is the approximate Zo?
• How much amplitude degradation will this t-line contribute to a 8 GT/s signal?
• If the transmission line is placed in a 28 ohm system (such as Rambus), will the amplitude degradation estimated above remain constant?
• Estimate alpha for 8 GT/s signal
Insertion loss example

• Since there are minimal reflections on this line, alpha can be estimated directly from the insertion loss
• S21~0.75 at 4 GHz (8 GT/s)

When the reflections are minimal, alpha can be estimated

• If S11 < ~ 0.2 (-14 dB), then the above approximation is valid
• If the reflections are NOT small, alpha must be extracted with ABCD parameters (which are reviewed later)
• The loss parameter is “1/A” for ABCD parameters
• ABCD will be discussed later.
Important concepts demonstrated
• The impedance can be determined by the magnitude of S11
• The electrical delay can be determined by the phase, or periodicity of S11
• The magnitude of the signal degradation can be determined by observing S21
• The total power dissipated by the network can be determined by adding the square of the insertion and return losses
A note about the term “Loss”
• True losses come from physical energy losses
• Ohmic (I.e., skin effect)
• Field dampening effects (Loss Tangent)
• Insertion and Return losses include effects such as impedance discontinuities and resonance effects, which are not true losses
• Loss free networks can still exhibit significant insertion and return losses due to impedance discontinuities

• Ease of measurement
• Much easier to measure power at high frequencies than open/short current and voltage
• S-parameters can be used to extract the transmission line parameters
• n parameters and n Unknowns

• Most digital circuit operate using voltage thresholds. This suggest that analysis should ultimately be related to the time domain.
• Many silicon loads are non-linear which make the job of converting s-parameters back into time domain non-trivial.
• Conversion between time and frequency domain introduces errors

s112 s122

s212 s222

s113 s123

s213 s223

s111 s121

s211 s221

• While it is possible to cascade s-parameters, it gets messy.
• Graphically we just flip every other matrix.
• Mathematically there is a better way… ABCD parameters
• We will analyzed this later with signal flow graphs

a11

a21

b12

b22

a13

a13

b11

b21

a12

a22

b13

b13

ABCD Parameters
• The transmission matrix describes the network in terms of both voltage and current waves

I2

I1

2

2

-

-

port

port

V1

V2

Network

Network

• The coefficients can be defined using superposition
Transmission (ABCD) Matrix
• Since the ABCD matrix represents the ports in terms of currents and voltages, it is well suited for cascading elements

I3

I2

I1

V3

V1

V2

• The matrices can be cascaded by multiplication

This is the best way to cascade elements in the frequency domain.

It is accurate, intuitive and simplistic.

Relating the ABCD Matrix to Common Circuits

Z

Assignment 6:

Convert these to s-parameters

Port 1

Port 2

Y

Port 1

Port 2

Z1

Z2

Port 1

Port 2

Z3

Y3

Y1

Y2

Port 2

Port 1

Port 1

Port 2

Converting to and from the S-Matrix
• The S-parameters can be measured with a VNA, and converted back and forth into ABCD the Matrix
• Allows conversion into a more intuitive matrix
• Allows conversion to ABCD for cascading
• ABCD matrix can be directly related to several useful circuit topologies
ABCD Matrix – Example #1

Port 1

Port 2

• Create a model of a via from the measured s-parameters
ABCD Matrix – Example #1

Port 1

Port 2

• The model can be extracted as either a Pi or a T network

L2

L1

CVIA

• The inductance values will include the L of the trace and the via barrel (it is assumed that the test setup minimizes the trace length, and subsequently the trace capacitance is minimal
• The capacitance represents the via pads
ABCD Matrix – Example #1
• Assume the following s-matrix measured at 5 GHz
ABCD Matrix – Example #1
• Assume the following s-matrix measured at 5 GHz
• Convert to ABCD parameters
ABCD Matrix – Example #1
• Assume the following s-matrix measured at 5 GHz
• Convert to ABCD parameters
• Relating the ABCD parameters to the T circuit topology, the capacitance and inductance is extracted from C & A

Z1

Z2

Port 1

Port 2

Z3

ABCD Matrix – Example #2
• Calculate the resulting s-parameter matrix if the two circuits shown below are cascaded

Port 1

Port 2

2

-

port

50

50

Network X

Network

Port 1

Port 2

2

-

port

50

50

Network Y

Network

2

-

port

2

-

port

50

Network Y

50

Network X

Network

Network

Port 2

Port 1

ABCD Matrix – Example #2
• Step 1: Convert each measured S-Matrix to ABCD Parameters using the conversions presented earlier
• Step 2: Multiply the converted T-matrices
• Step 3: Convert the resulting Matrix back into S-parameters using thee conversions presented earlier

• The ABCD matrix is very intuitive
• Describes all ports with voltages and currents
• Allows easy cascading of networks
• Easy conversion to and from S-parameters
• Easy to relate to common circuit topologies

• Difficult to directly measure
• Must convert from measured scattering matrix

The wave functions (a,b) used to define s-parameters for a two-port network are shown below. The incident waves is a1, a2 on port 1 and port 2 respectively. The reflected waves b1 and b2 are on port 1 and port 2. We will use a’s and b’s in the s-parameter follow slides

Signal Flow Graphs of S Parameters

a1

s21

b2

s22

s11

GS

GL

s12

b1

a2

“In a signal flow graph, each port is represented by two nodes. Node an represents the wave coming into the device from another device at port n, and node bn represents the wave leaving the device at port n. The complex scattering coefficients are then represented as multipliers (gains) on branches connecting the nodes within the network and in adjacent networks.”*

Example

Measurement equipment strives to be match i.e. reflection coefficient is 0

See: http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf

Mason’s Rule ~ Non-Touching Loop Rule
• T is the transfer function (often called gain)
• Tkis the transfer function of the kth forward path
• L(mk) is the product of non touching loop gains on path k taken mk at time.
• L(mk)|(k) is the product of non touching loop gains on path k taken mk at a time but not touching path k.
• mk=1 means all individual loops
What is really of most relevance to time domain analysis is the voltage transfer function.

It includes the effect of non-perfect loads.

We will show how the voltage transfer functions for a 2 port network is given by the following equation.

Notice it is not S21

Voltage Transfer function
Forward Wave Path

Vs

a1

s21

b2

s22

s11

GS

GL

s12

b1

a2

Reflected Wave Path

Vs

a1

s21

b2

s22

s11

GS

GL

s12

b1

a2

Voltage transfer function using ABCD

Let’s see if we can get this results another way

Extract the voltage transfer function
• Same as with flow graph analysis

a11

a21

b12

b22

a13

a13

b11

b21

a12

a22

b13

b13

s112 s122

s212 s222

s113 s123

s213 s223

s111 s121

s211 s221

• As promised we will now look at how to cascade s-parameters and solve with Mason’s rule
• The problem we will use is what was presented earlier
• The assertion is that the loss of cascade channel can be determine just by adding up the losses in dB.
• We will show how we can gain insight about this assertion from the equation and graphic form of a solution.
Creating the signal flow graph

a11

a21

b12

b22

a13

a13

s213

s211

1

s212

1

A11

B21

A12

B22

A13

B23

s221

s112

s222

s113

s223

b11

b21

a12

a22

b13

b13

s123

s121

1

s122

1

B11

A21

B12

A22

B13

A23

s112 s122

s212 s222

s113 s123

s213 s223

s111 s121

s211 s221

• We map output a to input b and visa versa.
• Next we define all the loops
• Loop “A” and “B” do not touch each other

B

A

C

Use Mason’s rule

s213

s211

1

s212

1

A11

B21

A12

B22

A13

B23

s221

s112

s222

s113

s223

s123

s121

1

s122

1

B11

A21

B12

A22

B13

A23

• There is only one forward path a11 to b23.
• There are 2 non touching looks

B

A

C

Mason’s Rule

B

C

B

A

A

Evaluate the nature of the transfer function

Assumption is that these are ~ 0

• If response is relatively flat and reflection is relatively low
• Response through a channel is s211*s212*213…
Jitter and dB Budgeting
• Change s21 into a phasor
• Insertion loss in db

=

=

i.e. For a budget just add up the db’s and jitter

Differential S-Parameters

a

1

a

2

=

a

a

a

3

4

3

b

a

b

3

4

4

• Differential S-Parameters are derived from a 4-port measurement
• Traditional 4-port measurements are taken by driving each port, and recording the response at all other ports while terminated in 50 ohms
• Although, it is perfectly adequate to describe a differential pair with 4-port single ended s-parameters, it is more useful to convert to a multi-mode port

a

b

S

S

S

S

a

2

1

11

12

13

14

1

b

S

4

-

port

2

S

S

S

b

21

b

2

22

23

24

1

b

S

S

S

S

3

33

34

31

32

b

S

S

4

S

S

43

41

44

42

Differential S-Parameters

a

dm1

a

dm2

=

a

cm1

a

cm2

• It is useful to specify the differential S-parameters in terms of differential and common mode responses
• Differential stimulus, differential response
• Common mode stimulus, Common mode response
• Differential stimulus, common mode response (aka ACCM Noise)
• Common mode stimulus, differential response
• This can be done either by driving the network with differential and common mode stimulus, or by converting the traditional 4-port s-matrix

b

DS

DS

DCS

DCS

dm1

11

12

11

12

b

DS

dm2

DS

DCS

DCS

21

22

21

22

b

CS

CS

CDS

CDS

cm1

11

12

11

12

b

CS

CDS

cm2

CS

CDS

21

21

22

22

Matrix assumes differential and common mode stimulus

Explanation of the Multi-Mode Port

Common mode conversion Matrix:

Differential Stimulus, Common mode response. i.e., DCS21 = differential signal [(D+)-(D-)] inserted at port 1 and common mode signal [(D+)+(D-)] measured at port 2

Differential Matrix:

Differential Stimulus, differential response

i.e., DS21 = differential signal [(D+)-(D-)]

inserted at port 1 and diff signal measured at port 2

b

a

DS

DS

DCS

DCS

dm1

dm1

11

12

11

12

b

a

DS

dm2

dm2

DS

DCS

DCS

21

22

21

22

=

b

a

CS

CS

CDS

CDS

cm1

cm1

11

12

11

12

b

a

CS

CDS

cm2

cm2

CS

CDS

21

21

22

22

differential mode conversion Matrix:

Common mode Stimulus, differential mode response. i.e., DCS21 = common mode signal [(D+)+(D-)] inserted at port 1 and differential mode signal [(D+)-(D-)] measured at port 2

Common mode Matrix:

Common mode stimulus, common mode

Response. i.e., CS21 = Com. mode signal

[(D+)+(D-)] inserted at port 1 and Com. mode

signal measured at port 2

Differential S-Parameters

• Converting the S-parameters into the multi-mode requires just a little algebra

Example Calculation, Differential Return Loss

The stimulus is equal, but opposite, therefore:

2

1

2

4

-

-

port

port

Network

Network

4

3

Assume a symmetrical network and substitute

Other conversions that are useful for a differential bus are shown

Differential Insertion Loss:

Differential to Common Mode Conversion (ACCM):

Similar techniques can be used for all multi-mode Parameters

• Describes 4-port network in terms of 4 two port matrices
• Differential
• Common mode
• Differential to common mode
• Common mode to differential
• Easier to relate to system specifications
• ACCM noise, differential impedance

• Must convert from measured 4-port scattering matrix
High Frequency Electromagnetic Waves
• In order to understand the frequency domain analysis, it is necessary to explore how high frequency sinusoid signals behave on transmission lines
• The equations that govern signals propagating on a transmission line can be derived from Amperes and Faradays laws assumimng a uniform plane wave
• The fields are constrained so that there is no variation in the X and Y axis and the propagation is in the Z direction
• This assumption holds true for transmission lines as long as the wavelength of the signal is much greater than the trace width

X

Direction of

propagation

Z

For typical PCBs at 10 GHz with 5 mil traces (W=0.005”)

Y

High Frequency Electromagnetic Waves
• For sinusoidal time varying uniform plane waves, Amperes and Faradays laws reduce to:

Amperes Law:

A magnetic Field will be induced by an electric current

or a time varying electric field

An electric field will be generated by a time varying magnetic flux

• Note that the electric (Ex) field and the magnetic (By) are orthogonal
High Frequency Electromagnetic Waves
• If Amperes and Faradays laws are differentiated with respect to z and the equations are written in terms of the E field, the transmission line wave equation is derived

This differential equation is easily solvable for Ex:

High Frequency Electromagnetic Waves
• The equation describes the sinusoidal E field for a plane wave in free space

Note the positive exponent

is because the wave is

traveling in the opposite direction

Portion of wave traveling

In the +z direction

Portion of wave traveling

In the -z direction

(determines the speed of light in a material)

and non-magnetic materials)

Since inductance is proportional to & capacitance is proportional

to , then is analogous to in a transmission line, which

is the propagation delay

High Frequency Voltage and Current Waves
• The same equation applies to voltage and current waves on a transmission line

Incident sinusoid

RL

Reflected sinusoid

z=-l

z=0

If a sinusoid is injected onto a transmission line, the resulting voltage

is a function of time and distance from the load (z). It is the sum of the

incident and reflected values

specifically represent

the time varying

Sinusoid, which was implied

in the previous derivation

Voltage wave reflecting

towards the source

Voltage wave traveling

High Frequency Voltage and Current Waves
• The parameters in this equation completely describe the voltage on a typical transmission line

= Complex propagation constant – includes all the transmission line parameters (R, L C and G)

(For the loss free case)

(lossy case)

= Attenuation Constant (attenuation of the signal due to transmission line losses)

(For good conductors)

= Phase Constant (related to the propagation delay across the transmission line)

(For good conductors and good dielectrics)

High Frequency Voltage and Current Waves
• The voltage wave equation can be put into more intuitive terms by applying the following identity:

Subsequently:

• The amplitude is degraded by
• The waveform is dependent on the driving function ( ) & the delay of the line
Interaction: transmission line and a load
• The reflection coefficient is now a function of the Zo discontinuities AND line length
• Influenced by constructive & destructive combinations of the forward & reverse waveforms

Zo

Zl

(Assume a line length of l (z=-l))

Z=-l

Z=0

This is the reflection coefficient looking into a t-line of length l

Interaction: transmission line and a load

• If the reflection coefficient is a function of line length, then the input impedance must also be a function of length

Zin

RL

Z=-l

Z=0

Note: is

dependent on

and

This is the input impedance looking into a t-line of length l

• In chapter 2, you learned how to calculate waveforms in a multi-reflective system using lattice diagrams
• Period of transmission line “ringing” proportional to the line delay
• Remember, the line delay is proportional to the phase constant
• In frequency domain analysis, the same principles apply, however, it is more useful to calculate the frequency when the reflection coefficient is either maximum or minimum
• This will become more evident as the class progresses

To demonstrate, lets assume a loss free transmission line

Remember, the input reflection takes the form

The frequency where the values of the real & imaginary

reflections are zero can be calculated based on the line length

Term 1

Term 2

Term 1=0

Term 2 =

Term 2=0

Term 1 =

Note that when the imaginary portion is zero, it means the phase

of the incident & reflected waveforms at the input are aligned. Also notice that value of “8” and “4” in the terms.

Example #1: Periodic Reflections

Calculate:

• Line length
• RL

(assume a very low loss line)

Er_eff=1.0

RL

Zo=75

Z=-l

Z=0

.

Coeff

Reflection

Imaginary

Example #1: Solution

01

-

2.5E

01

-

2.0E

Real

01

-

1.5E

-

01

1.0E

-

5.0E

02

0.0E+00

-

-

02

5.0E

01

-

1.0E

-

01

-

1.5E

-

-

01

-

2.0E

-

2.5E

-

01

0.0E+00

5.0E+08

1.0E+09

1.5E+09

2.0E+09

2.5E+09

Frequency

3.0E+09

Step 1: Determine the periodicity zero crossings or peaks & use the relationships on page 15 to calculate the electrical length

Example #1: Solution (cont.)
• Note the relationship between the peaks and the electrical length
• This leads to a very useful equation for transmission lines
• Since TD and the effective Er is known, the line length can be calculated as in chapter 2
Example #1: Solution (cont.)
• The load impedance can be calculated by observing the peak values of the reflection
• When the imaginary term is zero, the real term will peak, and the maximum reflection will occur
• If the imaginary term is zero, the reflected wave is aligned with the incident wave and the phase term = 1

Important Concepts demonstrated

• The impedance can be determined by the magnitude of the reflection
• The line length can be determined by the phase, or periodicity of the reflection