Which meter has a greater sensitivity and why? Meter A having a range of 0-10 V and a multiplier resistance of 18 kΩ and meter B with a range of 0-300 V and multiplier resistance of 298 kΩ? Both meter movements have a resistance of 2 kΩ.
2 Answers
To determine which meter has greater sensitivity, we'll need to calculate the sensitivity for each meter. Sensitivity in this context refers to the meter's ability to detect small changes in voltage, which can be related to the full-scale deflection (FSD) voltage and the total resistance of the meter's circuit.
### Sensitivity Calculation
Sensitivity is typically expressed as the smallest voltage change that results in a full-scale deflection. To calculate this, we use the following formula:
\[ \text{Sensitivity} = \frac{\text{Full-Scale Voltage}}{\text{Total Resistance}} \]
where the Total Resistance is the sum of the meter movement resistance and the multiplier resistance.
**1. Meter A:**
- **Range:** 0-10 V
- **Multiplier Resistance (R_m):** 18 kΩ
- **Meter Movement Resistance (R_movement):** 2 kΩ
**Total Resistance (R_total_A):**
\[ R_{total_A} = R_{m} + R_{m_{movement}} \]
\[ R_{total_A} = 18\text{kΩ} + 2\text{kΩ} \]
\[ R_{total_A} = 20\text{kΩ} \]
**Sensitivity (S_A):**
\[ S_A = \frac{\text{Full-Scale Voltage}}{R_{total_A}} \]
\[ S_A = \frac{10\text{V}}{20\text{kΩ}} \]
\[ S_A = 0.0005\text{A/V} \]
\[ S_A = 0.5\text{μA/V} \]
**2. Meter B:**
- **Range:** 0-300 V
- **Multiplier Resistance (R_m):** 298 kΩ
- **Meter Movement Resistance (R_movement):** 2 kΩ
**Total Resistance (R_total_B):**
\[ R_{total_B} = R_{m} + R_{m_{movement}} \]
\[ R_{total_B} = 298\text{kΩ} + 2\text{kΩ} \]
\[ R_{total_B} = 300\text{kΩ} \]
**Sensitivity (S_B):**
\[ S_B = \frac{\text{Full-Scale Voltage}}{R_{total_B}} \]
\[ S_B = \frac{300\text{V}}{300\text{kΩ}} \]
\[ S_B = 0.001\text{A/V} \]
\[ S_B = 1\text{μA/V} \]
### Comparison
- **Meter A Sensitivity:** 0.5 μA/V
- **Meter B Sensitivity:** 1 μA/V
**Conclusion:** Meter B has greater sensitivity compared to Meter A. This is because Meter B can detect a smaller change in voltage per unit of full-scale deflection compared to Meter A. This is reflected in its higher sensitivity value.
### Sensitivity Calculation
Sensitivity is typically expressed as the smallest voltage change that results in a full-scale deflection. To calculate this, we use the following formula:
\[ \text{Sensitivity} = \frac{\text{Full-Scale Voltage}}{\text{Total Resistance}} \]
where the Total Resistance is the sum of the meter movement resistance and the multiplier resistance.
**1. Meter A:**
- **Range:** 0-10 V
- **Multiplier Resistance (R_m):** 18 kΩ
- **Meter Movement Resistance (R_movement):** 2 kΩ
**Total Resistance (R_total_A):**
\[ R_{total_A} = R_{m} + R_{m_{movement}} \]
\[ R_{total_A} = 18\text{kΩ} + 2\text{kΩ} \]
\[ R_{total_A} = 20\text{kΩ} \]
**Sensitivity (S_A):**
\[ S_A = \frac{\text{Full-Scale Voltage}}{R_{total_A}} \]
\[ S_A = \frac{10\text{V}}{20\text{kΩ}} \]
\[ S_A = 0.0005\text{A/V} \]
\[ S_A = 0.5\text{μA/V} \]
**2. Meter B:**
- **Range:** 0-300 V
- **Multiplier Resistance (R_m):** 298 kΩ
- **Meter Movement Resistance (R_movement):** 2 kΩ
**Total Resistance (R_total_B):**
\[ R_{total_B} = R_{m} + R_{m_{movement}} \]
\[ R_{total_B} = 298\text{kΩ} + 2\text{kΩ} \]
\[ R_{total_B} = 300\text{kΩ} \]
**Sensitivity (S_B):**
\[ S_B = \frac{\text{Full-Scale Voltage}}{R_{total_B}} \]
\[ S_B = \frac{300\text{V}}{300\text{kΩ}} \]
\[ S_B = 0.001\text{A/V} \]
\[ S_B = 1\text{μA/V} \]
### Comparison
- **Meter A Sensitivity:** 0.5 μA/V
- **Meter B Sensitivity:** 1 μA/V
**Conclusion:** Meter B has greater sensitivity compared to Meter A. This is because Meter B can detect a smaller change in voltage per unit of full-scale deflection compared to Meter A. This is reflected in its higher sensitivity value.
To determine which meter has greater sensitivity, we need to evaluate the sensitivity of each meter. Sensitivity in this context refers to how much the meter's reading changes in response to a small change in voltage.
**Sensitivity** can be understood as the inverse of the meter's full-scale deflection voltage (V_FS), multiplied by the total resistance of the meter. In simpler terms, a meter with a higher sensitivity will show a larger change in reading per unit of voltage applied.
### Calculating Sensitivity
The sensitivity \( S \) of a meter can be defined as:
\[ S = \frac{1}{V_{\text{FS}} \cdot R_{\text{total}}} \]
where:
- \( V_{\text{FS}} \) is the full-scale voltage of the meter.
- \( R_{\text{total}} \) is the total resistance of the meter, including the multiplier resistance and the internal resistance of the meter movement.
#### For Meter A:
- **Range:** 0-10 V
- **Multiplier Resistance:** 18 kΩ
- **Internal Resistance:** 2 kΩ
**Total Resistance \( R_{\text{total}} \)** for Meter A:
\[ R_{\text{total}} = R_{\text{multiplier}} + R_{\text{movement}} = 18 \text{ kΩ} + 2 \text{ kΩ} = 20 \text{ kΩ} \]
**Sensitivity \( S_A \)**:
\[ S_A = \frac{1}{V_{\text{FS}} \cdot R_{\text{total}}} = \frac{1}{10 \text{ V} \cdot 20 \text{ kΩ}} \]
\[ S_A = \frac{1}{200 \text{ kΩ·V}} \]
\[ S_A = 5 \text{ μA/V} \]
#### For Meter B:
- **Range:** 0-300 V
- **Multiplier Resistance:** 298 kΩ
- **Internal Resistance:** 2 kΩ
**Total Resistance \( R_{\text{total}} \)** for Meter B:
\[ R_{\text{total}} = R_{\text{multiplier}} + R_{\text{movement}} = 298 \text{ kΩ} + 2 \text{ kΩ} = 300 \text{ kΩ} \]
**Sensitivity \( S_B \)**:
\[ S_B = \frac{1}{V_{\text{FS}} \cdot R_{\text{total}}} = \frac{1}{300 \text{ V} \cdot 300 \text{ kΩ}} \]
\[ S_B = \frac{1}{90 \text{ MΩ·V}} \]
\[ S_B \approx 11.1 \text{ μA/V} \]
### Comparison
- **Meter A:** Sensitivity \( = 5 \text{ μA/V} \)
- **Meter B:** Sensitivity \( \approx 11.1 \text{ μA/V} \)
**Conclusion:**
Meter B has a greater sensitivity than Meter A. This is because, despite having a larger range, Meter B's higher multiplier resistance results in a lower product of \( V_{\text{FS}} \cdot R_{\text{total}} \), which increases its sensitivity.
**Sensitivity** can be understood as the inverse of the meter's full-scale deflection voltage (V_FS), multiplied by the total resistance of the meter. In simpler terms, a meter with a higher sensitivity will show a larger change in reading per unit of voltage applied.
### Calculating Sensitivity
The sensitivity \( S \) of a meter can be defined as:
\[ S = \frac{1}{V_{\text{FS}} \cdot R_{\text{total}}} \]
where:
- \( V_{\text{FS}} \) is the full-scale voltage of the meter.
- \( R_{\text{total}} \) is the total resistance of the meter, including the multiplier resistance and the internal resistance of the meter movement.
#### For Meter A:
- **Range:** 0-10 V
- **Multiplier Resistance:** 18 kΩ
- **Internal Resistance:** 2 kΩ
**Total Resistance \( R_{\text{total}} \)** for Meter A:
\[ R_{\text{total}} = R_{\text{multiplier}} + R_{\text{movement}} = 18 \text{ kΩ} + 2 \text{ kΩ} = 20 \text{ kΩ} \]
**Sensitivity \( S_A \)**:
\[ S_A = \frac{1}{V_{\text{FS}} \cdot R_{\text{total}}} = \frac{1}{10 \text{ V} \cdot 20 \text{ kΩ}} \]
\[ S_A = \frac{1}{200 \text{ kΩ·V}} \]
\[ S_A = 5 \text{ μA/V} \]
#### For Meter B:
- **Range:** 0-300 V
- **Multiplier Resistance:** 298 kΩ
- **Internal Resistance:** 2 kΩ
**Total Resistance \( R_{\text{total}} \)** for Meter B:
\[ R_{\text{total}} = R_{\text{multiplier}} + R_{\text{movement}} = 298 \text{ kΩ} + 2 \text{ kΩ} = 300 \text{ kΩ} \]
**Sensitivity \( S_B \)**:
\[ S_B = \frac{1}{V_{\text{FS}} \cdot R_{\text{total}}} = \frac{1}{300 \text{ V} \cdot 300 \text{ kΩ}} \]
\[ S_B = \frac{1}{90 \text{ MΩ·V}} \]
\[ S_B \approx 11.1 \text{ μA/V} \]
### Comparison
- **Meter A:** Sensitivity \( = 5 \text{ μA/V} \)
- **Meter B:** Sensitivity \( \approx 11.1 \text{ μA/V} \)
**Conclusion:**
Meter B has a greater sensitivity than Meter A. This is because, despite having a larger range, Meter B's higher multiplier resistance results in a lower product of \( V_{\text{FS}} \cdot R_{\text{total}} \), which increases its sensitivity.