Difference between revisions of "Electronics written problems"

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

Revision as of 02:17, 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


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




Voltage divider.png

Current divider.png

Ladder circuit.png

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.


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.


Simple 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.")




Low pass filter.png

High pass filter.png

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 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.




Band pass filter.png

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.




Low pass filter.png

High pass filter.png