Exercise 8 – Measurements of time and frequency. 1. Aim of the exercise The aim of the exercise is to familiarize students with methods of measuring time parameters of electrical signals such as frequency, period and phase shift between the two signals. Additional error evaluation will be described. 2. Main topics Measurements of time parameters (time, frequency, and duty cycle) of electrical signals by using the oscilloscope and Digital Frequency Counter (DFC) Measurements of the phase shift between the two sinusoidal signals with a frequency counter and a digital oscilloscope. 3. Gained skills the ability to connect the circuit according to the measurement diagram the ability to measure correctly time parameters (period, frequency, duty cycle) and phase shift using a digital frequency counter and an oscilloscope. the ability to properly document the experiment results the ability to determine of measurement errors 4. Quantities, definition and units Time (t): fundamental physical quantity "The second is the duration of 9191631770 periods of the radiation corresponding to the transition between the hyperfine levels of the ground state of the ceasium-133” Frequency (f): "Frequency (f) O of a periodic phenomenon is the number (n) of occurrences of phenomenon in a time unit (τ); frequency is the reciprocal of the time (T) between successive occurrences of the phenomenon.” “"One hertz is the frequency of a period phenomenon of which the periodic time is one second; one hertz is one cycle per second”. Pulsatance, ω: "Angular frequency (Pulsatance) is the frequency of a periodic process expressed in radians per second; pulsatance is the rate of change of angle in time.” 5. Methods of measurements used in the experiment. There are many methods for measurements of frequency or time. In our experiment only a few of them are used: analog methods based on measurement of time with the oscilloscope, and direct method based on of measurement frequency and time with the multifunction digital counter. Oscilloscope method used in the experiments are extremely simple - they implement either internal (linear) time base, or external reference signal (Lissajous method). Having known the time base speed time/div (a value which may be read from oscilloscope’s screen), all we need to do is to measure the length of one or more cycles of the observed signal.. This method is fast but not very precise. Sinusoidal (external) time-base method (The Lissajous method). An other analog frequency measurement method involving the oscilloscope rely on using the oscilloscope as a kind of null indicator for comparison of a sine signal with unknown frequency with a reference sine signal whose frequency should be well defined and easily varied. One of the signals is fed to the Y channel of the oscilloscope, the second one to the X channel. An interaction of these two signals produces on the display more or less complicated snaky loops, whose form allows for determining the unknown frequency. A typical Lissajous figure is shown in Fig. 1. Fig. 1 Frequency measurement with the Lissajous method a) measurement diagram, b) Lissajous figure and frequency determination. If we assume that fy is the unknown frequency signal connected to the Y channel and fx is the reference signal connected to the X channel then we have 푛푥 1 푑휑 푓푦 = 푓푥 + 푛푦 2휋 푑푡 where nx, ny denote number of intersections of the Lissajous curve with horizontal and vertical axes, respectively, and dφ/dt is the rate of phase change (the trace rotation speed). The reference signal should be adjusted until the displayed figure is possibly stable (dφ/dt ≈ 0). It is sometimes attainable with difficulty, and needs both signal sources involved having adequate frequency stability. The method's error is roughly equal to relative calibration error of the reference source. Due to the mentioned stability problems, usefulness of the method is limited to rather low frequency applications. Digital counter method. Digital counter methods rely on continuing number of events (in this case – the number of cycles) with the counter open during precisely determined window. The periodic input signal of any shape, including sine waveform, is formed in an input shaper block to have standard form of possibly short pulses that are fed to the counter controlled by an accurate and stable quartz oscillator (see Fig.2). Input AND Counter Display fx shape gate r τ Fig. 2. Principle of direct digital measurement of frequency If the counter is opened for e.g. 1 second, then the number of pulses counted during this time directly gives the measured frequency. If we denote open gate time as τ, input signal period as Tx, and number of cycles counted as N, then 휏 = 푁 ∙ 푇푥 and the unknown frequency is equal to 푓푥 = 푁/휏 It is intuitively obvious that the accuracy of this method mainly depends on the accuracy of gate timing. It may be shown that the limiting error of direct frequency measurement method is equal to 1 1 훿푔푓푥 = 훿푔푓푟푒푓 + = 훿푔푓푟푒푓 + 푁 푓푥 ∙ 휏 where δgfref is the limiting error of quartz oscillator frequency.5gfk is the limiting. In conclusion, one may see that the limiting error of direct frequency measurement method decreases - as the number N increases, i.e. measured frequency rises, - gating time τ is longer. The accuracy of the method is getting worse if measured frequency goes down. The way out of this problem is to switch to digitally measure the period of the signal Tx instead of directly measure the frequency. In this method, the gate is opened for the time of one period of the input signal, and during this time pulses with the reference frequency fw produced by quartz oscillator are counted. The reference frequency fref may be changed by adjusting the frequency dividing factor m, so that fw = fref/m. Finally, the frequency we are looking for is equal to fx = 1/Tx. The limiting error of measured period is equal to 1 푚푓푥 훿푔푇푥 = 훿푔푓푟푒푓 + = 훿푔푓푟푒푓 + 푁 푓푟푒푓 and 훿푔푓푥 = 훿푔푇푥 The limiting error of this method of frequency measurement may be further decreased if we measure not the one period of input signal but time of few periods, say k periods. In this case, the limiting error is expressed as 1 푚푓푥 훿푔푓푥 = 훿푔푓푟푒푓 + = 훿푔푓푟푒푓 + 푘푁 푓푟푒푓 ∙ 푘 The universal frequency/time counter may also be used for measurement of time interval. In this case the first event opens the gate and the second one stops it. In between the counter count pulses with reference frequency coming from quartz oscillator. It is worth knowing that modem time-frequency counters are capable to select appropriate mode of operation automatically, thus minimizing the quantization error. Moreover, the timer-counter Keysight 53220A measure only the period of input signal. So, measured frequency is always calculated. A part of the limiting error of discussed measurements comes from the finite time of gate switching. The detailed discussion of that problem is beyond the scope of this text. 6. Instrumentation Laboratory module description Fig. 1. Module F01. G1 – sinusoidal waveform output, G2 – rectangular waveform output, PF – phase shifter 7. Measurements and tasks Table 1. Measurement functions of 53220A Keysight digital frequency counter Function Symbol Settings Period T FREQ/PERIOD PERIOD Frequency f FREQ/PERIOD FREQ Pulse counting n TOTALIZE GATED Duty cycle TIME INTERVAL DUTY CYCLE Pulse width TIME INTERVAL PULSE WIDTH Pulse distance t TIME INTERVAL PULSE WIDTH Phase shift TIME INTERVAL PHASE PHASE 1-2 Table 2. Measurement functions of 1052 Rigol oscilloscope Function Symbol Settings Period T MEASURE TIME PERIOD Frequency f MEASURE TIME FREQ Duty cycle MEASURE TIME +DUTY Pulse width MEASURE TIME +WIDTH Pulse distance t MEASURE TIME -WIDTH Phase shift MEASURE TIME PHAS 12 Note: Before the measurements: - connect power supply to the F01 Module. - set a frequency counter impedance to 50 Ω for inputs 1 and 2: (1 or 2) Impedance: 50 Ω. Task 1. Measurements of time parameters of sinusoidal signals 1.1 Use the DFC 53220A (Digital Frequency Counter) to measure time parameters (period and frequency) of sine signal from generator G1 output in F01 module. NOTE: Prior to the measurements assess the stability of the G1 generator and adjust the resolution of the frequency counter. To do this, set the number of digits displayed on the frequency counter display in a way to achieve observed instability of the result on the least significant digit ( Digits AutoDigits: Off, and then use the knob to set number of digits). To make reading the results easier, the manual trigger of measurements can be used. Once you activate following sequence of commands: Trigger Source Manual, each time you press the button Trigger, a single measurement will be performed. To go back to automatic trigger mode execution following sequence of commands: Trigger Source Internal. Measure appropriate parameters using following functions: a) PERIOD – measure the period of the signal with gating time of 1 s. If selected gating time is longer than measured period, the DFC measures an average period. b) FREQ – measure the frequency of the signal with gating time of 1 s. In this case, the frequency is measured with the indirect method. c) TOTALIZE:GATED – counting the number of pulses during 1 s of Gate Time. This case realizes the direct frequency measurement method (using frequency definition – counting of phenomena occurrences during reference time interval) Compare and comment obtained results from PERIOD, FREQ and TOTALIZE:GATED ! measurement method. 1.2 Use the oscilloscope to measure time parameters (period and frequency) of sine signal from generator G1 output in F01 module. Take measurements of the period and frequency with the use the manual procedure (length measurements) and the automatic measurement functions. Set the oscilloscope to minimize measurements errors.