Modelling an ultrasound A-scan
Teachers' guide
Here we will be modeling the amplitude ultrasound scan or A-scan used in ophthalmology. It is the simplest type of scanning, relying on a one-dimensional pulse-echo technique similar to that used in the echo location (sonar) of fish.
While this analogy with ultrasound A-scanning is good, you will need to emphasise that the two coaxial cables of differing electrical impedance are representing two different materials in the eye of differing acoustic impedance. By knowing the speed of the pulses and where reflections occur, they can then calculate the lengths of the cables in a similar way to which distances to parts of the eye would be determined. With ultrasound there are far fewer multiple reflections of note, although false echoes can occur between tissue and bone, tissue and air, and tissue and the transducer. In most cases there would also be appreciable scattering and so the amplitude of any multiple reflections would prove tiny. With the electrical pulses the students should be able to see the multiple reflections as there is less scattering and attenuation.
If students are shown the original output pulse they will then note that, on reflection at the cable junctions, there is a phase change with the pulse being inverted. This is of no consequence with the ultrasound scan as it is just the amplitude of the reflected pulse that is important.
As students will have met propagation of light by internal reflection down optical fibers, it may be wise to comment that the electrical pulses effectively travel in a straight line down the cables as if they were laid out straight, there is no ‘bouncing’ off of the walls of the cables. They may well be surprised at the speeds of pulses in such cables being a substantial fraction of the speed of light.
The activity, whilst modeling the ultrasound A-scan, is a useful one for seeing how to obtain times from an oscilloscope screen and, in general, how to set up and use an oscilloscope. Whilst the activity could be conducted on a traditional oscilloscope (ideally of the storage type) it is certainly easy to do with a PicoScope 2202. If you have the older ADC-200 version then the same activity can be conducted with it too, as indeed it could with a PicoScope 3000 Series device. Obviously it does need an oscilloscope with a fast timebase.
The speeds of the pulses down coaxial cables tends to be around 2 x 108 m/s. The two I used gave speeds of 1.95 x 108 m/s and 1.99 x 108 m/s which are rather close to each other. As the real speeds are not important I have suggested stating a speed of 1.90 x 108 m/s for one cable and 2.10 x 108 m/s for the other. The lengths they calculate will then not actually be the real ones, but there again I suspect no one is going to wish to unwind them and wind them back on the reels again, so that does not matter.
Typical results
Output pulse: 708 ns.
First main reflection: 1681 ns.
Second main reflection: 3736 ns.
Time for pulse to travel to end of cable A and back: (1681 - 708) ns = 973 ns.
Speed of pulse down cable A is 2.10 x 108 ms-1.
So length of cable A is given by:
Speed of pulse x ½ x time for pulse to travel = (2.10 x 108) x 0.5 x (973 x 10-9) = 102 m
Time for pulse to travel through cable A to the end of cable B and back: (3736 - 708) ns = 3028 ns. However, 973 ns is the time that this pulse was traveling through cable A, so the time in cable B must be (3028 - 973) ns = 2055 ns. Speed of pulse down cable B is 1.90 x 108 m·s–1. So length of cable B is given by:
Speed of pulse x ½ x time for pulse to travel = (1.90 x 108) x 0.5 x (2055 x 10–9) = 195 m
Answers to questions
Q1 Using the expression Ir/Ii = (Z2 - Z1)2/(Z2 + Z1)2 and substituting for Z1 = 1.38 x 106 kg m-2 s-1 andZ2 = 6.5 x 106 kg m-2 s-1 we then have:
Ir/Ii = (6.5 x 106 – 1.38 x 106)2/(6.5 x 106 + 1.38 x 106)2 = 0.42 (42%)
Q2 Answers will depend on the results the students got but, assuming the cables are very similar to those I recommended cable A should be near 100 m in length and cable B near 200 m in length.
Figure 1: set up of PicoScope 2202, pulse generator, coaxial cables and computer
Testing the pulse generator
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