# Presentation: 321609

## Introduction to RF for Particle AcceleratorsPart 1: Transmission Lines

- Dave McGinnis

## Transmission Line Topics

- Phasors
- Traveling Waves
- Characteristic Impedance
- Reflection Coefficient
- Standing Waves
- Impedance and Reflection
- Incident and Reflected Power

- Smith Charts
- Load Matching
- Single Stub Tuners
- dB and dBm
- Z and S parameters
- Lorentz Reciprocity
- Network Analysis
- Phase and Group Delay

## Terminology and Conventions

- Sinusoidal Source

- is a complex phasor

## Phasors

- In these notes, all sources are sine waves
- Circuits are described by complex phasors
- The time varying answer is found by multiplying phasors by          and taking the real part

## TEM Transmission Line Theory

- Charge on the inner conductor:

- where Cl is the capacitance per unit length

- Azimuthal magnetic flux:

- where Ll is the inductance per unit length

## Electrical Model of a Transmission Line

- Voltage drop along the inductor:

- Current flowing through the capacitor:

## Transmission Line Waves

- Limit as x->0

- Solutions are traveling waves

- v+ indicates a wave traveling in the +x direction

- v- indicates a wave traveling in the -x direction

## Phase Velocity and Characteristic Impedance

- vel is the phase velocity of the wave

- For a transverse electromagnetic wave (TEM), the phase velocity is only a property of the material the wave travels through

- The characteristic impedance Zo

- has units of Ohms and is a function of the material AND the geometry

## Pulses on a Transmission Line

- Pulse travels down the transmission line as a forward going wave only (v+). However, when the pulse reaches the load resistor:

- so a reverse wave v- and i- must be created to satisfy the boundary condition imposed by the load resistor

## Reflection Coefficient

- The reverse wave can be thought of as the incident wave reflected from the load

- Reflection coefficient

- Three special cases:
- 	RL = ∞	(open)		= +1
- 	RL = 0	(short)		 = -1
- 	RL = Zo			 = 0

- A transmission line terminated with a resistor equal in value to the characteristic impedance of the transmission line looks the same to the source as an infinitely long transmission line

## Sinusoidal Waves

- phase velocity

- wave number

- By using a _single_ frequency sine wave we can now define complex impedances such as:

- Experiment: Send a **SINGLE** frequency () sine wave into a transmission line and measure how the line responds

## Standing Waves

- At x=0

- Along the transmission line:

- traveling wave

- standing wave

## Voltage Standing Wave Ratio (VSWR)

- The VSWR is always greater than 1

## Reflection Coefficient Along a Transmission Line

- towards load
- towards generator

## Impedance and Reflection

- There is a one-to-one correspondence between G and ZL

## Impedance and Reflection: Open Circuits

- For an open circuit ZL= ∞ so L = +1

- Impedance at the generator:

- For d<<1

- looks capacitive

- For d = /2 or d=/4

- 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

## Impedance and Reflection: Short Circuits

- For a short circuit ZL= 0 so L = -1

- Impedance at the generator:

- For d<<1

- looks inductive

- For d = /2 or d=/4

- 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

## Incident and Reflected Power

- Voltage and Current at the generator (x=-d)

- The rate of energy flowing through the plane at  x=-d

- forward power

- reflected power

## 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:

- which implies:

- When ZL = Zo, the load is matched to the transmission line

## 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.

- 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.

## Normalized Impedance

- It will be easier if we normalize the load impedance to the characteristic impedance of the transmission line attached to the load.

- Since the impedance is a complex number, the reflection coefficient will be a complex number

## Smith Charts

- The impedance as a function of reflection coefficient can be re-written in the form:

- These are equations for circles on the (u,v) plane

## Smith Chart – Real Circles

- r=0

- r=1/3

- r=1

- r=2.5

## Smith Chart – Imaginary Circles

- x=1/3

- x=1

- x=2.5

- x=-1/3

- x=-1

- x=-2.5

## Smith Chart

## Smith Chart Example 1

- Given:

- What is ZL?

## Smith Chart Example 2

- Given:

- What is L?

## Smith Chart Example 3

- Given:

- What is Zin at 50 MHz?

## Admittance

- A matching network is going to be a combination of elements connected in series AND parallel.

- Impedance is NOT well suited when working with parallel configurations.

- For parallel loads it is better to work with admittance.

- Impedance is well suited when working with series configurations. For example:

## Normalized Admittance

- These are equations for circles on the (u,v) plane

## Admittance Smith Chart

- g=1/3

- b=-1

- b=-1/3

- g=1

- g=2.5

- g=0

- b=2.5

- b=1/3

- b=1

- b=-2.5

## 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.

## Admittance Smith Chart Example 1

- Plot 1+j1 on chart
- vector =
- Flip vector 180 degrees

- Given:

- What is ?

- Plot y

- Flip 180 degrees

- Read 

## Admittance Smith Chart Example 2

- Plot
- Flip vector by 180 degrees
- Read coordinate

- Given:

- What is Y?

- Plot 

- Flip 180 degrees

- Read y

## Matching Example

- Match 100 load to a 50 system at 100MHz

- A 100 resistor in parallel would do the trick but 1⁄2 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

## 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

## Matching Example

- y=0.5+j0
- Before we add the admittance, add a mirror of the r=1 circle as a guide.

- Admittance Chart

## 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

## 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

- Admittance Chart

## 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

## Matching Example

- Add positive imaginary admittance to get to z=1+j0

- Impedance Chart

## Matching Example

- This solution would have also worked

- Impedance Chart

## Matching Bandwidth

- 50 MHz

- 100 MHz

- Because the inductor and capacitor impedances change with frequency, the match works over a narrow frequency range

- Impedance Chart

## dB and dBm

- A dB is defined as a _**POWER**_ ratio. For example:

- A dBm is defined as log unit of power referenced to 1mW:

## Single Stub Tuner

- Match 100 load to a 50 system at 100MHz using two transmission lines connected in parallel

## Single Stub Tuner

- Adding length to Cable 1 rotates the reflection coefficient clockwise.
- Enough cable is added so that the reflection coefficient reaches the mirror image circle

- Impedance Chart

## Single Stub Tuner

- The stub is going to be added in parallel so flip to the admittance chart.
- The stub has to add a normalized admittance of 0.7 to bring the trajectory to the center of the Smith Chart

- Admittance Chart

## Single Stub Tuner

- Admittance Chart

- An open stub of zero length has an admittance=j0.0
- By adding enough cable to the open stub, the admittance of the stub will increase.
- 70 degrees will give the open stub an admittance of j0.7

## Single Stub Tuner

- Admittance Chart

## Single Stub Tuner

- Admittance Chart

- This solution would have worked as well.

## Single Stub Tuner Matching Bandwidth

- 50 MHz

- 100 MHz

- Because the cable phase changes linearly with frequency, the match works over a narrow frequency range

- Impedance Chart

## Two Port Z Parameters

- We have only discussed reflection so far. What about transmission? Consider a device that has two ports:

- The device can be characterized by a 2x2 matrix:

## Scattering (S) Parameters

- Since the voltage and current at each port (i) can be broken down into forward and reverse waves:

- We can characterize the circuit with forward and reverse waves:

## Z and S Parameters

- Similar to the reflection coefficient, there is a one-to-one correspondence between the impedance matrix and the scattering matrix:

## Normalized Scattering (S) Parameters

- The S matrix defined previously is called the _un-normalized_ scattering matrix. For convenience, define normalized waves:

- Where Zoi is the characteristic impedance of the transmission line connecting port (i)

- |ai|2 is the forward power into port (i)
- |bi|2 is the reverse power from port (i)

## Normalized Scattering (S) Parameters

- The normalized scattering matrix is:

- Where:

- If the characteristic impedance on both ports is the same then the normalized and un-normalized S parameters are the same.
- Normalized S parameters are the most commonly used.

## Normalized S Parameters

- The **s** parameters can be drawn pictorially

- s11 and s22 can be thought of as reflection coefficients
- s21 and s12 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

- s11

- s22

- s21

- s12

- a1

- a2

- b1

- b2

## Examples of S parameters

- 

- Zo

- G

- Transmission Line

- Short

- Amplifier

## Examples of S parameters

- Zo

- Isolator

- Circulator

## Lorentz Reciprocity

- If the device is made out of linear isotropic materials (resistors, capacitors, inductors, metal, etc..) then:

- or

- for

- 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

## Lossless Devices

- The s matrix of a lossless device is unitary:

- for all j

- for all i

- Lossless devices:		transmission line
- 				short
- 				circulator

- Non-lossless devices:		amplifier
- 				isolator

## 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.

## 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

- s11

- s22

- s21

- s12

- x11

- x22

- x21

- x12

- y11

- y22

- y21

- y12

- yx21

- yx12

- Connector Y

- Connector X

- We want to know the S parameters at these reference planes

- We measure the S parameters at these reference planes

## 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.

## Network Analyzer Calibration

- Since we measure four S parameters for each calibration standard, we need at least three independent standards.
- One possible set is:

- 

- Thru

- Short

- Delay*
- *~90degrees

## Phase Delay

- A pure sine wave can be written as:

- The phase shift due to a length of cable is:

- The _phase_ delay of a device is defined as:

## 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

## 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

## Group Delay

- The waves emanating from the source will look like

- Which can be re-written as:

## Group Delay

- The information travels at a velocity

- The group delay is defined as:

## Phase Delay and Group Delay

- Phase Delay:

- Group Delay:

## Transmission Line Topics

- Phasors
- Traveling Waves
- Characteristic Impedance
- Reflection Coefficient
- Standing Waves
- Impedance and Reflection
- Incident and Reflected Power

- Smith Charts
- Load Matching
- Single Stub Tuners
- dB and dBm
- Z and S parameters
- Lorentz Reciprocity
- Network Analysis
- Phase and Group Delay