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| Simplify the transfer functions using the following assumptions: | | Simplify the transfer functions using the following assumptions: |
− | * For the first circuit, assume that <math>R_1 C_1 \gg R_2 C_2</math>, and <math>R_2 \gg R_1</math> | + | * For the first circuit, assume that <math>R_1 C_1 \ll R_2 C_2</math>, and <math>R_2 \gg R_1</math> |
| * For the second circuit, assume that <math>R_1 C_1 = R_2 C_2</math>, and <math>R_2 \gg R_1</math> | | * For the second circuit, assume that <math>R_1 C_1 = R_2 C_2</math>, and <math>R_2 \gg R_1</math> |
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Revision as of 20:10, 31 October 2018
20.309: Biological Instrumentation and Measurement
This is Part 2 of Assignment 6.
Ideal elements
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For each of the ideal, two-terminal elements listed below, show the symbol, label the terminals, indicate the direction of current flow, write the constitutive equation, and find an expression for the impedance, $ Z(\omega)=\frac{V}{I} $. (To find the impedance, substitute $ V=Ae^{j\omega t} $ into the constitutive equation and solve for $ \frac{V}{I} $ as a function of $ \omega $.)
- Resistor
- Capacitor
- Inductor
- Voltage source
- Current source
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Resistive circuits
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For each of the circuits below, find the voltage at each node and the current through each element.
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Equivalent circuits
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For each of the circuits in the previous problem, find two equivalent circuits — the first one consisting of a single voltage source and a single resistor, and the second one consisting of one current source and one resistor. In both equivalent circuits, the I-V curve at the Vout the port should be identical to the original circuit.
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Measuring action potentials
The patch clamp is a technique for measuring voltages produced by electrically active cells such as neurons. A circuit model for a neuron connected to a patch clamp apparatus consists of a time-varying voltage source in series with an output impedance of 1011 Ω. There is an oscilloscope next to the neuron with an input impedance of 106 Ω and an input capacitance of 20 pFd. A new UROP in the lab attempts to measure the electrical spikes produced by a neuron (called action potentials) by connecting the patch clamp apparatus to the oscilloscope with a cable that has a capacitance of 80 pFd. Action potentials are about 100 mV in amplitude and about 1 ms in duration. You can model the noise in the oscilloscope as a random, additive, normally distributed voltage with a standard deviation of 10-3 V.
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- Neglecting the cable and oscilloscope capacitance, what is the magnitude of Vscope, the signal the student measures, after connecting the oscilloscope?
- Is the measurement successful? Why or why not?
- What is the signal to noise power ratio $ \left( \frac{V_{patch}}{V_{noise}} \right )^2 $ of the measurement (neglecting the capacitance)?
- Sketch Vneuron and Vscope assuming that Vneuron is a 1 ms duration, square pulse of magnitude 100 mV. You may neglect the oscilloscope's resistance in this part of the problem.
- How many times does the student curse during the measurement attempt?
- Ignoring capacitance, what is the minimum value of Rscope needed to make a high-fidelity measurement of an action potential?
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Easy Bode plots
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For each of the circuits below, find the transfer function $ H(\omega)=\frac{V_{out}}{V_{in}} $. On a log-log plot, sketch the magnitude of the transfer function versus frequency. Sketch the phase angle of the transfer function versus frequency on a semi-log plot. Suggest a descriptive name for each circuit (e.g. "low-pass filter.")
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Harder Bode plots
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For each of the circuits below, find the transfer function $ H(\omega)=\frac{V_{out}}{V_{in}} $.
Simplify the transfer functions using the following assumptions:
- For the first circuit, assume that $ R_1 C_1 \ll R_2 C_2 $, and $ R_2 \gg R_1 $
- For the second circuit, assume that $ R_1 C_1 = R_2 C_2 $, and $ R_2 \gg R_1 $
On a log-log plot, sketch the magnitude of the simplified transfer function versus frequency. Label cutoff frequencies. Sketch the phase angle of the transfer function versus frequency on a semi-log plot. Suggest a descriptive name for each circuit.
Hint: both circuits have the same topology. You can save yourself a little time by solving the circuit with four generic impedances, $ Z_1 $ … $ Z_4 $, and then substituting the particular values for each circuit at the end.
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Linear systems
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Assuming R1 = 1 Ω and C1 = 1 μFd, find an equation for $ V_{out}(t) $ for each circuit given the following inputs:
- $ v_{in}(t)=cos( 2 \pi * 0.1 t ) + cos( 2 \pi * 10 * t ) $
- $ v_{in}(t)=cos( 2 \pi t ) $
- $ v_{in}(t)=cos( 2 \pi * 10^{-6} t ) + cos( 2 \pi * 10^6 * t ) $
Feel free to make reasonable approximations. You should only get an urge to use a calculator for the first one.
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Second-order system
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Find the transfer function $ H(\omega)=\frac{V_{out}}{I_{in}} $ for the circuit below.
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Navigation
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