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 time constant 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 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. 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 t trivial solution of ) ( 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<<RC University of Utah PHYSICS6770 Laboratory 1 Week 1, First order systems First and second order systems 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: A thermocouple in the thermopile converts the temperature into a Thermopile consists of material with voltage which can be measured) heat capacity C which converts heat thermocouple ΔQ into temperature ΔT voltmeter thermopile T V ΔT = ΔQ/MC a fully absorbed laser beam with power Tenv = temp. of environment P introduces heat into the thermopile A = surface area of thermopile amount of heat emission M = mass of thermopile depends on the U = thermal conductivity at temperature difference of thermopile surface the thermopile to its 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.