5. Analogue Instruments

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1 By the end of this section you will be able to: Describe how the oscilloscope is built and operates, and its limitations. Describe the operation of the trigger circuit Describe the operation of the dual time base facility. Describe the operation of the lock-in amplifier 5.1. Oscilloscopes The oscilloscope is familiar, at least in its present developed form, which offers great facilities of automation, reliable triggering, and very often the ability to expand the time base so as to allow observation of very brief events. In this section we will examine the oscilloscope closely to understand its operation, and hence its limitations. Ch1 Dt 1 Vertical control Ch attenuator Vertical amplifiers Delay lines + or alt Y Dt attenuator Ext ti Internal Trigger Filter Filter Timebase generator 1 Timebase generator X Main Timebase Delayed Timebase Delay setting Figure 5-1 Structure of an oscilloscope Vertical system The signal to be observed is first attenuated in order to bring it into the dynamic range of the vertical amplifiers. The vertical amplifiers are engineered to accommodate the highest sensitivity of the scope at the minimum added noise. Keeping in mind that the noise figure of the attenuator is equal to its attenuation, it follows that this arrangement results into a constant signal to noise ratio at the vertical input of the oscilloscope. The input preamplifiers have a finite input impedance. It tends to be equal to 1ΜΩ shunted with 10pF. This is seldom a high enough impedance for a low frequency voltage measurement, and attenuation probes are used to raise the impedance level to a 10 or even 100 times higher value. The circuit they employ is a frequency independent resistive/capacitive voltage divider, as shown in figure. The frequency independence of the attenuation ratio is ensured by adjusting a trimmer capacitor so that a square signal is not distorted. Hence the familiar probe calibration procedure. CP IC-EEE Autumn

2 1pF V=.1 V IN V IN Z IN=.1Z 9MΩ 1MΩ 9pF Figure 5-: Probe attenuator circuit. Note that the 10:1 division ratio is frequency independent if the trimmer capacitor is properly adjusted One often needs to observe a small AC signal riding on a large DC level. In this case, the addition of a series capacitor in the signal path forms a high pass filter with the input impedance of the oscilloscope. AC coupling is dispersive (not frequency independent), so it distorts signals with a large harmonic content such as square waves. A high input impedance is not always desirable. At signal frequencies high enough that the electrical length of the interconnecting cables becomes important. The Z in =10MΩ//1pF input impedance of the scope is transformed by the cable of length L and impedance Z 0 to frequency dependent value Z in: ' Z in + jz 0 tan kl ω Z in = Z 0 with k = (1) Z + jz tan kl c 0 in Figure 5-3: Variation of the input impedance of a 1 meter long oscilloscope probe lead with frequency, between 0-00MHz The effective input impedance Z in is effectively indeterminate, since the precise electrical length of the cable is not known, and varies widely with frequency. Z in can be as low as a fraction of 1 ohm when L=λ/4. Needless to add that the input impedance variation introduces frequency dependent loading of the Device Under Test and hence dispersion in the measurement. To avert this ambiguous situation, high frequency oscilloscopes provide low input impedance preamplifiers with input impedance matched to the cables characteristic impedance ( typically 50Ω). As this may be an unacceptably low impedance level for a voltage measurement, high input impedance remote -local to the Device Under Test (DUT) - preamplifiers are often used in high frequency measurement situations. These have a high input impedance (typically the gate terminal of a MOSFET or JFET) and a 50Ω output impedance to drive the low impedance input to the scope. Such remote preamplifiers are called Active probes. CP IC-EEE Autumn

3 The oscilloscope s vertical amplifiers have a finite bandwidth. If we model tem as dominant pole amplifiers, they will have a constant gain G up to the dominant pole frequency of ω 0. As the impulse response of a 1 st order LPF is a decaying exponential, it is easy to show that an oscilloscope exhibits a finite, exponential, rise-time to the 90% of the value of a step input, equal to. τ rise = () ω0 This is one of the main limitations to the highest frequency signal that can be displayed, but not the only one Horizontal sweep system The horizontal sweep system of the oscilloscope consists of the trigger, a one shot sawtooth generator called the time-base generator which is of sufficient amplitude for the beam to span horizontally the screen. A horizontal magnification feature, often provided, simply adds a post amplifier to the time-base generator. The most important part of the oscilloscope is the trigger circuit, which consists of a very fast schmitt trigger, a number of filters, as well as the delay elements following the horizontal amplifiers. The delay elements are critical for the correct operation of the oscilloscope. They permit the sweep to start at the trigger event, or even before it. The trigger circuit effectively operates an enable control for the timebase generator. The quality of the trigger circuit is more often than not the only difference between a cheap and an expensive oscilloscope with apparently similar specifications. A number of other facilities are usually provided in modern oscilloscopes, including a control to chose between interlacing entire screens or just pieces of a screen between the two vertical channels (alternate and chop vertical modes) with the obvious bandwidth restrictions arising from the sampling theorem, and the holdoff feature, disabling the trigger for some interval after the trigger event, so as to prevent false triggering off artefacts Dual time-base The temporal extent of a feature one needs to observe is often too short compared to the interval between trigger events, hence unresolvable under normal sweep settings. Many oscilloscopes provide a dual time-base which is a second time-base generator triggered off the main time-base generator (figure 1). The delay setting is a reference level to compare to the primary time-base generator sawtooth waveform. The second time base (called expanded) allows the user to expand a small fraction of the original sweep to fill the entire display. It may be possible to view the fragment expanded on the original time-base settings as an intensified piece of the trace. In this setting the second time-base serves to step up the CRT electron gun current Trace intensity Limitations to the ultimate frequency response of the oscilloscope arise not only from the vertical amplifier bandwidth, but also from the rise times of the trigger amplifiers and noise (jitter) in the time-base generator. A further factor which limits the operation of the scope at high frequencies has to do with the trace intensity. Trace intensity is directly proportional to the number of electrons falling on a screen phosphor grain and hence proportional to the electron gun current and inversely proportional to the linear trace velocity on the screen. The latter is always larger than the horizontal sweep speed. Therefore, the number of electrons impinging on a grain of phosphor (of size d) on a screen of N x divisions and length L x at a time base T x per division when the electron gun current is I gun during one sweep is: CP IC-EEE Autumn

4 I n < N T d gun x x el At high horizontal rates this number gets much smaller than unity and the trace fades, a familiar phenomenon at the fastest time base settings of the oscilloscope. To compensate for trace fading at the fast time base settings an intensity control is provided. Also, switching to the secondary time base usually steps up the beam current/trace intensity, shortening the CRT life in the process. As there are practical limits to the magnitude of the achievable electron gun current, high speed oscilloscopes may employ photomultipliers on the screen to enhance the trace intensity. Such a screen, manufactured by Tektronix, is shown on figure 3. x (3) Figure 5-4: The microchannel plate photomultiplier array used to amplify beam intensity. 5.. Correlation techniques The main merit of direct observation of a signal with an oscilloscope is simplicity and intuitiveness. However oscilloscope measurement is inherently a wide band measurement. Direct observation is inevitably undermined by noise, and is only possible as long as the signal to noise ratio is acceptable. However, a measurement is often required under adverse noise conditions, where the noise is overwhelmingly larger than the signal. In such situations we resort to averaging and correlation techniques. A common representative of correlation techniques is the lock-in detection which we will examine in some detail, together with other common correlation techniques. An observable signal S is, in general, of the form: S = M + N (4) where M is the true quantity we are trying to evaluate with the measurement, and N is a noise component. S is in effect a random variable, distributed around a mean M, with a variance σ = N (5) The Central Limit Theorem states that the probability of a measurement returning a particular value S has a Gaussian distribution, regardless of the nature of the noise present in the system, and even its magnitude: CP IC-EEE Autumn

5 P ( S) ( S M ) σ e (6) A consequence of the central limit theorem is that the probability density of the noise average amplitude to have a value N after K observations varies as: P N σ ( N ) e K (7) so that the accumulated noise energy scales as N σ K (8) At the same time, the accumulated signal energy scales as S S K (9) and the signal to noise ratio of an ensemble of observations increases with the number of observations (signal to noise is ratio of powers!) S K (10) N and the measurement converges to the mean of the observable, M. Whether we resort to averaging or employ correlation techniques, we are effectively exploiting the statistics of the noise amplitude distribution Filtering or signal averaging We have seen that noise power content of the measurement will be characterised by a power spectral density function. When the signal is at a single frequency we can in principle restrict the bandwidth of the observation arbitrarily by use of filters. The signal to noise ratio is then: = S 1 = =Τ N( f) df SNR lim SNR B B 0 B if N(f) is finite then SNR tends to infinity as the bandwidth tends to zero and is proportional to the measurement time. We have seen that this approach is not valid for DC measurements, since the PSD of flicker noise is infinite at zero frequency, and consequently low frequency fluctuations and drift are difficult to filter out of a measurement. A filter is equivalent to averaging a large number of measurements Multichannel analysis In extremely noisy situations it is possible to directly measure the statistics of the measurement amplitude distribution. To do this, we divide the range of possible values of the observable S into a number of intervals ( bins ) and count how many (n i ) of K in total measurements fall in an interval S i ={S i -δ/, S i +δ/}, we find, 0 (11) n = KP S ) (1) i ( i CP IC-EEE Autumn

6 and the difference of the number of counts in adjacent bins scales with the number of measurements: n n +1 K (13) i This type of measurement is known as Multichannel Analysis and a continuous time version of it is called Boxcar Integration. Both are used extensively in atomic and high energy physics experiments where the noise to signal ratios can be immense, and spectral methods are of little use. The multichannel measurement is effectively a vanishing bandwidth measurement, the effective bandwidth being proportional the inverse of the total observation time Correlation The cross correlation between signals is a quantitative measure of the similarity between them. It is defined as: i * xy () = ( ) ( + ) R t x τ y τ t dτ (14) The Correlation theorem states that the Fourier transform of the correlator is the product of the Fourier transform of the first signal and the complex conjugate of the transform of the second signal. In particular, the autocorrelation of a signal is the correlation function of a signal itself; the Wiener-Khinchin theorem states that the Power Spectral Density of a signal is the Fourier transform of its autocorrelation. * * ( Iy) = Iy( Ix) and IR = Ix PSD IRxy = Ix xx = (15) We can exploit the correlation theorem to make cleaner measurements. Assume we have a clean measurement x( t ) corrupted by additive noise nt. ( ) The cross correlation of the signal and the noise is zero. Much like we discussed statistically independent noise sources, the signal is statistically independent of the noise. Any deterministic signal we can generate will also be statistically independent of the noise. As a result, if we can mix (multiply) the unknown signal with a known signal, the carrier C the result will also be statistically independent of the noise, but statistically dependent on the carrier. We can write, if C is the carrier, N is the noise and S is the signal, that the observable F, after the carrier signal multiplication is: F t = C t S t + N t ( ) ( ) ( ) ( ) * * * ( ) ( ( ) ( ) ( )) ( ) ( ) ( ) ( ) R t = C τ S τ + N τ C t+ τ dτ = C τ S τ C t+ τ dτ + R F, C NC, It is now clear that the noise-carrier correlator is ideally zero, and in practice small. Therefore, we can say: FC, ( ) ( ) ( ) ( ) * * R t C τ C t+ τ S τ dτ + ε A possible way to unravel this is to choose: * C ( τ ) C( t+ τ) = δ ( t τ) so that CP IC-EEE Autumn

7 FC, * * * * ( ) ( τ) ( + τ) ( τ) τ = δ ( τ) ( τ) τ = ( ) R t C C t S d t S d S t This way we can recover the signal, which may be buried in noise. All this, complicated as it may appear, is easy to do in practice. All sensors exhibit cross sensitivity to quantities unrelated to the observable. For example, a Hall magnetic field sensor has a high sensitivity on its bias current, and this is why the device is frequently used as a current measurement probe. By modulating one of those quantities we effectively mix the observable with a signal of known frequency. By subsequently demodulating using the same reference signal and a low pass filter, we can evaluate the cross correlator of the reference signal with the measurable quantity. The Lock-in amplifier, shown in figure 4, is an instrument used to perform a correlation measurement. It includes sensitive preamplifiers, long time constant low pass filters, as well as a delay line so that the phase of the observable may be determined. The mixer is a masking device, i.e. half the input waveform is allowed to pass and the other half is masked. A reference oscillator is also provided, which drives the mixer, may be phase locked to the external stimulus modulating the experiment, or may itself be used to modulate the experiment. The delay element serves to determine the phase of the measured quantity. Since the Fourier transform of the square wave used to mask the signal is unity at DC, the instrument measures the complex Fourier coefficient of the input at the modulating frequency. Sensor Observable Input Lock-in Amplifier LPF output Reference input Delay Reference output Phase lock Figure 5-5 The process of modulating and demodulating a measurement in order to remove noise. This is the block diagram of a measurement utilising a lock-in amplifier Optical Interferometry Reference generator Interferometry is a correlation technique used at very high frequencies, when the wavelength of the measurable quantity may be small, and the bandwidth of the measurement is negligible. In that case, the signal measured is called monochromatic or coherent. We can usually arrange add a signal to a delayed copy of itself. If the fractional bandwidth is very small, by modulating the delay we can modulate the amplitude of the sum between zero and twice the signal amplitude. A lock-in amplifier can then be used to detect the amplitude variation, which is proportional to the original amplitude. The delay itself may be modulated by physically moving a mirror or by changing the refractive index of a wave propagation medium inserted in the signal path. Interferometry leads to some of the most sensitive measurements known, and is capable of resolving distances of the order of a nanometre, and time intervals smaller than a picosecond. CP IC-EEE Autumn

8 Signal (RF or optical) Senso Input Delay To Lock-in amplifier Delay control Figure 5-6: The principle of interferometry. Two monochromatic or nearly monochromatic high frequency electrical signals are set to interfere with a controlled variable delay Mechanical intereferometry Reference Interferometry, or mixing is not restricted only to optical signals. After all, a wave is a spatial variation of an electromagnetic field. We can therefore detect the interference between periodic mechanical structures. An example is the Vernier scale where two rulers with different ruling periods (typically 1 and 0.9, respectively) are compared to resolve intervals with 10 times the resolution of the coarse scale. The vernier principle is exploited in precision displacement measurements. The rulers may be photographically printed on two glass plates. By overlapping the plates and using them to interrupt a number of light beams we can automate the vernier measurement. By modulating the beam position and using lock-in detection the sensitivity of a measurement can be greatly enhanced. Alternatively, the rulers may be magnetically printed, and their position detected by oscillating or rotating coils. Both techniques are in wide industrial use to measure (and control!) the position of machine tools to an accuracy of a few microns. CP IC-EEE Autumn