Problem 4 IYPT 2027: 4. Singing capacitor

Problem Description & Analysis

Certain types of ceramic capacitors emit an audible sound when an alternating voltage is applied across them. Investigate the characteristics of the sound produced and how they depend on relevant parameters, such as voltage, frequency, and capacitor properties.

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دامنهٔ ولتاژ V (V) 300 V

فاصلهٔ صفحات d (mm) 0.3 mm

مساحت صفحه A (cm²) 25 cm²

فرکانس تحریک (Hz) 1000 Hz

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Python Code Lab

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Python — Editable

# آزمایش خازن آوازخوان — بلندی صدا برحسب فرکانس تحریک
import numpy as np
import matplotlib.pyplot as plt

# --- بخش ۱: داده‌های شما ---
# فرکانس سیگنال ورودی به خازن (Hz)
f_drive = np.array([100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1200, 1500])
# سطح صوت اندازه‌گیری‌شده با میکروفون (dB SPL)
SPL_dB  = np.array([42, 48, 55, 63, 71, 78, 82, 79, 74, 68, 58, 50])

# --- بخش ۲: نظریه: پاسخ اسیلاتور هارمونیک تحریک‌شده در فرکانس 2f_drive ---
V   = 300.0
d   = 3e-4
A   = 25e-4
eps0 = 8.854e-12
def SHO_response(f, f_res, Q):
    r = 2*f / f_res
    return 1.0 / np.sqrt((1 - r**2)**2 + (r/Q)**2)

# فیت دستی: جستجو بین چند مقدار
best = None
for fr in np.linspace(500, 2000, 60):
    for Qtry in np.linspace(5, 60, 30):
        y = SHO_response(f_drive, fr, Qtry)
        scale = np.max(SPL_dB) - np.max(20*np.log10(y))
        pred  = 20*np.log10(y) + scale
        err   = np.sum((pred - SPL_dB)**2)
        if best is None or err < best[0]:
            best = (err, fr, Qtry, scale, pred)
_, f_res, Q, scale, pred = best

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(figsize=(8, 4.6))
ax.plot(f_drive, SPL_dB, 'ro', ms=9, label='measured')
f_smooth = np.linspace(50, 1600, 400)
y_smooth = 20*np.log10(SHO_response(f_smooth, f_res, Q)) + scale
ax.plot(f_smooth, y_smooth, 'b-', lw=2,
        label=f'fit: f_res={f_res:.0f} Hz, Q={Q:.1f}')
ax.set_xlabel('drive frequency (Hz)')
ax.set_ylabel('SPL (dB)')
ax.set_title('Singing capacitor — SPL vs drive frequency')
ax.legend(); ax.grid(True, alpha=0.3)
plt.tight_layout()

# --- بخش ۴: نتایج ---
print(f"Fitted plate mechanical resonance ≈ {f_res:.0f} Hz")
print(f"Quality factor Q                 ≈ {Q:.1f}")
print(f"Loudest at drive f ≈ f_res/2      ≈ {f_res/2:.0f} Hz (force at 2f)")
F_peak = 0.5*eps0*A*(V/d)**2
print(f"Peak electrostatic force         ≈ {F_peak*1000:.2f} mN")

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Problem Description & Analysis

Interactive Simulation (Python)

Python Code Lab

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