When multiple capacitors are connected in series in a circuit, their individual capacitances combine to act as an equivalent net capacitance across the overall string. The total capacitance in a series circuit is always less than the smallest capacitor in the chain.

Understanding how to calculate the equivalent capacitance for capacitors in series is key to properly designing and analyzing circuits containing series-connected capacitors. In this guide, we will cover:

- How series capacitors behave in AC and DC circuits
- Rules to calculate net capacitance in series strings
- Equivalent circuit models for analysis
- Capacitor voltage division in series chains
- Practical applications and examples
- Troubleshooting excessive voltage across capacitors

Gaining a robust knowledge of the principles governing capacitors in series will provide you the ability to optimize capacitor banks for filters, timing circuits, voltage dividers, and other applications. Let’s start by reviewing the fundamentals.

## Capacitor Basics

A capacitor consists of two conducting plates separated by an insulating dielectric material. When a voltage differential exists between the plates, an electric field forms across the dielectric which stores charge.

The capacitance (C) depends on the plate area (A), dielectric thickness (d), and dielectric constant (εr):

`C = εr*A/d`

Key capacitor behaviors:

- Opposes changes in voltage
- Stores energy in electric field
- Can pass high frequencies but blocks DC
- Charges and discharges over time

Now let’s examine how these properties combine when capacitors are connected in series.

## Capacitors in Series – AC Behavior

When AC voltages are applied across a string of series capacitors, the alternating current can pass through each capacitor freely.

In AC circuits, series capacitors look like a single equivalent capacitor with a capacitance equal to the lowest value capacitor in the chain. This makes sense intuitively – the smallest capacitor will impede the AC the most, so it determines the overall impedance.

For example, for 10nF, 22nF and 47nF capacitors in series:

- The 10nF capacitor has the least capacitive reactance
- This capacitor’s impedance limits the AC through the chain
- So the series string acts like a single 10nF capacitor

Modeling the series capacitors as a single equivalent capacitor is an effective AC circuit analysis technique. But this model does not hold true for DC conditions.

## Capacitors in Series – DC Behavior

Direct current cannot pass through a capacitor – it becomes an open circuit. So what happens for DC applied to series capacitors?

With DC voltage applied, each capacitor in the series chain charges up to the applied voltage. Essentially, each capacitor acts like its own individual voltage divider.

The key observations:

- Each capacitor charges up to the same DC voltage
- The series string blocks direct current
- The equivalent capacitance decreases compared to a single capacitor

This equivalent capacitance decrease follows specific rules…

## Rules to Calculate Equivalent Capacitance

To find the net equivalent capacitance of capacitors in series, two key rules apply:

### Rule 1: Net Capacitance Decreases

The overall capacitance of a series string is always less than the smallest capacitor:

`Ceq < Cmin`

For example, 10nF, 22nF, and 47nF capacitors in series always results in:

`Ceq < 10nF`

This matches the AC behavior where the smallest capacitor determines the impedance. But for DC, the exact Ceq formula follows the next rule…

### Rule 2: Reciprocal Summation

The total capacitance for series capacitors is calculated using:

`1/Ceq = 1/C1 + 1/C2 + 1/C3 ...`

- Where C1, C2, C3, etc. are the individual capacitances

Applying this reciprocal summation rule to a series string of:

10nF + 22nF + 47nF

Gives:

`1/Ceq = 1/10nF + 1/22nF + 1/47nF Ceq = 7.48 nF`

So connecting these three capacitors in series results in an equivalent capacitance of 7.48nF, always less than the smallest capacitor.

This reciprocal summation formula allows calculating the net capacitance of any series combination. Understanding these fundamental rules is key to working with series capacitor circuits. Next let’s look at the equivalent circuit model.

## Equivalent Circuit Model

Based on the capacitance rules, the standard equivalent circuit model for any series capacitor combination replaces the chain with a single capacitor:

Where:

- Ceq is the equivalent capacitance calculated from the reciprocal summation of individual capacitances
- RLeak is the insulation resistance representing leakage through the dielectric

This model allows applying simpler capacitor formulas for analyzing series chains in circuit simulations or calculations.

Note that while this model holds for DC conditions, simply using the smallest capacitor value directly can suffice for AC-only analysis as mentioned earlier.

Now let’s examine how the voltage divides across series capacitors…

## Voltage Division Across Series Capacitors

Because series capacitors all charge up to the total applied voltage, the voltage divides proportionally across each capacitor depending on their capacitive reactance.

The voltage division follows this formula:

`VC1 = VTotal * (XC2 || XC3 || ... ) / (XC1 + XC2 + XC3 ...) VC2 = VTotal * (XC1 || XC3 || ... ) / (XC1 + XC2 + XC3 ...)`

Where:

- VC1, VC2, etc. are the voltages across each capacitor
- XC1, XC2, etc. are the capacitive reactances
- VTotal is the total voltage applied to the series string

The capacitor with the lowest capacitive reactance receives the highest voltage.

*This voltage division characteristic is important when specifying capacitor voltage ratings in a series circuit. The capacitor with the smallest value must have a voltage rating exceeding the total applied voltage.*

Now let’s look at some examples of using series capacitors.

## Applications of Series Capacitors

Some common applications that leverage series capacitor behaviors:

### AC Coupling and DC Blocking

Connecting a series capacitor allows an AC signal to pass while blocking the DC component. This AC coupling is useful for isolating stages:

C1 passes the AC input signal but blocks DC from reaching the amplifier. R1 slowly discharges C1.

### Voltage Transformation in Power Systems

Connecting high voltage ceramic capacitors in series allows creating a string to withstand very high voltages for power electronic applications and transmission grids.

### High Voltage DC Link for Inverters

Stacking film or electrolytic capacitors in series enables creating a high voltage DC bus for feeding inverters that require very high DC voltages:

### Timing Circuits

Combining different capacitor values in series provides an RC time constant that can be used for timer and oscillator circuits.

These examples highlight applications where connecting capacitors in series supports key design goals. But series capacitors also introduce potential issues that engineers should be aware of…

## Troubleshooting Issues with Series Capacitors

While connecting capacitors in series has benefits, some problems can emerge that require mitigation:

### Voltage Spikes from Mismatched Values

If the capacitor voltage ratings are mismatched, the lowest rated capacitor may experience overvoltage spikes and get damaged. Always check division with ratings.

### Unbalanced Voltage Sharing

Differences in leakage currents or capacitor aging can lead to uneven voltage division, overstressing one capacitor. Voltage balancing resistors helps mitigate this.

### Open Capacitor Faults

An open fault on one capacitor will shift more voltage onto the remaining capacitors, potentially exceeding their rating. Protection circuits should detect and respond to open faults.

Properly sizing components and implementing monitoring helps prevent these common modes of failure in series capacitor banks and strings.

## Conclusion

In summary, the key rules when working with capacitors in series are:

- AC signals only “see” the smallest capacitance
- DC cannot pass, so equivalent capacitance decreases
- Equivalent capacitance uses the reciprocal summation formula
- Voltage divides according to the capacitive reactance

Carefully applying these principles allows you to properly analyze, design, and troubleshoot circuits utilizing series capacitors in filter networks, timing circuits, voltage dividers, and other applications.

## Frequently Asked Questions

### How is total capacitance calculated for capacitors in series?

Use the reciprocal summation rule: 1/Ceq = 1/C1 + 1/C2 + 1/C3… This gives the equivalent capacitance for a series combination.

### Why is capacitance less when capacitors are in series versus parallel?

In series, capacitors charge to the same voltages so their fields oppose and capacitance decreases. In parallel, the fields sum because the caps share the same voltage.

### How does capacitor voltage divide across a series string?

Voltage divides according to the capacitive reactance (XC). The lower XC capacitor has more voltage across it.

### When should you use series capacitors versus parallel capacitors?

Use series to decrease net capacitance and withstand high voltages. Use parallel to increase capacitance for higher charge storage.

### How do you calculate impedance for capacitors in series?

You can simply use the value of the smallest capacitor in the series string to model the AC impedance, since it limits current flow.