Impedance Measurements in a Differential Transmission Line By: David Migas

Motivation We would like to improve the current setup of the CMS’s inner tracker. – It is difficult to manipulate – It is bulky, and interferes with the other parts of the detector – The data transmission rate is limited

When the present pixel detector is replaced… Micro twisted pair cables may be used – They will eliminate much material from the CMS – Allow the setup to be more easily manipulated – Simplify the setup – Allow data to be sent at faster rates

Coupled twisted pair cable Two small insulated wires twisted together Twisting it reduces environmental noise additions. We find that a wave can be expressed as a combination of: – Odd Mode: Two equal and opposite waves – Even Mode: Two equal waves.

The odd mode Since we take the difference of the two signals – Much of the noise is subtracted out – Smaller signals may be sent – The signal doesn’t need to be read with respect to a ground

In a real coupled wire, the function for the odd mode looks like this: Given a sinusoidally varying time dependant part, we find: – A sinusoidally varying position dependant part that is modified by a decaying exponential To use these cables efficiently, reflections at any boundary must be minimized.

At the boundary of a cable and a resistor… We have two conditions: In a more convenient format, these read:. From continuity of potential From conservation of current

Since… Our two equations can now be written as: andwithin the wire andin the resistor, is equivalent to And we letand

When we eliminate r, we find: When we eliminate t, we find: So, in order to have the signal be perfectly transmitted, with none reflected, we simply match the resistance to the impedance.

Taking a closer look at the reflection coefficient, Or, more simply put: Where X is the amplitude of the wave measured at the boundary.

We can’t measure vr or vi, only their sum, X. The only data we need is: – The amplitude of the wave at the terminated end – The resistance used to terminate it It tells us: – The opposite of the slope will be the impedance – Half the y-intercept will be the amplitude at the end.

Before these cables are used, we need to know three things: – The delay – The signal loss – The impedance

The Experimental Setup

The Wire Cable 1 Properties (micrometers)Cable 2 Properties (micrometers) Inner diameter (Copper) 250 ± 4 μm Inner diameter (Copper) 125 ± 3 μm Inner Insulator (Polyesterimid S180) 17 μm (min) Inner Insulator (Polyesterimid S180) 19 μm (min) Outer Insulator (Polyamid FS) 13 μm (min) Outer Insulator (Polyamid FS) 9 μm (min) Outer Diameter316 μm (max)Outer Diameter168 μm (max) Resistance0.348 ± Ohm/mResistance1.393 ± 0.08 Ohm/m The 250 micrometer cable was created in the lab The 125 micrometer cable was prefabricated

The Setup A signal is injected into the wire One probe measures this end of the wire The wire is terminated by resistors into the ground A second probe measures this end

The Signal A pulse generating program was created by Tony and Nick The first hump on the teal trace is the actual pulse

Reading the Oscilloscope 1 – The initial signal 2 – The initial signal at the termination end of the wire plus the reflected signal 3 – The reflected signal after travelling through the wire

The Variables The diameter of the wire – 125 um and 250 um The termination resistance: – 10, 22, 24, 30, 51, 75, 91, 1000, 10,000 The length of the wire – 250 um – 2 meters (twice) – 125 um – 1, 2 and 4 meters

Observables recorded The amplitude of signal 1 and 2 The time delay between them

Upon interpreting the data… Impedance Delay Signal Loss

The Results

Impedance 250 micrometer – 22.4 ±.5 and 24 ± 1 Ohms – Average: 23 ± 1 Ohms 125 micrometer – 30.5 ±.9, 28 ± 1, and 29 ± 2 Ohms (1,2,4) – Average: 29 ± 2 Ohms

Delay 250 micrometer – 2 meter ±.4 ns (twice) 125 micrometer – 1 meter ±.4 ns – 2 meter ±.4 ns – 4 meter ±.4 ns

Signal Remaining 250 micrometer – 2 meter -.77 ±.03 and.70 ± micrometer – 1 meter -.83 ±.01 – 2 meter -.71 ±.02 – 4 meter -.59 ±.01