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| </center> | | </center> |
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| + | ==Measuring action potentials== |
| + | The [https://en.wikipedia.org/wiki/Patch_clamp 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 10<sup>11</sup> Ω. There is an oscilloscope next to the neuron with an ''input impedance'' of 10<sup>6</sup> 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. The oscilloscope has a noise floor of 10<sup>-3</sup> V. |
| + | [[File:Patch clamp circuit model.png|750px]] |
| + | |
| + | {{Template:Assignment Turn In|message= |
| + | * Neglecting the cable and oscilloscope capacitance, what is the magnitude of V<sub>scope</sub>, 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'' <math>\left( \frac{V_{patch}}{V_{noise}} \right )^2</math> of the measurement (neglecting the capacitance)? |
| + | * Sketch V<sub>neuron</sub> and V<sub>scope</sub> assuming that V<sub>neuron</sub> 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 R<sub>scope</sub> needed to make a high-fidelity measurement of an action potential? |
| + | }} |
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| ==Equivalent circuits== | | ==Equivalent circuits== |
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| </center> | | </center> |
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− | ==Measuring action potentials==
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− | The [https://en.wikipedia.org/wiki/Patch_clamp 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 10<sup>11</sup> Ω. There is an oscilloscope next to the neuron with an ''input impedance'' of 10<sup>6</sup> 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. The oscilloscope has a noise floor of 10<sup>-3</sup> V.
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− | [[File:Patch clamp circuit model.png|750px]]
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− |
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− | {{Template:Assignment Turn In|message=
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− | * Neglecting the cable and oscilloscope capacitance, what is the magnitude of V<sub>scope</sub>, the signal the student measures, after connecting the oscilloscope?
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− | * Is the measurement successful? Why or why not?
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− | * What is the ''signal to noise power ratio'' <math>\left( \frac{V_{patch}}{V_{noise}} \right )^2</math> of the measurement (neglecting the capacitance)?
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− | * Sketch V<sub>neuron</sub> and V<sub>scope</sub> assuming that V<sub>neuron</sub> is a 1 ms duration, square pulse of magnitude 100 mV. You may neglect the oscilloscope's resistance in this part of the problem.
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− | * How many times does the student curse during the measurement attempt?
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− | * Ignoring capacitance, 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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− | }}
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| {{Template:Electronics boot camp part 1 navigation}} | | {{Template:Electronics boot camp part 1 navigation}} |
| {{Template:20.309 bottom}} | | {{Template:20.309 bottom}} |
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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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. The oscilloscope has a noise floor 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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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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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 \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 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.
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1
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2
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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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1
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2
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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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