Certainly! Let’s work through the derivation step-by-step: --- ### **1. Understanding the 12-Pulse Rectifier** A **12-pulse rectifier** is often formed by connecting two 6-pulse rectifier bridges (often using transformers with phase-shifted secondaries, e.g., star and delta). This improves the quality of the DC output and reduces harmonics in the current drawn from the AC supply. - **Each 6-pulse bridge** introduces characteristic harmonics of order \( h = 6n \pm 1 \) (where \( n = 1, 2, 3, \ldots \)). - **By phase-shifting and combining two bridges**, certain harmonics are canceled. --- ### **2. Harmonic Orders for a 12-Pulse Rectifier** For an **m-pulse** rectifier, the characteristic harmonics in the AC line current are: \[ h = mp \pm 1, \quad p = 1, 2, 3, \ldots \] For a **12-pulse** rectifier (\( m = 12 \)): \[ h = 12p \pm 1, \quad p = 1, 2, 3, \ldots \] So, the lowest order harmonics are: \( 11, 13, 23, 25, 35, 37, \ldots \) --- ### **3. Primary Phase Current Waveform** The **primary (line) current** \( i(t) \) can be represented as a Fourier series: \[ i(t) = \sum_{h} I_h \sin(h \omega t + \phi_h) \] where \( h \) are the harmonic orders present, \( I_h \) is the amplitude of the \( h \)-th harmonic, \( \omega \) is the supply angular frequency, and \( \phi_h \) is the phase angle. --- ### **4. Harmonic Amplitude Calculation** For a **pulsed rectifier**, the amplitude of the \( h \)-th harmonic (for a generic multi-pulse rectifier) is: \[ I_h = \frac{2 I_{\text{dc}}}{h \pi} \left| \sin\left( \frac{m \pi}{2h} \right) \right| \] - \( I_{\text{dc}} \) is the average DC current. - \( m \) is the pulse number (here, \( m = 12 \)). But for a 12-pulse rectifier, only harmonics where \( h = 12p \pm 1 \) exist. --- ### **5. Final Equation for Harmonic Profile** Thus, the **harmonic content of the primary phase current** is: \[ \boxed{ i(t) = \sum_{p=1}^{\infty} \frac{2 I_{\text{dc}}}{(12p \pm 1) \pi} \sin\left( \frac{12 \pi}{2 (12p \pm 1)} \right) \sin\left[(12p \pm 1) \omega t\right] } \] Where: - \( p = 1, 2, 3, \ldots \) - Only \( h = 12p \pm 1 \) harmonics are present (i.e., 11, 13, 23, 25, ...). - \( I_{\text{dc}} \) is the average DC current. --- ## **Summary Table** | Pulse Number \( m \) | Harmonic Orders in AC Line Current | |---------------------|------------------------------------| | 6 | \( 6n \pm 1 \) (5, 7, 11, 13, ...) | | 12 | \( 12n \pm 1 \) (11, 13, 23, 25, ...) | --- ## **Final Answer** ### **Equation for the harmonic profile of the primary phase current of a 12-pulse rectifier:** \[ \boxed{ i(t) = \sum_{p=1}^{\infty} \frac{2 I_{\text{dc}}}{(12p \pm 1) \pi} \sin\left( \frac{12 \pi}{2 (12p \pm 1)} \right) \sin\left[(12p \pm 1) \omega t\right] } \] - The harmonic orders present are \( h = 12p \pm 1 \), i.e., 11, 13, 23, 25, ... - The amplitude of each harmonic is as shown above. - This equation gives the **harmonic profile** of the primary phase current for a 12-pulse rectifier. --- **References:** - Padiyar, K. R., "HVDC Power Transmission Systems" - Mohan, Undeland & Robbins, "Power Electronics: Converters, Applications, and Design"