Lecture 6: Measurements of Inductance, Capacitance, Phase, and Frequency 1

Outline In these slides, we review some methods for the measurement of:  Inductance  Capacitance  Phase  Frequency 2

Inductance measurement The measurement of inductance is of interest since, for example, the output of some sensors, such as the inductive displacement sensor, has the form of a change in inductance. There are AC bridge circuits commercially available which can be used to measure inductance accurately. Let us study an approximate but simple method for measuring inductance. 3

In this method, the unknown inductance L is connected in series with a variable resistance R in a circuit excited with a sinusoidal voltage of a known frequency f. The resistance R is adjusted until V R = V L. At this point, the two impedances X R = X L. This gives where r is the inductor resistance. 4 Inductance measurement

Capacitance measurement Measurement of capacitance is of interest since the output of some sensors, such as capacitive level gauge and capacitive displacement sensor, is in the form of a change in capacitance. Similarly, capacitance can be measured accurately by AC bridge circuits commercially available. But let study two simple methods for capacitance measurement. 5

Method 1: the unknown capacitor C is connected in series with a variable resistance R in a circuit excited with a sinusoidal voltage of a known frequency, f. The resistance R is adjusted until V R = V C. At this point, X R = X C and thus: Method 2: put the capacitor in an RC circuit and measure its time constant: τ = RC 6 Capacitance measurement

Phase measurement Some instruments or transducers, such as transit-time ultrasonic flowmeter and radar level sensor, convert the measured variable into a phase change. The simple method for measuring the phase difference between two signals is to use a 2-channel oscilloscope. The two signals are both displayed (vs. time) and a suitable time base (time/div) is chosen such that the time between the crossing points of the two signals can be measured. 7

Measurement of phase using X–Y mode Another useful technique for approximate phase measurement is to plot the two signals (of equal magnitude) on the X–Y mode available in oscilloscope. The plot obtained is called Lissajous figures and is usaully an ellipse. 8

If the X and Y inputs are given by: At t = 0: But, from the Figure, when Solution of the previous equation gives the phase shift φ. However, to determine which signal is leading the other, we need to display the two signals plotted (against time) on a 2-channel oscilloscope. 9 Measurement of phase using X–Y mode

Digital counter-timer The most accurate instrument for measuring the phase difference between two signals is the digital counter-timer. 10

Digital counter-timer First, the two signals are amplified/attenuated so that they have the same amplitude. Then, the time that elapses between the two signals crossing some reference threshold value is measured. The crossing points of the two signals through the reference threshold voltage level are applied to a gate that starts and then stops pulses from the oscillator into an electronic counter. The elapsed time, and hence phase difference, between the two input signals is then measured in terms of the counter display. 11

Frequency measurement Frequency measurement is required as part of those devices that convert measured physical quantity into a frequency change, such as turbine flowmeter and Doppler-shift ultrasonic flowmeter. To measure the frequency of a signal, we can display it on oscilloscope, measure its period T, and then calculate the frequency f = 1/T. However, the accuracy of this method is limited to ±5% of the reading. Two additional methods for measuring frequency are:  Digital counter-timer.  Phase-locked loop (PLL). 12

Digital counter-timer Is the most accurate instrument for measuring frequency. The essential component within a counter-timer is an oscillator that provides a very accurately known and stable reference frequency, which is typically either 100 kHz or 1MHz. The oscillator output is transformed by a pulse-shaper circuit into a train of pulses and applied to an electronic gate. 13

Successive pulses at the reference frequency alternately open and close the gate. The input signal, of unknown frequency, is similarly transformed into a train of pulses and applied to the gate. The number of these pulses passing through the gate during the time that it is open is proportional to the frequency of the unknown signal. To measure much lower frequencies, a series of decade frequency dividers are provided. These increase the time between the reference frequency pulses by factors of ten, and a typical instrument can have gate pulses separated in time by between 1 µs and 1 second. 14 Digital counter-timers

Phase-locked loop (PLL) A PLL is a circuit consisting of a phase-sensitive detector, a voltage controlled oscillator (VCO), and amplifiers, connected in a closed-loop system. In a VCO, the oscillation frequency is proportional to the applied voltage. 15

Before describing the operation of the PLL, we need to review the concept of phase difference of two sinusoids shown below. If the phase difference remains constant in successive cycles, this indicates that the frequency of the two sinusoids is equal. 16 Phase-locked loop (PLL)

PLL Operation 17 The phase-sensitive detector compares the phase of the amplified input signal with the phase of the VCO output. Any phase difference generates an error signal, which is amplified and fed back to the VCO. This adjusts the frequency of the VCO until the error signal goes to a steady-state value (small but non-zero). Thus the VCO becomes locked to the frequency of the input signal. The d.c. output from the VCO is then proportional to the input signal frequency.