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Single stage High Pass circuit

Analysis

The claimed transfer function of this circuit is

\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 $j=\sqrt -1$ is the imaginary unit.

What is the magnitude of the transfer function?
What is the phase response of the circuit?
What class (low-pass, high-pass, band-pass, band-stop) of filter is this?
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.

Show the plot of the measurement below:

Comparison

Respond to the following questions:

Does the shape of the frequency response match your expectation from the analysis? Is there any point that stands out as odd?
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 $C=4.7nF$ . 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.

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)

\begin{split}\begin{matrix} T(f) = \ \frac{V_{out}(f)}{V_{in}(f)} = \frac{1}{1 + j2\pi fRC}\ \\ \end{matrix}\end{split}

Where $j=\sqrt -1$ 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.

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.

Show the plot of the measurement below:

Comparison

Respond to the following questions:

Use the expression you derived to calculate the value of R from the measured value of $f_c$ .

To calculate the value of $R$ from the measured value of $f_c$ , we can rearrange the transfer function equation as follows: $T(f) = \frac{1}{1 + j2pi fRC}$ . Solving for $R$ , we get $R = \frac{1}{2pi f_cC}$ . Substitute the measured value of $f_c$ and the known value of $C$ to obtain the calculated value of $R$ .

The previous analysis all presumed we knew the value of $f, C$ perfectly. In reality, the values of there are only approximately known.

If the capacitance value $C$ can vary ±20%, what is the bounds on the error of the calculated value of $R$ ?

If the capacitance value varies by $\pm 20$ %, the bounds on the error of the calculated value of $R$ can be determined by considering the sensitivity of $R$ with respect to $C$ . Using the formula for sensitivity, the upper bound of the error in $R$ can be calculated as $\Delta R = \left| \frac{dR}{dC} \right| \cdot 0.2C$ , where $\frac{dR}{dC}$ is the derivative of $R$ with respect to $C$ . 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 $R$ can be determined by considering the sensitivity of $R$ with respect to $f$ . Using the formula for sensitivity, the upper bound of the error in $R$ can be calculated as $\Delta R = \left| \frac{dR}{df} \right| \cdot 0.001f$ , where $\frac{dR}{df}$ is the derivative of $R$ with respect to $f$ . Similarly, the lower bound can be obtained by considering the negative change (-0.1%) in frequency.

c. If the both $C,f$ as above simultaneously, what is the total bounding on the error of the calculated value of $R$ ? (Hint: This should be a rectangular area)

If both $C$ and $f$ vary simultaneously, the total bounding on the error of the calculated value of $R$ 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 $R$ due to the variations in $C$ and $f$ .

(Optional) In the same line of thought, assume that the values of $C$ and $f$ are described statistically by gaussian distributions with mean and variances provided below:

C \sim \mathcal{N}(4.7,1)\ \text{nF} \\

f \sim \mathcal{N}\left( f_{C},1 \right)\ \text{Hz}

If $C$ and $f$ are described by Gaussian distributions with means and variances, the resulting probability distribution of $R$ can be calculated by applying the rules of propagation of uncertainty. By considering the distributions of $C$ and $f$ as well as their respective derivatives with respect to $R$ , the probability distribution of $R$ can be obtained. This distribution will provide information about the likelihood of different values of $R$ based on the statistical characteristics of $C$ and $f$ .