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CHAPTER 7 FILTERS, LOADING AND OP-AMPS INTRODUCTION Sometimes we make measurements and what is measured is a combination of what we wished to measure and noise. This noise could be caused by the electronic circuitry, by external factors that affect the measurement system, or by external factors that affect the variable we are trying to measure. If this noise is in a different frequency range to the signal we can filter it out. A full discussion of noise, its sources and how its effects may be reduced are given in Chapter 11 of these course notes. This is one example of the use of filters. However, any system will have a frequency response function associated with it and will amplify some frequency components and attenuate others. So in some sense all systems are filters. A thermocouple behaves as a first order system. If we plot its frequency response we will see that the magnitude becomes very small at high frequencies. So this is an example of a system that removes high frequencies from the signal. Very fast temperature fluctuations may not be measured with a thermocouple. We say that the thermocouple behaves like a low pass filter; it only allows low frequency components in signals to pass through. Some systems behave like high pass filters, they attenuate low frequencies and pass high frequencies. Some accelerometers made of quartz crystal have a frequency response that is small at very low frequencies, is flat for a range of frequencies, peaks in the region of the natural frequency of the crystal and then becomes very small at high frequencies. This type of system is a second order system combined with a high pass filter. Band pass filters can be constructed by combining a low pass filter in series with a high pass filter as shown in Figure 1. These are often used in instrumentation to filter out low and high frequency noise, and also as part of a demodulation instrument to extract one channel of data. You use a band pass filter when you tune into a radio station. x(t) y(t) Figure 1: High and low pass filters combined to produce a band pass filter It would be good if we could say that the frequency response function of the band pass filter was the product of the frequency response functions of the high pass and low pass filters. H(jω) High Pass Filter L c G (jω) Low Pass Filter U c
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7-2 However, this, in general, is not the case. This is because when systems are joined together they interact with one another. The frequency response function of the band pass filter is: 1 2 12 G(j ) H (j ) H (j ) L (j ) (1) where 12 L (j ) is the term that arises because of the interaction. We will refer to this as the loading term. Every time we add another subsystem in series we need to include another loading term. So, if in our measurement system we had three components plugged together in series, e.g., a thermocouple, an amplifier and a filter, and their frequency response functions were: 1 2 3 H (j ),H (j ) and H (j ) , respectively, the frequency response function of the combined system would be: 1 2 3 12 23 H (j ) H (j ) H (j ) L (j ) L (j ) (2)
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