Radar Principles: Measure the Transformer Hum and its Frequency
Undergraduate Electrical Engineering Course
Upon successful completion of this virtual laboratory experiment, the student will be able to:
Power transformers produce a characteristic audible hum during operation. This acoustic noise originates primarily from two physical phenomena:
Transformer cores are constructed from laminated silicon steel. When subjected to an alternating magnetic field, these ferromagnetic materials experience minute dimensional changes — a phenomenon called magnetostriction. The core material expands and contracts twice during each full cycle of the AC supply because the magnetic flux reaches peak magnitude twice per cycle (positive and negative peaks).
Therefore, for a 50 Hz power system, the fundamental hum frequency is 100 Hz. For a 60 Hz system, it is 120 Hz. These frequencies fall within the human audible range (20 Hz – 20 kHz).
Current-carrying windings experience Lorentz forces proportional to the product of current and flux density. Since both current and flux vary sinusoidally at the line frequency, the force varies at twice the line frequency, again producing 100 Hz (or 120 Hz) vibration. Additionally, loose laminations or clamping structures can resonate at harmonic frequencies.
Real transformer hum is not a pure sinusoid. Due to the non-linear B-H characteristic of the core material and saturation effects, the vibration waveform contains significant harmonic components:
| Harmonic | 50 Hz System | 60 Hz System | Typical Relative Amplitude |
|---|---|---|---|
| Fundamental (2nd electrical harmonic) | 100 Hz | 120 Hz | 1.00 (0 dB) |
| 2nd harmonic (4th electrical) | 200 Hz | 240 Hz | 0.10 to 0.30 (-20 to -10 dB) |
| 3rd harmonic (6th electrical) | 300 Hz | 360 Hz | 0.05 to 0.15 (-26 to -16 dB) |
| 4th harmonic (8th electrical) | 400 Hz | 480 Hz | 0.02 to 0.08 (-34 to -22 dB) |
Using an oscilloscope, measure the period T between consecutive peaks of the acoustic waveform. The frequency is:
Accuracy limitation: Dependent on time-base calibration and visual interpolation. Typical accuracy: ±1% to ±2%.
A digital frequency counter counts the number of zero-crossings (or positive peaks) within a known gate time tg:
Accuracy limitation: ±1 count error (±1/tg Hz). For tg = 1 s, resolution is 1 Hz.
The Fast Fourier Transform converts a time-domain signal x(t) into its frequency-domain representation X(f):
Key parameters:
Radar Connection: FFT is the backbone of modern pulse-Doppler radar, synthetic aperture radar (SAR), and frequency-modulated continuous-wave (FMCW) radar for target detection and velocity estimation.
Why study transformer hum in a radar course? The fundamental signal processing chain is identical:
Transformer hum provides a safe, low-frequency, easily observable platform to master these techniques before applying them to GHz-range radar signals.
In radar, a moving target shifts the frequency of the returned echo by the Doppler frequency:
where vr is radial velocity, fc is carrier frequency, and c is the speed of light. Precise measurement of small frequency shifts (often Hz or sub-Hz) requires the same FFT-based spectral estimation techniques used to resolve closely-spaced harmonics in transformer hum analysis.
Power Transformer Model
Core vibration frequency = 2 × Line Frequency
Your laboratory report must be organized into the following sections:
| Criterion | Weight | Expectations |
|---|---|---|
| Theoretical Understanding | 20% | Correct equations, clear explanations, radar connection established |
| Experimental Procedure | 15% | Clear, reproducible steps; all four experiments completed |
| Data Presentation | 20% | Tables, graphs, screenshots are labeled, accurate, and well-organized |
| Analysis & Discussion | 25% | Critical thinking, error analysis, meaningful radar analogies |
| Report Quality | 10% | Grammar, formatting, professional appearance |
| Quiz Performance | 10% | Post-lab quiz score (Section 6) |
Answer all questions. After submitting, review the correct answers and explanations to reinforce your understanding.
Correct Answer: b) Magnetostriction of the core laminations
Explanation: Magnetostriction is the reversible dimensional change in ferromagnetic materials when subjected to a magnetic field. In transformers, the core laminations expand and contract twice per AC cycle, generating the characteristic hum at twice the line frequency. While electromagnetic forces (Lorentz) also contribute, magnetostriction is the dominant mechanism, typically accounting for 60–80% of the total acoustic energy.
Correct Answer: c) 120 Hz
Explanation: Because magnetostrictive strain is proportional to the square of flux density (or follows an even-symmetric curve), the core experiences maximum compression/extension at both the positive and negative peaks of the AC cycle. Thus, the mechanical vibration occurs at twice the electrical line frequency: fhum = 2 × 60 Hz = 120 Hz.
Correct Answer: d) Even harmonics of the line frequency (100, 200, 300 Hz...)
Explanation: The magnetostriction curve is approximately an even function of flux density B. Since B(t) = Bmaxsin(ωt), the strain ε ∝ B² contains only DC and 2ω components. Therefore, the resulting vibration spectrum contains the fundamental hum (2f) and its integer multiples (4f, 6f, 8f...), which correspond to even harmonics of the line frequency. Odd line-frequency harmonics are suppressed in balanced three-phase units.
Correct Answer: b) The sampling rate must be at least twice the maximum frequency present
Explanation: The Nyquist-Shannon sampling theorem states that to avoid aliasing, a signal must be sampled at a rate fs strictly greater than twice its maximum frequency component (fs > 2fmax). In practice, fs ≥ 2.5fmax is often used to allow for non-ideal anti-aliasing filter roll-off. This principle is equally critical in radar digital receivers to prevent target ambiguity.
Correct Answer: a) Rectangular
Explanation: The rectangular window (equivalent to no windowing) has a main lobe width of 2 bins (zero-to-zero) but exhibits side lobes at only -13 dB relative to the peak. This creates significant spectral leakage. Hanning, Hamming, and Blackman windows progressively widen the main lobe (2, 2, 3 bins respectively at -3dB) but suppress side lobes to -31 dB, -41 dB, and -57 dB, reducing leakage. This trade-off is fundamental in radar for resolving closely-spaced targets vs. detecting weak targets near strong clutter.
Correct Answer: c) Target radial velocity
Explanation: The Doppler effect causes the frequency of the radar echo to shift proportionally to the radial velocity component of the target: fd = 2vrfc/c. By measuring this frequency shift using the same FFT-based spectral analysis techniques practiced in this lab, radar systems determine whether a target is approaching (positive shift) or receding (negative shift), and at what speed. Range is determined by time-of-flight, not Doppler.
Correct Answer: b) 102 Hz
Explanation: The frequency corresponding to FFT bin k is calculated as f = k × Δf, where Δf is the frequency resolution. Here, f = 51 × 2 Hz = 102 Hz. This is consistent with a 50 Hz line frequency system where the fundamental hum is 100 Hz; the small discrepancy (2 Hz) illustrates the quantization error inherent in finite-length FFT analysis. In radar, this same bin-to-frequency mapping is used to convert spectral peaks to velocity estimates.
Correct Answer: d) Spectral leakage spreads energy across multiple bins
Explanation: When the FFT window length is not an exact integer multiple of the signal period, the periodic extension implicit in the FFT creates discontinuities at the window boundaries. These discontinuities introduce additional frequency components, causing the signal energy to "leak" into adjacent bins. This is why window functions (Hanning, Hamming, etc.) are applied — they taper the signal to zero at the edges, minimizing the discontinuity and reducing leakage, though at the cost of slightly reduced frequency resolution.
Correct Answer: a) 300 Hz
Explanation: The fundamental hum frequency in a 50 Hz system is 100 Hz (2 × 50 Hz). Harmonics are integer multiples of this fundamental. Therefore, the 3rd harmonic is 3 × 100 Hz = 300 Hz. Note that this corresponds to the 6th electrical harmonic of the line frequency. The 150 Hz option (3rd electrical harmonic) is a common distractor but is typically absent in transformer hum spectra due to the even-symmetric nature of magnetostriction.
Correct Answer: c) Increasing the total observation time (more samples at the same rate)
Explanation: FFT frequency resolution is given by Δf = fs/N = 1/T, where T is the total observation time. Resolution improves only by increasing T. Option (a) increases fs and N proportionally, leaving Δf unchanged. Option (b) worsens resolution. Option (d) changes the window shape but not the underlying resolution limit. In radar, longer coherent integration times (equivalent to more samples) directly improve Doppler resolution, enabling finer velocity discrimination.
Correct Answer: b) Approaching, because the received frequency increased
Explanation: A positive Doppler shift (higher received frequency) indicates the target is moving toward the radar, compressing the wavelength of the reflected wave. Conversely, a negative shift indicates recession. The magnitude can be used to calculate radial velocity: vr = fd·c / (2fc) = (2000)(3×10⁸) / (2×10×10⁹) = 30 m/s. This frequency-domain measurement principle is identical to identifying the dominant peak in the transformer hum spectrum.
Correct Answer: d) Both rely on identical spectral analysis and frequency measurement techniques
Explanation: While transformer hum (≈100 Hz acoustic) and radar echoes (MHz–GHz electromagnetic) differ vastly in physical nature and frequency, the signal processing pipeline is conceptually identical: signal acquisition → amplification → ADC → FFT spectral analysis → peak detection → frequency/velocity estimation. Transformer hum provides a safe, audible, low-frequency platform to master FFT windowing, resolution, leakage, and SNR concepts before applying them to radar Doppler processing, where these same principles determine target detection and tracking performance.