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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+ | ==Equivalent circuits== | ||
+ | {{Template:Assignment Turn In|message= | ||
+ | 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 V<sub>out</sub> the port should be identical to the original circuit. | ||
+ | }} | ||
==Measuring action potentials== | ==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 [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. |
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+ | 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]] | [[File:Patch clamp circuit model.png|750px]] | ||
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* 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? | * 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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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 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. | ||
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+ | ''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.'' | ||
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==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 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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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 |
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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 |
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3 | 4 |
Harder Bode plots
1 | 2 |
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Linear systems
1 | 2 |
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Second-order system
Find the transfer function $ H(\omega)=\frac{V_{out}}{I_{in}} $ for the circuit below. |
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