Difference between revisions of "Electronics written problems"
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− | This is Part | + | This is Part 1 of [[Electronics boot camp I: passive circuits and transfer functions| Assignment 6]]. |
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− | == | + | ==Resistive circuits== |
{{Template:Assignment Turn In|message= | {{Template:Assignment Turn In|message= | ||
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. | ||
}} | }} | ||
− | < | + | <center> |
− | + | {| | |
− | [[File:Voltage divider.png| | + | !1 |
− | + | !2 | |
− | + | |- | |
− | [[File:Current divider.png| | + | |[[File:Voltage divider.png|250px]] |
− | + | |[[File:Current divider.png|250px]] | |
− | + | |- | |
− | [[File:Ladder circuit.png| | + | !colspan = "2" |3 |
+ | |- | ||
+ | |colspan = "2" |[[File:Ladder circuit.png|375px]] | ||
+ | |} | ||
+ | </center> | ||
==Equivalent circuits== | ==Equivalent circuits== | ||
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}} | }} | ||
− | ==Measuring | + | ==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> 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. 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]] |
− | * What is the ''signal to noise power ratio''? | + | |
+ | {{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? | * 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? |
+ | }} | ||
− | == | + | ==Easy Bode plots== |
{{Template:Assignment Turn In|message= | {{Template:Assignment Turn In|message= | ||
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.") | ||
}} | }} | ||
− | < | + | <center> |
− | + | {| | |
− | [[File:Low pass filter.png| | + | !1 |
− | + | !2 | |
− | + | |- | |
− | [[File:High pass filter.png| | + | | [[File:Low pass filter.png|250px]] |
+ | | [[File:RL Low pass filter.png|250px]] | ||
+ | |- | ||
+ | !3 | ||
+ | !4 | ||
+ | |- | ||
+ | | [[File:High pass filter.png|250px]] | ||
+ | | [[File:RL High pass filter.png|250px]] | ||
+ | |} | ||
+ | </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 \ | + | * 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 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. | + | 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> … <math>Z_4</math>, and then substituting the particular values for each circuit at the end.'' | ||
}} | }} | ||
− | + | {| | |
− | + | |- | |
− | [[File:Band pass filter.png| | + | !1 |
− | + | !2 | |
− | + | |- | |
− | [[File:Second order low pass filter.png| | + | |[[File:Band pass filter.png|375px]] |
+ | |[[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 Ω and C1 = 1 | + | Assuming R<sub>1</sub> = 1 Ω 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( | + | * <math>v_{in}(t)=cos( 0.1 t ) + cos( 10 t )</math> |
− | * <math>v_{in}(t)=cos( | + | * <math>v_{in}(t)=cos( t )</math> |
− | * <math>v_{in}(t)=cos( | + | * <math>v_{in}(t)=cos( 10^{-6} t ) + cos( 10^6 t )</math> |
− | Feel free to make reasonable approximations. You should only | + | Feel free to make reasonable approximations. You should only get an urge to use a calculator for the first one. |
}} | }} | ||
− | + | {| | |
− | + | |- | |
− | [[File:Low pass filter.png| | + | !1 |
− | < | + | !2 |
− | < | + | |- |
− | [[File: | + | |[[File:Low pass filter.png|250px]] |
+ | |[[File:High pass filter.png|250px]] | ||
+ | |} | ||
+ | |||
+ | ==Second-order system== | ||
+ | {{Template:Assignment Turn In|message= | ||
+ | Find the transfer function <math>H(\omega)=\frac{V_{out}}{I_{in}}</math> for the circuit below. | ||
+ | }} | ||
+ | <center> | ||
+ | [[File:Second-order parallel RLC circuit.png|400px]] | ||
+ | </center> | ||
+ | |||
+ | {{Template:Electronics boot camp part 1 navigation}} | ||
{{Template:20.309 bottom}} | {{Template:20.309 bottom}} |
Latest revision as of 18:45, 15 April 2019
This is Part 1 of Assignment 6.
Ideal elements
Resistive circuits
For each of the circuits below, find the voltage at each node and the current through each element. |
1 | 2 |
---|---|
3 | |
Equivalent circuits
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.
Easy Bode plots
1 | 2 |
---|---|
3 | 4 |
Harder Bode plots
1 | 2 |
---|---|
Linear systems
1 | 2 |
---|---|
Second-order system
Find the transfer function $ H(\omega)=\frac{V_{out}}{I_{in}} $ for the circuit below. |
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