First and second order systems. Part 1: First order systems: RC low pass filter and Thermopile. Goals: Department of Physics

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1 slide 1 Part 1: First order systems: RC low pass filter and Thermopile Goals: Understand the behavior and how to characterize first order measurement systems Learn how to operate: function generator, oscilloscope, current amplifier, lock-in amplifier, HeNe laser, photodiode, thermopile, acousto-optic modulator

2 slide 2 First order system: A first order measurement system is a system whose dynamics is described by a first order differential equation. The transfer function assumes the form G(s) = 1/(1+τs) with τ being the time constant of the system

3 slide 3 First order System A simple example of a first order system is an electrical RC filter consisting of a capacitor with capacitance C and a resistor with resistance R. There are other types of systems which have the same input/output response as an RC filter. Examples include mechanical systems which are viscously damped and fluidic systems. A cantilever in very thick viscous oil would be an example but not a cantilever in air! In this section, you will look at the time domain and frequency domain input / output characteristics of an electrical and thermal first order system.

4 slide 4 First order system: Example 1, RC filter The voltage input / output relation for an RC filter shows all of the characteristics of a 1st order system: V in R I C V out When the output voltage V out is measured with an ideal voltage meter, the currents I in the resistor R and the capacitor C are equal. Since we obtain a relationship between the input and output voltage. The dynamical equation of this system is a first order ordinary differential equation (ODE) which is why the RC filter is a first order measurement system

5 slide 5 First order system: Example 1, RC filter I V in R C V out To find a solutions of an inhomogeneous ODE, find all the solutions of the homogeneous ODE and then add them to one solution of the inhom. ODE by matching the boundary conditions V in V in (t) homogeneous ODE: solution: So what would be the step function response of the system? 0 t V in V out (t) time constant? RC 0 t Boundary conditions: (i) (ii) trivial solution of inhom. ODE Add solutions of hom+inhom. ODE boudnary conditions: 0

6 slide 6 First order system: Example 1, RC filter I So what is the Transfer function G(s) of this measurement system? V in R C V out = Laplace transformation (see tables)

7 slide 7 First order system: Example 1, RC low pass filter AN RC filter is an electrical circuit consisting of a series of a resistor R and a capacitor C as illustrated in the circuit diagram. I V in R C V out Low pass filter Low pass filter means, when input and output are as defined in the sketch, the dynamics of the output voltage will follow the dynamics of the input voltage only for low frequency components. Frequency filters are used for signal processing for measurement systems but also for consumer electronics (audio amplifiers).

slide 8 First order system: Example 1, RC low pass filter Set up the low pass filter so that the input is driven by an oscillator of variable frequency and constant amplitude.

8 slide 8 First order system: Example 1, RC low pass filter Set up the low pass filter so that the input is driven by an oscillator of variable frequency and constant amplitude. Look at the output simultaneously with an oscilloscope and a two phase lock-in amplifier. Function Generator Stanford Research DS345 Lock-In amplifier Stanford Research SR830 Oscilloscope Tektronix TDS1012 BNC cable I BNC adapter V in R Low pass filter C V out BNC adapter Build the low pass on an electronics breadboard with R=10kΩ and C=10nF.

9 slide 9 First order system: Example 1, RC low pass filter Explanation of the experimental setup The function generator produces a frequency which is connected via coaxial cable and BNC adaptors to the input of the low pass, the reference input of the lock-in amplifier and channel 1 of the oscilloscope. The output of the low pass is then connected to the input of the lock-in amplifier and channel two of the oscilloscope. The lock-in amplifier will measure the magnitude of the input signal and the phase relation between the input and the reference signal and, therefore, the phase shift introduced by the low pass setup. The same information can be obtained in a different way with the oscilloscope which displays the time dependence of both the input and the output of the low pass in one display. While for low frequencies, both signals will be similar in magnitude and phase, a phase shift and intensity drop should becomes visible for high frequencies. Note that high and low frequencies refer to the inverse of the time constant of the low pass.

10 slide 10 First order system: Example 1, RC low pass filter 1. Familiarize yourself with the oscilloscope, the function generator and the lock-in amplifier. If necessary, read their manuals in order to understand how they work. 2. Measure the time domain response of the RC filter by applying a step function to the low pass filter. This can be accomplished by applying a square wave (between 0 and 1 volt) with a period which is much longer than the RC time constant of the low pass filter. Use the storage feature of the oscilloscope to measure the output voltage V out (t) for a step input. Compare your measurements with the theory presented on the previous slides. V in (t) t V out (t) t The period T of the oscillation should be much larger than the time constant (T >> RC)

11 slide 11 First order system: Example 1, RC low pass filter 3. Measure the frequency domain response by applying a 1 volt rms (= root of means squre value) sinusoidal voltage V in at the input of the low pass filter. Measure the amplitude and phase of the sinusoidal output V out (f) as a function of frequency f between 10Hz and 100kHz using a two-phase lock-in amplifier. Perform 20 measurements at appropriate frequencies over this range to characterize the response. Also observe and record your general observations of the amplitude and phase shift of the low pass output signal V out (f) (channel 2 ) relative to the drive signal (channel 1) on the oscilloscope as f is changed. This can best be accomplished by triggering on channel 1 (drive signal) and comparing the signal on channel 2 with that of channel 1. You need not record all of these measurements, but convince yourself that these measurements are consistent with those made with the lock-in amplifier. Then quantitatively compare your lock-in measurements with the theory for 1st order systems. Explain any discrepancies. V in (t) t V out (t) t T<<RC

12 slide 12 First order system: Example 2, thermopile 4. Repeat the measurements in parts 2 and 3 qualitatively using an optical beam (heat source) which illuminates a thermopile (= heat measurement system which transduces heat into temperature). Experimental setup: Thermopile consists of material with heat capacity C which converts heat ΔQ into temperature ΔT a fully absorbed laser beam with power P introduces heat into the thermopile thermopile ΔT = ΔQ/MC T heat emission thermocouple amount of heat emission depends on the temperature difference of the thermopile to its environment = UA(T-T env ) A thermocouple in the thermopile converts the temperature into a voltage which can be measured) T env voltmeter V = temp. of environment A = surface area of thermopile M = mass of thermopile U = thermal conductivity at thermopile surface The change of the heat in the thermopile is the sum of the absorbed and emitted heat: dq/dt + [UA/MC] Q = P + UAT env Inhom. first order ODE with time constant τ = MC/UA

13 slide 13 First order system: Example 2, thermopile Explanation of the setup: The thermopile is a sensor consisting of a mass with well known heat capacity and a black surface which means a surface which absorbs light excellently at the wavelength for which it is made. If irradiated, the light is absorbed and converted into heat. The heat will then increase the temperature so that the temperature gradient between the thermopile and its environment appears. Heat will then flow along this gradient and, under constant irradiation, a steady state absorption and thus asteady state heat flow and thus, a steady state temperature gradient will exist. Hence, by measurement of the temperature in the thermopile, the heat flow and therefore, the irradiation power can be measured. Thermopiles are used for the highly accurate measurements of light intensities such as for the calibration of lasers powers. In our experiment, the temperature is measured with a thermocouple.

14 slide 14 First order system: Example 2, thermopile Explanation of the experiment: While the thermopile is a quantitatively very accurate light intensity sensor, it has the drawback that is very slow. When the irradiative power changes, the system has to return back to a steady state (with an exponential decay function since it is a first order system). The time constants for this can be quite long (ms to s range) which make the thermopile a bad choice as light detector in fast measurement applications. The main task of this experiment is to show that the thermopile is a first order measurement system and to determine the time constant τ for the given device.

slide 15 First order system: Example 2, thermopile 4.

The principles of the AOM will be discussed later in the semester. For now it can be considered as a black box.

15 slide 15 First order system: Example 2, thermopile 4.1 The important part of this experiment is to have a setup where the laser intensity can be modulate either periodically with a sine-oscillation or stepwise. For this, the following setup is used: The beam modulation is accomplished with an acousto optic modulator (AOM). The principles of the AOM will be discussed later in the semester. For now it can be considered as a black box. When an appropriate input beam is applied, two output beams, one transmitted and one deflected beam will appear. The intensity of the deflected beam is in a certain range proportional to the input voltage at the AOM driver. mirror AOM shutter mirror beam expander laser AOM driver Photodiode or thermopile

16 slide 16 First order system: Example 2, thermopile 4.2. Carry out the following tasks: (i) Turn on the HeNe laser. Lasers will be discussed later in class and can be considered here as black boxes which are used as light sources for the experiment. (ii) For your safety, absolutely obey the laser safety rules discussed in class! (iii) Align the setup such that an expanded laser beam with beam diameter of less than the thermopile acceptance aperture is directed towards the thermopile. Laser beam expanders will be discussed later in class and can be considered here as black boxes which change the diameter of a laser beam. (iv) Connect the power supply of the AOM to the AOM driver and apply (V=28V DC) (v) Note that the absolute maximum modulation voltage of the AOM is 1V. Any voltage beyond this value may damage the instrument. However, this maximum value is far above the linear range of the AOM a reasonable voltage for the calibration process is 100mV

17 slide 17 First order system: Example 2, thermopile 4.2. Carry out the following tasks: (vi) Calibrate the modulated laser intensity: For this, remove the thermopile and put a calibrated photodiode 818UV in its place. The photodiode can be considered a black box, whose short circuit current is proportional to the light power which is irradiated onto its photosensitive area. The used photodiode produces 300μA/W at an irradiation wavelength of λ=633nm (the color of the HeNe laser) (vii)you should have to following setup: laser AOM expander photodiode AOM driver Electrometer (Keithley 614) Oscilloscope (Tektronix TDS1012) FunctionGenerator Stanford Research DS345

18 slide 18 First order system: Example 2, thermopile 4.2. Carry out the following tasks: (viii)for the calibration, turn off the room light in order to reduce background illumination. (ix) The calibration can be done in three steps: (a) check whether the AOM responds: Change the modulation voltage slightly and check whether the current of the photodiode changes. (b) Check for linearity: Apply an oscillating voltage to the AOM driver and check within which range the measured current is proportional to this voltage. (c) Make the absolute calibration: Using the calibration factor for the photodiode given above, determine how the voltage at the AOM driver input converts into laser power at the photodiode within the linear range. In order to do this, measure for a few different input voltage values the photodiode current. This calibration will allow to apply a defined power to the thermopile (once it is put into the setup) by application of an appropriate AOM driver input voltage.

slide 19 First order system: Example 2, thermopile 4.2. Carry out the following tasks: (x) After the calibration, replace the photodiode by the thermopile and start the actual experiment.

19 slide 19 First order system: Example 2, thermopile 4.2. Carry out the following tasks: (x) After the calibration, replace the photodiode by the thermopile and start the actual experiment. The thermopile requires a preamplifier, which is a 1010 low noise amp. Note: This amplifier is driven by built in batteries and, therefore it is constantly on. Because of this, there must be a short circuit cap on the input as long as the amplifier is not in use in order to prevent a floating input potential which could damage the amplifier. (xi) Adjust the DC offset of the preamp. if necessary.

20 slide 20 First order system: Example 2, thermopile 4.2. Carry out the following tasks: (xii)connect the lock-in amplifier and the oscilloscope such that the same measurement as done in part 2 and 3 can be executed. Note that the setup has differences since for the thermopile measurement system, a much slower response time and, therefore, longer time constant τ is expected: (a) Use the internal function generator of the lock-in amplifier as frequency source (b) Do not use the AC coupling of the lock-in amplifier since it represents a high pass with 160mHz transit frequency. That may be too high for the given measurement system Lock-In amplifier Stanford Research SR830 BNC adapter to AOM driver Oscilloscope Tektronix TDS1012 BNC cable BNC adapter to 1010 preamp.

21 slide 21 First order system: Example 2, thermopile 4.3. Repeat the measurements in parts 2 and 3: Show that the two different approaches approximately give the same 1st order system time constant, using the theory provided in the text. Note that while the system is qualitatively equivalent to the RC circuit in part one, there are strong differences quantitatively since the system is significantly slower. Take this into account for the proper choice of the frequency range through which the measurements are carried out. DO NOT record your data for many frequencies in to order to keep the entire measurement time within the four hour measurement time. For this part of the lab, the goal is only confirm the same qualitative properties as for the RC pass is to show. Collect only as much data as needed for the determination of τ.

Part 2: Second order systems: cantilever response

- cantilever response slide 1 Part 2: Second order systems: cantilever response Goals: Understand the behavior and how to characterize second order measurement systems Learn how to operate: function generator,