Electronics written problems

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20.309: Biological Instrumentation and Measurement

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This is Part 2 of Assignment 6.

Ideal elements


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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.


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


Easy 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.")


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


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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 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


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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

Second-order system


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


Second-order parallel RLC circuit.png

Measuring action potentials

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 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 10-3 V. Patch clamp circuit model.png


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  • 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 input impedance that a measurement device must have in order to make a high-fidelity measurement of an action potential?