Difference between revisions of "Electronics written problems"

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__NOTOC__
 
__NOTOC__
  
This is Part 2 of [[Electronics boot camp I: passive circuits and transfer functions| Assignment 6]].
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This is Part 1 of [[Electronics boot camp I: passive circuits and transfer functions| Assignment 6]].
 
<br />
 
<br />
  
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For each of the circuits below, find the voltage at each node and the current through each element.
 
For each of the circuits below, find the voltage at each node and the current through each element.
 
}}
 
}}
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<center>
 
{|
 
{|
 
!1
 
!1
 
!2
 
!2
 
|-
 
|-
|[[File:Voltage divider.png|350px]]
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|[[File:Voltage divider.png|250px]]
|[[File:Current divider.png|350px]]
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|[[File:Current divider.png|250px]]
 
|-
 
|-
 
!colspan = "2" |3
 
!colspan = "2" |3
 
|-
 
|-
|colspan = "2" |[[File:Ladder circuit.png|525px]]
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|colspan = "2" |[[File:Ladder circuit.png|375px]]
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|}
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</center>
  
 
==Equivalent circuits==
 
==Equivalent circuits==
 
{{Template:Assignment Turn In|message=  
 
{{Template:Assignment Turn In|message=  
 
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.
 +
}}
 +
 +
==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. One potential problem with the patch clamp technique is that a device must be physically attached to the neuron being measured. Connecting anything to a neuron might alter its behavior. The measurement device itself can ''distort'' the signal. This problem of ''loading'' the system to be measured affects many kinds of measurements (not just electronic ones). In this problem you will consider a simple model of the distortion in a patch clamp measurement.
 +
 +
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; 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<sup>-3</sup> V.
 +
[[File:Patch clamp circuit model.png|750px]]
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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)?
 +
* 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?
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* 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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For each of the circuits below, find the transfer function <math>H(\omega)=\frac{V_{out}}{V_{in}}</math>. 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.")
 
For each of the circuits below, find the transfer function <math>H(\omega)=\frac{V_{out}}{V_{in}}</math>. 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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<center>
 
{|
 
{|
 
!1
 
!1
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| [[File:RL High pass filter.png|250px]]
 
| [[File:RL High pass filter.png|250px]]
 
|}
 
|}
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</center>
  
 
==Harder Bode plots==
 
==Harder Bode plots==
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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>
  
 
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.
 
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, <math>Z_1</math> &hellip; <math>Z_4</math>, and then substituting the particular values for each circuit at the end.''
 
}}
 
}}
 
{|
 
{|
 
|-
 
|-
 
!1
 
!1
|-
 
|[[File:Band pass filter.png|515px]]
 
|-
 
 
!2
 
!2
 
|-
 
|-
|[[File:Second order low pass filter.png|515px]]
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|[[File:Band pass filter.png|375px]]
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|[[File:Second order low pass filter.png|375px]]
 
|}
 
|}
  
 
==Linear systems==
 
==Linear systems==
 
{{Template:Assignment Turn In|message=
 
{{Template:Assignment Turn In|message=
Assuming R<sub>1</sub> = 1 &Omega; and C1 = 1 &mu;Fd, find an equation for <math>V_{out}(t)</math> for each circuit given the following inputs:
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Assuming R<sub>1</sub> = 1 &Omega; and C1 = 1 F, find an equation for <math>V_{out}(t)</math> for each circuit given the following inputs:
* <math>v_{in}(t)=cos( 2 \pi * 0.1 t ) + cos( 2 \pi * 10 * t )</math>  
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* <math>v_{in}(t)=cos( 0.1 t ) + cos( 10 t )</math>  
* <math>v_{in}(t)=cos( 2 \pi t )</math>  
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* <math>v_{in}(t)=cos( t )</math>  
* <math>v_{in}(t)=cos( 2 \pi * 10^{-6} t ) + cos( 2 \pi * 10^6 * t )</math>  
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* <math>v_{in}(t)=cos( 10^{-6} t ) + cos( 10^6 t )</math>  
 
Feel free to make reasonable approximations. You should only get an urge to use a calculator for the first one.
 
Feel free to make reasonable approximations. You should only get an urge to use a calculator for the first one.
 
}}
 
}}
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!2
 
!2
 
|-
 
|-
|[[File:Low pass filter.png|350px]]
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|[[File:Low pass filter.png|250px]]
|[[File:High pass filter.png|350px]]
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|[[File:High pass filter.png|250px]]
 
|}
 
|}
  
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}}
 
}}
 
<center>
 
<center>
[[File:Second-order parallel RLC circuit.png|425px]]
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[[File:Second-order parallel RLC circuit.png|400px]]
 
</center>
 
</center>
 
==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> &Omega;. There is an oscilloscope next to the neuron with an ''input impedance'' of 10<sup>6</sup> &Omega; 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 input impedance that a measurement device must have in order to make a high-fidelity measurement of an action potential?
 
}}
 
  
  
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{{Template:Electronics boot camp part 1 navigation}}
 
{{Template:20.309 bottom}}
 
{{Template:20.309 bottom}}

Latest revision as of 18:45, 15 April 2019

20.309: Biological Instrumentation and Measurement

ImageBar 774.jpg


This is Part 1 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.


Measuring action potentials

The patch clamp is a technique for measuring voltages produced by electrically active cells such as neurons. One potential problem with the patch clamp technique is that a device must be physically attached to the neuron being measured. Connecting anything to a neuron might alter its behavior. The measurement device itself can distort the signal. This problem of loading the system to be measured affects many kinds of measurements (not just electronic ones). In this problem you will consider a simple model of the distortion in a patch clamp measurement.

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. Patch clamp circuit model.png


Pencil.png
  • 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?


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 \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.


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

Linear systems


Pencil.png

Assuming R1 = 1 Ω and C1 = 1 F, find an equation for $ V_{out}(t) $ for each circuit given the following inputs:

  • $ v_{in}(t)=cos( 0.1 t ) + cos( 10 t ) $
  • $ v_{in}(t)=cos( t ) $
  • $ v_{in}(t)=cos( 10^{-6} t ) + cos( 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

Second-order system


Pencil.png

Find the transfer function $ H(\omega)=\frac{V_{out}}{I_{in}} $ for the circuit below.


Second-order parallel RLC circuit.png


Navigation

Back to 20.309 Main Page.