20.309: Biological Instrumentation and Measurement
This is Part 1 of Assignment 6.
Ideal elements

For each of the ideal, twoterminal 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

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

Equivalent circuits

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 IV curve at the V_{out} 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 timevarying voltage source in series with an output impedance of 10^{11} Ω. There is an oscilloscope next to the neuron with an input impedance of 10^{6} 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.

 Neglecting the cable and oscilloscope capacitance, what is the magnitude of V_{scope}, 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 V_{neuron} and V_{scope} assuming that V_{neuron} 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 R_{scope} needed to make a highfidelity measurement of an action potential?

Easy Bode plots

For each of the circuits below, find the transfer function $ H(\omega)=\frac{V_{out}}{V_{in}} $. On a loglog plot, sketch the magnitude of the transfer function versus frequency. Sketch the phase angle of the transfer function versus frequency on a semilog plot. Suggest a descriptive name for each circuit (e.g. "lowpass filter.")

Harder Bode plots

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 loglog 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 semilog 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.

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Linear systems

Assuming R_{1} = 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.

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Secondorder system

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

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