To repair a power supply for a stereo amplifier, an...

To repair a power supply for a stereo amplifier, an electronics technician needs a $100-\mu \mathrm{F}$ capacitor capable of withstanding a potential difference of $90 \mathrm{V}$ between the plates. The only available supply is a box of five $100-\mu \mathrm{F}$ capacitors, each having a maximum voltage capability of $50 \mathrm{V} .$ Can the technician substitute a combination of these capacitors that has the proper electrical characteristics? If so, what will be the maximum voltage across any of the capacitors used? (Suggestion: The technician may not have to use all the capacitors in the box.)

Key Concepts

Voltage Division in Series

In a series arrangement, the total applied voltage is divided among the capacitors in proportion to their impedances (for capacitors, this is related to their capacitance). When identical capacitors are used, the voltage divides equally among them. This equal sharing is crucial for ensuring that none of the capacitors is exposed to a voltage exceeding its rated value, thus preserving the integrity of each component.

Voltage Rating Considerations

Every capacitor has a maximum voltage rating beyond which it may fail. When substituting capacitors to meet a higher voltage requirement, it is important to design the configuration such that each capacitor sees only a fraction of the total voltage applied, keeping the voltage across any single capacitor at or below its rated limit. This is often achieved by combining capacitors in series to spread the voltage evenly, and then using parallel combinations to achieve the required capacitance.

Series Connection of Capacitors

When capacitors are connected in series, the total voltage rating of the combination is increased because the applied voltage divides across each capacitor. In a series circuit, the same charge passes through each capacitor, but the voltage drop across each depends on its capacitance. This means that by connecting capacitors with lower voltage ratings in series, you can effectively reach a higher overall voltage rating while ensuring that each individual capacitor experiences only a fraction of the total voltage.

Parallel Connection of Capacitors

Connecting capacitors in parallel causes their capacitances to add while the voltage across each capacitor remains the same as that applied to the overall circuit. This configuration is useful for achieving a desired total capacitance when individual capacitors have lower capacitance values than required. However, it does not change the voltage rating of the individual capacitors.

Equivalent Capacitance Calculation

The effective capacitance of capacitors connected in series is less than any individual capacitor's capacitance, calculated using the reciprocal formula. For identical capacitors connected in series, the equivalent capacitance is the capacitance of one capacitor divided by the number of capacitors in that series chain. When series groups are connected in parallel, the overall capacitance of the network is the sum of the capacitances of each series group, allowing designers to tailor both the voltage rating and capacitance to desired specifications.

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