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| * What is the magnitude of the signal the student measures after connecting the oscilloscope? | | * What is the magnitude of the signal the student measures after connecting the oscilloscope? |
| * Does the student succeed? Why or why not? | | * Does the student succeed? Why or why not? |
− | * What is the ''signal to noise power ratio'' <math>\left( \frac{V_{patch}}{V_{noise}} \right )^2</math>? | + | * What is the ''signal to noise power ratio'' <math>\left( \frac{V_{patch}}{V_{noise}} \right )^2</math> of the measurement? |
| * How many times does the student curse during the measurement attempt? | | * How many times does the student curse during the measurement attempt? |
| * What is the minimum input impedance that a measurement device must have in order to make a high-fidelity measurement of an action potential. | | * What is the minimum input impedance that a measurement device must have in order to make a high-fidelity measurement of an action potential. |
Revision as of 02:16, 18 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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Solving 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
Circuit model of a patch clamp (not including capacitance).
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. A simple model for the oscilloscope is a 106 Ω resistor to ground. A new UROP in the lab attempts to measure the electrical spikes produced by the neuron (called action potentials) using the oscilloscope. The oscilloscope has a noise floor of 10-3 V.
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- What is the magnitude of the signal the student measures after connecting the oscilloscope?
- Does the student succeed? Why or why not?
- What is the signal to noise power ratio $ \left( \frac{V_{patch}}{V_{noise}} \right )^2 $ of the measurement?
- How many times does the student curse during the measurement attempt?
- What is the minimum input impedance that a measurement device must have in order to make a high-fidelity measurement of an action potential.
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Simple 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 \gg 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 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.
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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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