Capacitors Series Combination Explained

This question requires us to determine the maximum safe voltage that can be applied to a series combination of two capacitors with different voltage ratings. Here are the given values:

• Capacitor 1: $C_1 = 2 \, \mu\text{F}$, Maximum Safe Voltage $V_{\text{max,1}} = 10 \, \text{V}$
• Capacitor 2: $C_2 = 4 \, \mu\text{F}$, Maximum Safe Voltage $V_{\text{max,2}} = 20 \, \text{V}$

Understanding Voltage Distribution in Series Capacitors

When capacitors are connected in series, the total voltage applied across the combination is divided among the individual capacitors. The voltage across each capacitor is inversely proportional to its capacitance. This means that the capacitor with the smaller capacitance value will experience a higher voltage across it compared to the capacitor with a larger capacitance value, assuming they are identical otherwise.

The formulas to calculate the voltage across each capacitor ($V_1$ and $V_2$) in a series combination are:

• Voltage across Capacitor 1: $V_1 = V_{\text{total}} \times \frac{C_2}{C_1 + C_2}$
• Voltage across Capacitor 2: $V_2 = V_{\text{total}} \times \frac{C_1}{C_1 + C_2}$

Here, $V_{\text{total}}$ represents the total voltage applied across the series combination.

Calculating the Maximum Safe Working Voltage

For the series combination to operate safely, the voltage across each individual capacitor must not exceed its specified maximum safe working voltage. We need to find the maximum $V_{\text{total}}$ that satisfies both conditions:

• Condition 1: The voltage across Capacitor 1 ($V_1$) must be less than or equal to its maximum rating ($V_{\text{max,1}} = 10 \, \text{V}$).
• Condition 2: The voltage across Capacitor 2 ($V_2$) must be less than or equal to its maximum rating ($V_{\text{max,2}} = 20 \, \text{V}$).

Let's calculate the maximum permissible $V_{\text{total}}$ based on each condition:

Analysis for Capacitor 1:

• The condition is $V_1 \le 10 \, \text{V}$.
• Using the voltage distribution formula: $V_{\text{total}} \times \frac{C_2}{C_1 + C_2} \le 10 \, \text{V}$
• Substitute the given capacitance values: $V_{\text{total}} \times \frac{4 \, \mu\text{F}}{2 \, \mu\text{F} + 4 \, \mu\text{F}} \le 10 \, \text{V}$
• Simplify the fraction: $V_{\text{total}} \times \frac{4}{6} \le 10 \, \text{V}$ or $V_{\text{total}} \times \frac{2}{3} \le 10 \, \text{V}$
• Solve for $V_{\text{total}}$: $V_{\text{total}} \le 10 \, \text{V} \times \frac{3}{2}$
• This gives us: $V_{\text{total}} \le 15 \, \text{V}$

This calculation shows that to protect Capacitor 1, the total applied voltage cannot exceed 15 V.

Analysis for Capacitor 2:

• The condition is $V_2 \le 20 \, \text{V}$.
• Using the voltage distribution formula: $V_{\text{total}} \times \frac{C_1}{C_1 + C_2} \le 20 \, \text{V}$
• Substitute the given capacitance values: $V_{\text{total}} \times \frac{2 \, \mu\text{F}}{2 \, \mu\text{F} + 4 \, \mu\text{F}} \le 20 \, \text{V}$
• Simplify the fraction: $V_{\text{total}} \times \frac{2}{6} \le 20 \, \text{V}$ or $V_{\text{total}} \times \frac{1}{3} \le 20 \, \text{V}$
• Solve for $V_{\text{total}}$: $V_{\text{total}} \le 20 \, \text{V} \times 3$
• This gives us: $V_{\text{total}} \le 60 \, \text{V}$

This calculation shows that to protect Capacitor 2, the total applied voltage cannot exceed 60 V.

Determining the Overall Maximum Safe Voltage

The maximum safe working voltage for the entire series combination is limited by the capacitor that reaches its voltage limit first. This means we must choose the lower of the two calculated voltage limits to ensure both capacitors remain within their safe operating ranges.

Therefore, the maximum safe working voltage for the series combination is the minimum of the limits derived from each capacitor:

• $V_{\text{max, safe, combination}} = \min(V_{\text{limit from C1}}, V_{\text{limit from C2}})$
• $V_{\text{max, safe, combination}} = \min(15 \, \text{V}, 60 \, \text{V})$
• $V_{\text{max, safe, combination}} = 15 \, \text{V}$

So, the maximum safe working voltage for the series combination of the 2 μF/10V and 4 μF/20V capacitors is 15 V.