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Minimum 2nd. lecture Measurement is the estimation of the quantity of certain value (with known uncertainty) by comparison with the standard unit. Main terms: estimation measurement uncertaintity standard unit SI system of units: 7 main units and derived units from them. All quantities and their units are collected in the ISO (International Standard Organization) standard. direct (method of) measurement method of measurement in which the value of a measurand is obtained directly, without the necessity for supplementary calculations based on a functional relationship between the measurand and other quantities actually measured indirect (method of) measurement method of measurement in which the value of a quantity is obtained from measurements made by direct methods of measurement of other quantities linked to the measurand by a known relationship 3. lecture: Errors Error is the difference between the reading (measured) value and real value. Sometimes not the difference but its magnitude. Systematic error Its behaviour is deterministic, it is present in the same way in every repeated measurement. Example! (shift) Stochastic error Its behaviour is random from measurement to measurement. Example! (scattering) The error could be given in two ways: Absolute way, like: 125 V +-2V It means that real value more than 123 but less then 127 Relative to the real or to the measured value: 125V+-2% Meaning of the error could be different: - limit or range average error standard deviation (appr. 68% probability that real value is within 1 SD, 95% that within 2 SD, 99,8% within 3 SD) others Accuracy: Error of the instrument, when you measure once. It depends on range. from manual: 0-20V ----- 0.5% +1digit First term is relative SD of the reading value. Second term is the error of the last digit seeing on the display. Example: Reading value is 15.6V within 0-20V range. 1% 0,156 0,2V 15.6V+-0,2V with 68% prob. Error of counting: (1/156)*100%=0,6% means app. 0,1V. Within 1 SD there is no significant difference between the 2 results. If you repeat the same measurement N-times, you can calculate the result and assume the error. Result equals to mathematical mean: x 1 N i 1 xi N SD of the measured data: N 1 2 xi x N 1 i 1 SDdata SD of the result: SDresult SDdata N Result should be given as the mean and +- SD_result and should be rounded according to the magnitude of the error. 4. lecture: evaluation of data Find out the connection between 2 physical quantity, like for example voltage and current of a resistor. Choose one quantity as a parameter of the measurement, change step by step and measure another. You will get pairs of data. Make a table. Make a graph: horizontal axe is the parameter, vertical axe is another measured quantity. Try to fit your data with some analytical function. It means finding a simple function which gives similar result like your measurement at the same values of the parameters. How good is the fitting? least squares method: x_i measured data at ith value of the parameter. y_i calculated result using by the analytical function. at the best fit quantity N 2 xi yi 2 i 1 should has got a minimum. During fitting you are searching for a best fit, by changing the type of the analytical function or by changing its parameters. 5th lecture: Propagation of the errors A little error in the measured quantity can cause big error in the result of the calculation. If error means error limit: In the case of multiplication and divison of the quantities the relative error of the result equals to the sum of the relative errors of the each quantities. You have 1V 0.5% and 1A 1% (typical) R=U/I, 1,5% In the case of addition and substraction the absolute erorr of the result is the sum of the absolute error of the quantities. You have I_1 1A +- 1mA, I_2 2A, +-1mA in result I_1+I_2 you get +-3mA absolute error. deviation (scattering) Addition and subtraction of a number will not change standard deviation. Multiplication or division by a number will change standard deviaton in the same way. Addition or subtraction of two quantities (A,B) with certain absolute standard deviations: SD A B SD A2 SDB2 In the case of multiplication and division similar formula is true but for relative SD. Worst case analysis is a general method for taking into account propagation of errors. Multimeter basics: More Material AC – basics Alternating Current (AC) Alternating Current (AC) flows one way, then the other way, continually reversing direction. An AC voltage is continually changing between positive (+) and negative (-). AC from a power supply This shape is called a sine wave. The rate of changing direction is called the frequency of the AC and it is measured in hertz (Hz) which is the number of forwards-backwards cycles per second. Mains electricity in the UK has a frequency of 50Hz. This triangular signal is AC because it changes between positive (+) and negative (-). An AC supply is suitable for powering some devices such as lamps and heaters but almost all electronic circuits require a steady DC supply (see Properties of electrical signals An electrical signal is a voltage or current which conveys information, usually it means a voltage. The term can be used for any voltage or current in a circuit. The voltage-time graph on the right shows various properties of an electrical signal. In addition to the properties labelled on the graph, there is frequency which is the number of cycles per second. The diagram shows a sine wave but these properties apply to any signal with a constant shape. Amplitude is the maximum voltage reached by the signal. It is measured in volts, V. Peak voltage is another name for amplitude. Peak-peak voltage is twice the peak voltage (amplitude). When reading an oscilloscope trace it is usual to measure peak-peak voltage. Time period is the time taken for the signal to complete one cycle. It is measured in seconds (s), but time periods tend to be short so milliseconds (ms) and microseconds (µs) are often used. 1ms = 0.001s and 1µs = 0.000001s. Frequency is the number of cycles per second. It is measured in hertz (Hz), but frequencies tend to be high so kilohertz (kHz) and megahertz (MHz) are often used. 1kHz = 1000Hz and 1MHz = 1000000Hz. frequency = 1 time period and time period = 1 frequency Mains electricity in the UK has a frequency of 50Hz, so it has a time period of 1/50 = 0.02s = 20ms. Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation: , where is the peak voltage (unit: volt), is the angular frequency (unit: radians per second) o The angular frequency is related to the physical frequency, (unit = hertz), which represents the number of cycles per second, by the equation . is the time (unit: second). The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value of value is −1, an AC voltage swings between and usually written as or , is therefore is +1 and the minimum . The peak-to-peak voltage, . The relationship between voltage and the power delivered is where represents a load resistance. Rather than using instantaneous power, , it is more practical to use a time averaged power (where the averaging is performed over any integer number of cycles). Therefore, AC voltage is often expressed as a root mean square (RMS) value, written as , because For a sinusoidal voltage: The factor is called the crest factor, which varies for different waveforms. For a triangle waveform centered about zero For a square waveform centered about zero For an arbitrary periodic waveform of period : Example To illustrate these concepts, consider a 230 V AC mains supply used in many countries around the world. It is so called because its root mean square value is 230 V. This means that the time-averaged power delivered is equivalent to the power delivered by a DC voltage of 230 V. To determine the peak voltage (amplitude), we can rearrange the above equation to: For 230 V AC, the peak voltage is therefore , which is about 325 V. The peakto-peak value of the 230 V AC is double that, at about 650 V. Analog oscilloscope Of the four basic blocks of the oscilloscope, the most visible of these blocks is the display with its cathode-ray tube (CRT). The vertical amplifier Conditions the input signal so that it can be displayed on the CRT. The vertical amplifier provides controls of volts per division, position, and coupling, allowing the user to obtain the desired display. Must have a high enough bandwidth to ensure that all of the significant frequency components of the input signal reach the CRT. The trigger is responsible for starting the display at the same point on the input signal every time the display is refreshed. The final piece of the simplified block diagram is the time base. Vertical and horizontal positioning Phase shift: http://www.youtube.com/watch?v=30J5U0ThRUc Moving coil meters for DC measurements The rotation of the coil (and the pointer attached to it) is due to the torque M, which dependson the flux density B of the magnet, on dimensions d and l of the coil, on number of turns z of the coil and of course on the measured current I: M = Bzdl * I Returning torque of the spring k*alfa, where alfa is the angle of the pointer. Microammeter, voltmeter, ammeter A moving coil meter measures directly the small current. (I alfa) Voltmeter: Ux proportional with I, R_d is high. Ammeter: R_d is much higher then R_b shunt resistor How shunt is working?(extending the range n-times) Moving coil ammeter I_max=20mA Desired range: 200 mA What to do? R_inner is small, 9 Moving coil ammeter I_desired, 200mA R_inner is small, 9 I_ammeter, 20mA R_shunt is small R_inner*I_ammeter=I_shunt*R_shunt (parallel branches, U equal) I_shunt=I_desired-I_ammeter R_shunt=R_inner/(n-1), where n=I_desired/I_ammeter HereR_shunt=1 ohm How serial resistor is working (extending the range n-times) Inner resistances Moving coil ammeter R_inner is 1M Measured load 10k U_max=20V=I_max/R_inner Desired range: 2kV mA What to do? Moving coil ammeter R_inner is 1M, R_serial Measured load 10k Apply more resistivity in serial with the inner resistivity of the voltmeter U_desired=I_max/(R_inner+R_serial) R_serial=R_inner*(n-1) 8. lecture: Oscilloscope basics Digital Oscilloscope Analog to digital conversion basics The two major components in a high-speed digitizer's analog front end are the analog input path and the analog-to-digital converter (ADC). The analog input path attenuates, amplifies, filters, and/or couples the signal to optimize the digitization by the ADC. The ADC samples the conditioned waveform and converts the analog input signal to digital values that represent the conditioned input signal. Figure 1 Reprezentation of periodic signals in the time and frequency domain Fourier theorem: Each periodic singnal can be produced by the sum of the sinus like signals. sinusoid signal: 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0 5 It has amplitude, frequeny, phase-shift. sum of them: 10 15 20 2 1 0 -1 -2 0 5 10 15 20 time domain: horizontal axe: time, vertical axe: signal ferquency domain: horizontal axe: the frequency of the sinusoidal components of the signal, vertical axe: their amplitude 800 600 400 200 0 0 20 40 x10 -3 60 80 100 Bandwidth Bandwidth describes the frequency range in which the input signal can pass through the analog front end with minimal amplitude loss - from the tip of the probe or test fixture to the input of the ADC. Bandwidth is specified as the frequency at which a sinusoidal input signal is attenuated to 70.7% of its original amplitude, also known as the -3 dB point. The following figure shows the typical input response for a 100 MHz highspeed digitizer. Figure 2 For example, if you input a 1 V, 100 MHz sine wave into high-speed digitizer with a bandwidth of 100 MHz, the signal will be attenuated by the digitizer’s analog input path and the sampled waveform will have an amplitude of approximately 0.7 V. Figure 3 It is recommended that the bandwidth of your digitizer be 3 to 5 times the highest frequency component of interest in the measured signal to capture the signal with minimal amplitude error (bandwidth required = (3 to 5)*frequency of interest). rise time Another important topic related to bandwidth is rise time. The rise time of an input signal is the time for a signal to transition from 10% to 90% of the maximum signal amplitude and is inversely related to bandwidth by the following formula, based on the one pole model, R-C limited input response. Figure 5 This means that the rise time of a 100 MHz digitizer input path is 3.5 ns. It is recommended that the rise time of the digitizer input path be 1/3 to 1/5 the rise time of the measured signal to capture the signal with minimal rise time error. The theoretical rise time measured (Trm) can be calculated from the rise time of the digitizer (Trd) and the actual rise time of the input signal (Trs). Figure 6 For example, the rise time measurement when measuring a signal with 12 ns rise time with a 100 MHz digitizer is approximately 12.5 ns. sample rate Sample rate is the speed at which the digitizer’s ADC converts the input signal, after the signal has passed through the analog input path, to digital values that represent the voltage level. This means that the digitizer will sample the signal after any attenuation, gain, and/or filtering has been applied by the analog input path, and convert the resulting waveform to digital representation. The sample rate of a high-speed digitizer is based on the sample clock that tells the ADC when to convert the instantaneous analog voltage to the digital values. National Instruments high-speed digitizers support a variable effective sample rate derived from the maximum sample rate of the device. For example, the NI 5112 has a maximum sample rate of 100 Megasamples/second (MS/s) and can be set to rates of (100MS/s)/n, where n = 1,2,3,4,.... Nyquist theorem Nyquist Theorem: Sample rate > 2 * highest frequency component (of interest) of the measured signal The Nyquist theorem states that a signal must be sampled at a rate greater than twice the highest frequency component of the signal to accurately reconstruct the waveform; otherwise, the high-frequency content will alias at a frequency inside the spectrum of interest (passband). An alias is a false lower frequency component that appears in sampled data acquired at too low a sampling rate. The following figure shows a 5 MHz sine wave digitized by a 6 MS/s ADC. The dotted line indicates the aliased signal recorded by the ADC and is sampled as a 1 MHz signal instead of a 5 MHz signal. Figure 8: Sine Wave Demonstrating the Nyquist Frequency The 5 MHz frequency aliases back in the passband, falsely appearing as a 1 MHz sine wave. To prevent aliasing in the passband, you can use a lowpass filter to limit the frequency of the input signal or increase your sampling rate. Resolution The best way to understand the concept of resolution is by comparison with a yardstick. Divide a 1 meter yardstick into millimeters. What is the resolution? The smallest “tick” on the yardstick is the resolution. Yes, you might be able to interpolate between these, but in the absence of this sophisticated guessing process the resolution is 1 part out of 1000. The resolution of a n-bit analog-to-digital Converter (ADC) is a function of how many parts the maximum signal can be divided into. The formula to calculate resolution is 2^n. For example, a 12 bit ADC has a resolution of 2^12 = 4,096. Therefore, our best resolution is 1 part out of 4,096, or 0.0244% of the full scale. An ADC takes an analog signal and turns it into a binary number. Thus, each binary number from the ADC represents a certain voltage level. Resolution is the smallest input voltage change a digitizer can capture. Resolution can be expressed in bits (LSB), in proportions, or in percent of full scale. Resolution limits the precision of a measurement. The higher the resolution (number of bits), the more precise the measurement. An 8-bit ADC divides the vertical range of the input amplifier into 256 discrete levels. With a vertical range of 10 V, the 8-bit ADC cannot ideally resolve voltage differences smaller than 39 mV. In comparison, a 14-bit ADC with 16,384 discrete levels can ideally resolve voltage differences as small as 610 µV. Let us examine how a sine wave would look if it is passed through ADCs with different resolutions. We will compare a 3-bit ADC and a 16-bit ADC. A 3-bit ADC can represent 8 discrete voltage levels. A 16-bit ADC can represent 65,536 discrete voltage levels. As you can see, the representation of our sine wave with 3-bit resolution looks more like a step function than a sine wave. However, the 16-bit ADC gives us a clean looking sine wave. Note that if you are using a 3-bit ADC, minute voltage fluctuations in the incoming signal will not be detected. Figure 1. A 5 kHz Sine Wave being sampled by a 3-bit versus a 16-bit ADC Another way to think of resolution is by considering your television screen. The higher the resolution of the screen, the more pixels you have to show the picture, so you get a better picture. Summary Analog to digital conversion 1. Bandwidth Bandwidth is specified as the frequency at which a sinusoidal input signal is attenuated to 70.7% of its original amplitude, also known as the -3 dB point. Higher bandwidth gives smaller rise time due to the presens of the more high frequency components of the measured signal. 2. Sampling rate How many samples / second, sampling requency, 100 Ms/s 3. Resolution Number of digital values using for digital conversion and the maximum value, calculate! Featuring of DSO data-storage Record length refers to the amount of memory dedicated to storing digitized samples for postprocessing or display for a single acquisition. In a digitizer, record length limits the maximum duration of a single-shot acquisition. For example, with a 1,000-sample record and a sample rate of 20 MHz, the duration of the acquisition is 50 µs (the number of points multiplied by the acquisition time per sample, or 1,000 x 50 ns). With a 100,000-sample record and a sample rate of 20 MHz, the duration of acquisition is 5 ms (100,000 x 50 ns). In many cases, measurement quality depends on the digitizer's ability to take a sustained acquisition while maintaining high sampling rates. In these cases, the amount of acquisition memory determines the fidelity of the acquired signal. High-speed digitizers with deep onboard acquisition memory have the ability to take enhanced time and frequency-domain measurements. Vertical range and Offset Vertical range is the peak-to-peak voltage span that a digitizer can measure at the input connector. Most digitizers have several choices for vertical range. Vertical offset is the voltage the vertical range is centered on. Vertical offset positions a waveform around a DC value. Using this offset allows you to examine small changes in the input signal, which can improve the accuracy of your measurement. For example, imagine that you are acquiring the waveform shown in Figure 1 that outputs 0.75 V to 1.25 V. Without using vertical offset, you would need to specify a range of 2.5 V (±1.25 V) to capture the waveform. However, with vertical offset, you would only need to specify a range of 0.5 V (1.25 V - 0.75 V). Exchange the comments for the figure below! Coupling (AC,DC, Ground) On many digitizers, you can configure the input channels to be DC coupled, AC coupled, or GND coupled. DC coupling allows DC and low-frequency components of a signal to pass through without attenuation. In contrast, AC coupling removes DC offsets and attenuates low frequency components of a signal. Activating AC coupling inserts a capacitor in series with the input. This feature can be exploited to zoom in on AC signals with large DC offsets, such as switching noise on a 12 V power supply. GND coupling disconnects the input and internally connects the channel to ground to provide a ground, zero-voltage reference. Block diagram of the digital multimeter http://www.ti.com/solution/digital_multimeter_bench_system Design Considerations A Digital Multimeter (DMM) is a precision analog instrument used to measure AC and DC voltage, AC and DC current, capacitance, and resistance. Five system level blocks are common to bench DMM designs: Signal Conditioning and A/D conversion of the input signal, LCD/LED/Keypad, Control and Data Processing, Memory/Peripheral devices, and Power Management. Implementation specifics will obviously vary depending on the feature set of the meter. The common core subsystems are: Analog Front-End Signals are initially passed through a signal conditioning subsystem which amplifies or attenuates the analog signal in preparation for further conditioning, depending on whether the measurement is AC volts, DC volts, current, or resistance. Precision amplifiers and analog to digital converters are used to facilitate resolutions from 5 1/2 to 8 1/2 digits on modern DMMs. LCD/LED/Keypad Bench DMMs offer dual, 7 segment displays. The PC is typically used to provide a fullfeatured graphical user interface for bench DMMs. Control and Data Processing Executes measuring processes and controls interface with memory and peripheral devices. The Digital Signal Processor (DSP) is used to linearize data from the ADC and perform calibration. It also performs real-time analysis of acquired signals such as min/max, averaging, and conversion to engineering units. Modern DMMs allow users to directly measure temperature using a variety of sensors. Memory/Peripheral Devices Measurement results are stored in EEPROM or FLASH memory and can be uploaded to a PC via USB, Ethernet, RS-232, or IEEE-488 (GPIB) interfaces. Power Management and Conversion Converts the input battery power to run various functional blocks. TI has a wide range of products like amplifiers, ADCs, Power, Interface, and Processors to meet the requirements of bench Digital Multimeter designers. Questions / Topics: What is measurement? (uncertainity) SI quantities, SI system of units, some derived quantities What is the error? (absolute, relative) Classification of errors Average of the absolute error, error limit, Standard deviation Evaluation of simple measurement Usage of DMM (also in practical) Usage of oscilloscope (also in practical) Net resistance for serial and parallel resistors (in practice, measurement and calculation) Settings on DC power supply (in practice) Measuring AC signal with oscilloscope, settings on function generator (in practice) Mathematical description of AC signal: What is: amplitude, periodic time, frequency, maximum value RMS: meaning, formula How moving coil meters are working? Extending the maximum range of the voltmeter Extending the maximum range of the ammeter, the sunt resistor The inner resistance: measuring high and low resistances Explain the main oscilloscope functions: time/div; volt/div; vertical shift, trigger, coupling (AC,DC), vertical gain How CRT is working? Fourier theorem Bandwidth Rise time Sampling rate/what is sampling Bit resolution or resolution of ADC Offset Nyquist or sampling theorem Time diagramm and frequency diagramm of the signal. Block diagramm and main functions of the digital multimeter Block diagramm and main functions of the Digital Storage Oscilloscope