Difference between revisions of "Electronics written problems"

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For each of the circuits in the previous problem, find two equivalent circuits &mdash; 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 V<sub>out</sub> the port should be identical to the original circuit.
 
For each of the circuits in the previous problem, find two equivalent circuits &mdash; 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 V<sub>out</sub> the port should be identical to the original circuit.
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==Measuring action potentials==
 
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[[File:Patch clamp circuit model.png|thumb|right|Circuit model of a patch clamp (not including capacitance).]]
 
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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> &Omega;. There is an oscilloscope next to the neuron with an ''input impedance'' of 10<sup>6</sup> &Omega;. A simple model for the oscilloscope is a 10<sup>6</sup> &Omega; 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<sup>-3</sup> V.
 
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* What is the magnitude of the signal the student measures after connecting the oscilloscope?
 
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* Does the student succeed? 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?
 
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* How many times does the student curse during the measurement attempt?
 
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* 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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|[[File:High pass filter.png|350px]]
 
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==Measuring action potentials==
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[[File:Patch clamp circuit model.png|thumb|right|Circuit model of a patch clamp (not including capacitance).]]
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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> &Omega;. There is an oscilloscope next to the neuron with an ''input impedance'' of 10<sup>6</sup> &Omega;. A simple model for the oscilloscope is a 10<sup>6</sup> &Omega; 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<sup>-3</sup> V.
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{{Template:Assignment Turn In|message=
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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'' <math>\left( \frac{V_{patch}}{V_{noise}} \right )^2</math> 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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Revision as of 19:16, 18 October 2018

20.309: Biological Instrumentation and Measurement

ImageBar 774.jpg


This is Part 2 of Assignment 6.

Ideal elements


Pencil.png

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


Resistive circuits


Pencil.png

For each of the circuits below, find the voltage at each node and the current through each element.


1 2
Voltage divider.png Current divider.png
3
Ladder circuit.png

Equivalent circuits


Pencil.png

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.


Easy Bode plots


Pencil.png

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.")


1 2
Low pass filter.png RL Low pass filter.png
3 4
High pass filter.png RL High pass filter.png

Harder Bode plots


Pencil.png

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.


1
Band pass filter.png
2
Second order low pass filter.png

Linear systems


Pencil.png

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.


1 2
Low pass filter.png High pass filter.png

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.


Pencil.png
  • 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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