1. Fourier Series of a 50% Duty Cycle Square Wave
For a square wave with amplitude A:
\[ x(t) = \frac{4A}{\pi}\left[ \sin(\omega t) + \frac{1}{3}\sin(3\omega t) + \frac{1}{5}\sin(5\omega t) + \cdots \right] \]
Observations:
- Only odd harmonics exist (1st, 3rd, 5th, 7th, ...)
- Amplitude of nth harmonic:
\[ X_n = \frac{4A}{n\pi} \quad (n = 1,3,5,7...) \]
\[ X_n \propto \frac{1}{n} \]
2. Power of Each Harmonic
Power depends on the square of RMS value.
\[ X_{n(rms)} = \frac{X_n}{\sqrt{2}} \]
\[ P_n \propto X_{n(rms)}^2 \]
\[ P_n \propto \left(\frac{1}{n}\right)^2 \]
\[ \boxed{P_n \propto \frac{1}{n^2}} \]
3. Meaning of This Result
- 1st harmonic → Highest power
- 3rd harmonic → 1/9 of fundamental power
- 5th harmonic → 1/25 of fundamental power
- 7th harmonic → 1/49 of fundamental power
Power decreases rapidly as harmonic number increases.
4. Example (Relative Power)
If fundamental power = 100 W:
| Harmonic | Relative Power |
|---|---|
| 1st | 100 W |
| 3rd | 11.1 W |
| 5th | 4 W |
| 7th | 2.04 W |
5. Why Are They Not Equal?
- Fourier coefficient ∝ 1/n
- Power ∝ (coefficient)²
- Therefore, power ∝ 1/n²
Higher-frequency harmonics contain very small power.
Final Conclusion
\[ \boxed{\text{Harmonics of a rectangular pulse do NOT contain equal power.}} \]
\[ \boxed{P_n \propto \frac{1}{n^2}} \]
Harmonic Power in a Sine Wave
For a pure sine wave:
\[ x(t) = A \sin(\omega t) \]
It contains only one frequency component — the fundamental frequency.
1. Harmonic Content of a Sine Wave
- Fundamental (1st harmonic) → Present
- 2nd harmonic → 0
- 3rd harmonic → 0
- 4th harmonic → 0
- All higher harmonics → 0
Therefore, a sine wave has no harmonics beyond the fundamental.
2. Power Distribution
\[ X_{rms} = \frac{A}{\sqrt{2}} \]
If applied to a resistive load \( R \):
\[ P = \frac{X_{rms}^2}{R} \]
\[ P = \frac{A^2}{2R} \]
This is the total power.
\[ P_{total} = P_1 \]
\[ P_2 = P_3 = P_4 = ... = 0 \]
3. Comparison with Square Wave
| Signal Type | Harmonics Present | Power Distribution |
|---|---|---|
| Sine wave | Only fundamental | All power in 1st harmonic |
| Square wave | Infinite odd harmonics | Power decreases as 1/n² |
4. Important Concept
- Zero harmonic distortion
- THD = 0%
- Single frequency component
That is why power systems try to maintain sinusoidal voltage.
Final Conclusion
\[ \boxed{\text{All the power of a sine wave is in the fundamental harmonic only.}} \]