physics6770 laboratory 1 week 1, first order systems first...

PHYSICS6770 Laboratory 1

University of UtahDepartment of Physics

First and second order systemsWeek 1, First 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




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




slide 3

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.

First order System




slide 4

First order system: Example 1, RC filterThe voltage input / output relation for an RC filter shows all of the characteristics of a 1st order system:

R CVin Vout

I

R CVin Vout

I

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




slide 5

homogeneous ODE: solution:

First order system: Example 1, RC filter

R CVin Vout

I

R CVin Vout

I

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

So what would be the step function response of the system?

Vin(t)

t0

Vin

Vou

t(t)

t0

Vin

? trivial solution of inhom. ODE

Add solutions of hom+inhom. ODE

Boundary conditions: (i)(ii)

time constant RC

boudnaryconditions: 0 ⇒

⇒




slide 6

First order system: Example 1, RC filter

R CVin Vout

I

R CVin Vout

ISo what is the Transfer function G(s)of this measurement system?

= Laplace transformation(see tables)




slide 7

First order system: Example 1, RC low pass filterAN RC filter is an electrical circuit consisting of a series of a resistor R and a capacitor C as illustrated in the circuit diagram.

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).

R CVin Vout

I

R CVin Vout

I

Low pass filter




slide 8

First order system: Example 1, RC low pass filterSet up the low pass filter so that the input is driven by an oscillator of variablefrequency and constant amplitude. Look at the output simultaneously with anoscilloscope and a two phase lock-in amplifier.

R CVin Vout

I

R CVin Vout

I

Lock-In amplifierStanford Research SR830

Function GeneratorStanford Research DS345

OscilloscopeTektronix TDS1012

Low pass filter

BNC cable

BNC adapterBNC adapter

Build the low pass on an electronics breadboard with R=10kΩ and C=10nF.




slide 9

First order system: Example 1, RC low pass filterExplanation 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.




slide 10

First order system: Example 1, RC low pass filter1. 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.

Vou

t(t)

tVin(t)

t

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




slide 11

First order system: Example 1, RC low pass filter3. 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.

Vou

t(t)

tVin(t)

t

T<<RC




slide 12

First order system: Example 2, thermopile4. 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:

ΔT = ΔQ/MCT

Tenv = temp. of environmentA = surface area of thermopileM = mass of thermopileU = thermal conductivity at

thermopile surface

Thermopile consists of material with heat capacity C which converts heat

ΔQ into temperature ΔT

amount of heat emission depends on the temperature difference of the thermopile to its environment = UA(T-Tenv)

V

A thermocouple in the thermopile converts the temperature into a voltage which can be measured)

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 time constant τ = MC/UA

a fully absorbed laser beam with power P introduces heat into the thermopile

thermopilevoltmeter

thermocouple

heat

em

issi

on




slide 13

First order system: Example 2, thermopileExplanation 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.




slide 14

First order system: Example 2, thermopileExplanation 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, thermopile4.1 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:

lasermirror

mirror

AOM

beamexpander

AOM driver Photodiode or thermopile

shutter

The beam modulation is accomplished with an acoustooptic 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.




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




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)

laser AOM expander photodiode

AOM driver

Electrometer(Keithley 614)

Oscilloscope(Tektronix TDS1012)

FunctionGeneratorStanford Research

DS345

(vii)You should have to following setup:




slide 18

First order system: Example 2, thermopile4.2. 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


(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.




slide 20


(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 amplifierStanford Research SR830

OscilloscopeTektronix TDS1012

BNC cable

BNC adapterto AOM driver

BNC adapterto 1010 preamp.




slide 21

First order system: Example 2, thermopile4.3. 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.

physics6770 laboratory 1 week 1, first order systems first...

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