PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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 of the system

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems slide 3 First order System

A simple example of a first order system is an electrical RC filter consisting of a with C and a 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.

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems slide 4 First order system: Example 1, RC filter I The voltage input / output relation for an

RC filter shows all of the characteristics Vin R C Vout of a 1st order system:

When the output voltage Vout 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 Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems slide 5 First order system: Example 1, RC filter I

Vin R C Vout

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 homogeneous ODE: solution: So what would be the step function response of the system? time Boundary conditions: (i) Vin Vin constant (ii)

) RC trivial solution of ) (t

(t inhom. ODE out in V V ? Add solutions of t t hom+inhom. ODE 0 0 boudnary conditions: 0 ⇒ ⇒ Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems slide 6 First order system: Example 1, RC filter I So what is the Transfer function G(s) V R of this measurement system? in C Vout

= Laplace transformation (see tables)

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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

Vin R C Vout

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).

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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 Lock-In Oscilloscope Generator amplifier Tektronix TDS1012 Stanford Research Stanford Research DS345 SR830 BNC cable

I BNC adapter BNC adapter V R C V in Low pass filter out

Build the low pass on an electronics breadboard with R=10kΩ and C=10nF. Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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. Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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 Vout(t) for a step input. Compare your measurements with the theory presented on the previous slides. ) ) (t (t in

t out t V V

The period T of the oscillation should be much larger than the time constant (T >> RC)

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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 Vin at the input of the low pass filter. Measure the amplitude and phase of the sinusoidal output Vout(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 Vout(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. ) ) (t (t in

t out V V t Department of Physics T<

environment = UA(T-Tenv) heat emission The change of the heat in the thermopile is the sum of the absorbed and emitted heat:

dQ/dt + [UA/MC] Q = P + UATenv Inhom. first order ODE with Department of Physics time constant τ = MC/UA University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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.

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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.

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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 mirror laser optic modulator (AOM). The principles of the AOM will be AOM discussed later in the semester. For now it can be considered as Photodiode or shutter AOM driver a black box. When an thermopile appropriate input beam is applied, two output beams, one transmitted and one deflected mirror beam will appear. The intensity of the deflected beam is in a certain range proportional to the beam input voltage at the AOM driver. expander

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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 Electrometer Oscilloscope (Keithley 614) (Tektronix AOM driver TDS1012)

FunctionGenerator Stanford Research DS345 Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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.

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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.

Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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 Lock-In (b) Do not use the AC coupling of Oscilloscope the lock-in amplifier since it amplifier Tektronix TDS1012 represents a high pass with Stanford Research 160mHz transit frequency. That SR830 BNC cable may be too high for the given measurement system BNC adapter BNC adapter to AOM driver to 1010 preamp. Department of Physics University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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.

Department of Physics University of Utah