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Single Stage High Pass circuit

Build the Single stage RC circuit shown in Fig. 3, with R=10kΩ, C=0.47μFR = 10k\Omega ,\space C = 0.47\mu F.
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Fig. 3: (left) schematic of the single stage RC circuit, (right) implementation on breadboard

Analysis

The claimed transfer function of this circuit is
T(f)= Vout(f)Vin(f)=j2πfRC1+j2πfRC #(2)\begin{split}\begin{matrix} T(f) = \ \frac{V_{out}(f)}{V_{in}(f)} = \frac{j2\pi fRC}{1 + j2\pi fRC}\ \#(2) \\ \end{matrix}\end{split}
Where j=−1j=\sqrt -1 is the imaginary unit.
  1. What is the magnitude of the transfer function?
  1. What is the phase response of the circuit?
  1. What class (low-pass, high-pass, band-pass, band-stop) of filter is this?
  1. What is the -3dB frequency?

Measurement

Using the Red Pitaya’s Bode Analyzer tool, measure the frequency response (|T(f)|) as described in section 3.1.2.
  1. Show the plot of the measurement below:

Comparison

Respond to the following questions:
  1. Does the shape of the frequency response match your expectation from the analysis? Is there any point that stands out as odd?
  1. Find the -3dB point in the circuit, and compare this value to the one you previously calculated.

Single stage RC circuit – Unknown parameter estimation

Build the Single stage RC circuit shown in Fig. 4, with the potentiometer and C=4.7nFC=4.7nF. Use another resistor to provide electrical contact. Ensure that the potentiometer pins used are the two furthest pins, as this will be the total resistance of the device.
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Fig. 4: (left) schematic of the single stage RC circuit, (right) implementation on breadboard

Analysis

The claimed transfer function of this circuit is the same as in 3.1 (reprinted here for courtesy)
T(f)= Vout(f)Vin(f)=11+j2πfRC \begin{split}\begin{matrix} T(f) = \ \frac{V_{out}(f)}{V_{in}(f)} = \frac{1}{1 + j2\pi fRC}\ \\ \end{matrix}\end{split}
Where  j=−1j=\sqrt -1 is the imaginary unit. However now the value of R is unknown. Since we already know the expected behavior of the system, we can estimate the value of R by measuring the transfer function again.
  1. Derive the expression for the -3dB frequency as a function of R.

Measurement

Using the Red Pitaya’s Bode Analyzer tool, measure the frequency response (|T(f)|) as described in section 3.1.2. Pay special attention to include the cutoff frequency in the sweep.
  1. Show the plot of the measurement below:

Comparison

Respond to the following questions:
  1. Use the expression you derived to calculate the value of R from the measured value of fcf_c.
To calculate the value of RR from the measured value of fcf_c, we can rearrange the transfer function equation as follows: T(f)=11+j2pifRCT(f) = \frac{1}{1 + j2pi fRC}. Solving for RR, we get R=12pifcCR = \frac{1}{2pi f_cC} . Substitute the measured value of fcf_c and the known value of CC to obtain the calculated value of RR.
  1. The previous analysis all presumed we knew the value of f,Cf, C perfectly. In reality, the values of there are only approximately known.
    1. If the capacitance value CC can vary ±20%, what is the bounds on the error of the calculated value of RR?
    2. If the capacitance value varies by ±20\pm 20%, the bounds on the error of the calculated value of RR can be determined by considering the sensitivity of RR with respect to CC. Using the formula for sensitivity, the upper bound of the error in RR can be calculated as ΔR=∣dRdC∣⋅0.2C\Delta R = \left| \frac{dR}{dC} \right| \cdot 0.2C, where dRdC\frac{dR}{dC} is the derivative of RR with respect to CC. Similarly, the lower bound can be obtained by considering the negative change (-20%) in capacitance.
      b. If the frequency f value can vary ±0.1%, what is the bounds on the error of the calculated value of R?
      If the frequency value varies by ±\pm 0.1%, the bounds on the error of the calculated value of RR can be determined by considering the sensitivity of RR with respect to ff. Using the formula for sensitivity, the upper bound of the error in RR can be calculated as ΔR=∣dRdf∣⋅0.001f\Delta R = \left| \frac{dR}{df} \right| \cdot 0.001f, where dRdf\frac{dR}{df} is the derivative of RR with respect to ff. Similarly, the lower bound can be obtained by considering the negative change (-0.1%) in frequency.
      c. If the both C,fC,f as above simultaneously, what is the total bounding on the error of the calculated value of  RR? (Hint: This should be a rectangular area)
      If both CC and ff vary simultaneously, the total bounding on the error of the calculated value of RR can be obtained by combining the individual bounds calculated in parts (a) and (b). The total bounding will form a rectangular area defined by the maximum and minimum values of the error in RR due to the variations in CC and ff.
  1. (Optional) In the same line of thought, assume that the values of CC and ff are described statistically by gaussian distributions with mean and variances provided below:
C∼N(4.7,1) nFC \sim \mathcal{N}(4.7,1)\ \text{nF} \\f∼N(fC,1) Hzf \sim \mathcal{N}\left( f_{C},1 \right)\ \text{Hz}
If CC and ff are described by Gaussian distributions with means and variances, the resulting probability distribution of RR can be calculated by applying the rules of propagation of uncertainty. By considering the distributions of CC and ff as well as their respective derivatives with respect to RR, the probability distribution of RR can be obtained. This distribution will provide information about the likelihood of different values of RR based on the statistical characteristics of CC and ff.