Let's break down the problem step by step: **Given:** - Power factor of fault, \( \text{pf} = 0.4 \) - Recovery voltage = 0.90 × full line value - Breaking current is **symmetrical** - Frequency of restriking voltage, \( f = 16\,\text{kHz} \) - Circuit breaker: 3-pole, 132 kV - Neutral is grounded, **fault does not involve ground** (so, phase-to-phase fault) - **To find:** Average rate of rise of restriking voltage (RRRV) --- ## **Step 1: Find the Recovery Voltage (\( V_{rec} \))** Since the fault does **not** involve ground and the neutral is grounded, **phase-to-phase** voltage applies. For a 3-phase system: \[ \text{Line-to-line voltage (V_{LL})} = 132\,\text{kV} \] The **recovery voltage** is given as **0.90 times the full-line value**: \[ V_{rec} = 0.90 \times V_{LL} = 0.90 \times 132\,\text{kV} = 118.8\,\text{kV} \] --- ## **Step 2: Find the Peak Value of Recovery Voltage** Restriking voltage is generally considered **across the contacts**, so **peak value** is used: \[ V_{m} = \sqrt{2} \times V_{rec} = \sqrt{2} \times 118.8\,\text{kV} = 1.414 \times 118.8 = 168.0\,\text{kV} \] But sometimes the RMS value is used, but for **RRRV calculation**, **use recovery voltage as given** (i.e., 118.8 kV), unless the question specifies otherwise. --- ## **Step 3: Find the Time for First Rise (Quarter Cycle of Oscillation)** Restriking voltage rises **from 0 to maximum** in **quarter of one cycle** of the oscillation. **Frequency of oscillation, \( f = 16\,\text{kHz} \):** \[ T = \frac{1}{f} = \frac{1}{16,000} = 62.5\,\mu\text{s} \] **Quarter-cycle time:** \[ t_{q} = \frac{T}{4} = \frac{62.5}{4} = 15.625\,\mu\text{s} \] --- ## **Step 4: Calculate the Average Rate of Rise of Restriking Voltage (RRRV)** \[ \text{Average RRRV} = \frac{\text{Recovery Voltage}}{\text{Quarter-cycle time}} \] \[ \text{Average RRRV} = \frac{118,800\,\text{V}}{15.625 \times 10^{-6}\,\text{s}} = 7,603,200,000\,\text{V/s} \] Or, \[ \boxed{7.6 \times 10^6\,\text{V/ms}} \] Or, \[ \boxed{7.6\,\text{kV}/\mu\text{s}} \] Or, \[ \boxed{7.6\,\text{MV/ms}} \] --- ### **Final Answer (with explanation):** **Step-by-step summary:** 1. Recovery voltage across CB = \( 0.9 \times 132\,\text{kV} = 118.8\,\text{kV} \). 2. Frequency of restriking oscillation = \( 16\,\text{kHz} \), so quarter cycle = \( 15.625\,\mu\text{s} \). 3. Average RRRV = \( \frac{118,800}{15.625 \times 10^{-6}} = 7.6 \times 10^6\,\text{V/ms} \) or \( 7.6\,\text{kV}/\mu\text{s} \). --- ## **Final Answer:** \[ \boxed{ \text{The average rate of rise of restriking voltage (RRRV) is } 7.6\,\text{kV}/\mu\text{s} \text{ or } 7.6 \times 10^6\,\text{V/ms} \text{ or } 7.6\,\text{MV/ms} } \]