Active Filter — Frequency & Phase Response

Experiment: Study of Active Filter Circuits

1. Aim

To study the frequency and phase response of an active band-pass filter built around an operational amplifier, to measure the voltage gain as a function of frequency, to identify the resonant (centre) frequency, and to automate the frequency sweep using a Python program.

2. Apparatus / Components Required

• SEELab3 or ExpEYES-17 unit
• Resistors: $R_1 = 10\text{ k}\Omega$, $R_2 = 1\text{ k}\Omega$, $R_3 = 100\text{ k}\Omega$
• Capacitors: $C_1 = 0.1\text{ }\mu F$, $C_2 = 0.01\text{ }\mu F$
• Operational Amplifier: OP07 (or equivalent precision op-amp) with $\pm 6\text{ V}$ supply
• Breadboard and connecting wires
• PC with Python 3 (libraries: expeyes, numpy, scipy, matplotlib)

3. Theory & Principle

3.1 Filters

A filter is a circuit whose gain depends on the frequency of the input signal. The four basic types are:

• Low-pass filter (LPF): passes frequencies below a cut-off frequency $f_c$; attenuates higher frequencies.
• High-pass filter (HPF): passes frequencies above $f_c$; attenuates lower frequencies.
• Band-pass filter (BPF): passes a band of frequencies centred on a resonant frequency $f_0$; attenuates both below and above.
• Band-stop (Notch) filter: attenuates a specific narrow band of frequencies.

3.2 Active vs Passive Filters

A passive filter uses only R, L, C components. An active filter incorporates an operational amplifier, which provides gain, impedance buffering, and eliminates the loading effect between stages — allowing sharper and more controllable roll-off without the use of bulky inductors.

3.3 Voltage Gain and the Bode Plot

The voltage gain at each frequency is defined as:

Expressed in decibels (dB):

A plot of gain (dB) versus frequency on a logarithmic frequency axis is called a Bode plot. The frequency at which the gain drops to $\frac{1}{\sqrt{2}}$ of its peak value (i.e., $-3\text{ dB}$ below the peak) defines the bandwidth edges $f_1$ and $f_2$. The SEELab app plots the amplitude ratio of output and input signals vs frequency.

3.4 The Designed Circuit

The active band-pass filter in this experiment uses an OP07 op-amp with the following component values:

The circuit was designed using the Okawa Online Filter Design Tool. The theoretical resonant frequency is:

Design constraint: Choose component values such that the peak gain $A_v < 1$ (i.e., $< 0\text{ dB}$) to prevent output clipping. The WG frequency range of SEELab3/ExpEYES-17 is 5 Hz to 5000 Hz — all measurements must lie within this range.

3.5 Phase Response

In addition to amplitude, the output signal is phase-shifted relative to the input by an angle $\phi(f)$ that varies with frequency. At the resonant frequency of a band-pass filter, the phase shift is $0°$. Below $f_0$ the output leads the input; above $f_0$ it lags. Extracting phase requires fitting the time-domain waveforms to a sinusoidal model.

4. Circuit Diagram / Setup

1. Assemble the active band-pass filter on a breadboard using the component values listed above. Refer to the schematic displayed in the SEELab3 app.
2. Power the op-amp from the $\pm 12\text{ V}$ supply rails (or the available dual supply). Connect GND of the supply to the SEELab3 GND.
3. WG → Filter input: Connect the Wave Generator output (WG) to both the filter input and channel A2 (to monitor $V_{in}$).
4. Filter output → A2: Connect the op-amp output to channel A1 (to monitor $V_{out}$).

Steady State Response (Phone App)

Steady State Setup For ExpEYES17

5. Procedure

5a. Manual / App-Based Measurement

1. Open the SEELab3 app and select the “Filter” experiment.
2. Screen 1 — Waveform View: Set an initial frequency of $\approx 100\text{ Hz}$. The display shows $V_{in}$ (A1) and $V_{out}$ (A2) simultaneously. Slide vertically on the WG icon to change the frequency.
3. Sweep the frequency from $\approx 100\text{ Hz}$ to $5000\text{ Hz}$ and observe how the output amplitude changes. Note the frequency at which $V_{out}$ is maximum — this is close to $f_0$.
4. If output amplitude clips anywhere within the range, reduce input signal to 1V amplitude.
5. Screen 2 — Record Data: Enter the start frequency, stop frequency, and step size. Click “Record”. The software automatically sweeps through all frequencies and plots:
   • Gain plot: $A_v$ (or dB) vs $f$
   • Phase plot: $\phi$ vs $f$
6. Read off the peak frequency ($f_0$), the $-3\text{ dB}$ bandwidth edges ($f_1$, $f_2$), and the peak gain from the plot.

5b. Python Automation

The Python program replicates the manual sweep programmatically:

# filter-resp.py
# Measures voltage gain vs frequency and saves to CSV

import expeyes.eyesj as eyes
import numpy as np
import time

p = eyes.open()          # connect to SEELab3 / ExpEYES

start_freq  = 100        # Hz
stop_freq   = 5000       # Hz
step_freq   = 50         # Hz
amplitude   = 80         # WG amplitude (0–100 %)

results = []             # list to collect [freq, Vin, Vout, gain]

freq = start_freq
while freq <= stop_freq:
    p.set_sine(freq, amplitude)          # set WG frequency
    time.sleep(0.1)                      # allow signal to settle
    Vin  = p.get_voltage('A1')           # measure input amplitude
    Vout = p.get_voltage('A2')           # measure output amplitude
    gain = Vout / Vin if Vin != 0 else 0
    results.append([freq, Vin, Vout, gain])
    print(f"f={freq:5d} Hz  Vin={Vin:.3f}  Vout={Vout:.3f}  Gain={gain:.4f}")
    freq += step_freq

# Save data to CSV
with open('filter-resp.csv', 'w') as f:
    f.write('frequency,Vin,Vout,gain\n')
    for row in results:
        f.write(','.join(f'{v:.6f}' for v in row) + '\n')

print("Data saved to filter-resp.csv")
p.close()

The analysis program reads the saved CSV and fits a Gaussian curve to the gain-frequency data to extract $f_0$ and bandwidth:

# filter-resp-analyse.py
# Reads CSV, plots frequency response, fits Gaussian to gain peak

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

data = np.loadtxt('filter-resp.csv', delimiter=',', skiprows=1)
freq = data[:, 0]
gain = data[:, 3]

# Gaussian model: A * exp( -(f - f0)^2 / (2*sigma^2) )
def gaussian(f, A, f0, sigma):
    return A * np.exp(-0.5 * ((f - f0) / sigma) ** 2)

# Initial guesses
p0 = [max(gain), freq[np.argmax(gain)], 200]
popt, _ = curve_fit(gaussian, freq, gain, p0=p0)
A_fit, f0_fit, sigma_fit = popt

# -3 dB bandwidth: for Gaussian, BW ≈ 2 * sqrt(2*ln2) * sigma
bw = 2 * np.sqrt(2 * np.log(2)) * abs(sigma_fit)

print(f"Peak gain  : {A_fit:.4f}")
print(f"Centre freq: {f0_fit:.1f} Hz")
print(f"Bandwidth  : {bw:.1f} Hz")
print(f"Q factor   : {f0_fit / bw:.2f}")

plt.figure(figsize=(8, 4))
plt.plot(freq, gain, 'o', markersize=4, label='Measured')
plt.plot(freq, gaussian(freq, *popt), '-', label=f'Gaussian fit  $f_0$={f0_fit:.1f} Hz')
plt.axhline(A_fit / np.sqrt(2), color='gray', linestyle='--', label='-3 dB level')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Voltage Gain $A_v$')
plt.title('Active Band-Pass Filter — Frequency Response')
plt.legend()
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('freq-resp.png', dpi=150)
plt.show()

6. Observation Table

6a. Manual Frequency Sweep

6b. Bandwidth and Quality Factor

7. Results and Discussion

• The filter exhibited a band-pass characteristic with maximum gain at $f_0 =$ ____ Hz, against a theoretical value of $527.8\text{ Hz}$.
• The peak voltage gain was $A_{v,\text{max}} =$ ____ (____ dB), which is less than unity as designed.
• The $-3\text{ dB}$ bandwidth was found to be $\Delta f =$ ____ Hz, giving a quality factor $Q =$ ____.
• The Gaussian fit to the Python-acquired data gave $f_0 =$ ____ Hz and $Q =$ ____, in close agreement with the app-based measurement.
• The phase response crossed $0°$ at the resonant frequency, confirming band-pass behavior.
• If the 100K is replaced with 20K ohms, the filter exhibited a band-pass characteristic with maximum gain at $f_0 =$ ____ Hz, against a theoretical value of $____\text{ Hz}$.

8. Precautions

1. Gain less than unity: Design the filter so the peak gain $A_v < 1$. A gain $> 1$ will cause the op-amp output to clip against the supply rails, distorting the waveform and making amplitude measurements unreliable.
2. Frequency range: The WG of SEELab3/ExpEYES-17 operates only between 5 Hz and 5000 Hz. Design $f_0$ well within this range; a value near $500\text{ Hz}$ is ideal to allow observation of both roll-off slopes.
3. Op-amp supply: Ensure the OP07 is powered from a stable dual supply ($\pm 12\text{ V}$ or $\pm 9\text{ V}$) with the supply GND tied to the SEELab3 GND. A floating supply ground will corrupt all measurements.
4. Settle time in Python sweep: Insert a short delay (time.sleep) after changing the WG frequency before measuring amplitudes, so the circuit reaches steady state. Too short a delay at low frequencies introduces significant error.
5. Step size selection: Use a fine step size (e.g., $20$–$50\text{ Hz}$) near the resonant frequency to capture the peak accurately; a coarser step is sufficient at the extremes of the sweep.

9. Troubleshooting