What Formula Is Used To Calculate Two Capacitors In Series

Series Capacitance Calculator

Calculate the total capacitance when two capacitors are connected in series using the standard formula.

What is the Two Capacitors in Series Formula?

The two capacitors in series formula is used to calculate the total or equivalent capacitance (C_total) when two capacitors, C1 and C2, are connected end-to-end (in series) in an electrical circuit. When capacitors are connected in series, the total capacitance is less than the capacitance of any individual capacitor in the series combination. This is because the effective plate separation increases, and the area for charge storage is limited by the smallest capacitor.

The formula is derived from the fact that the charge (Q) stored on each capacitor in series is the same, while the total voltage (V) across the series combination is the sum of the voltages across each capacitor (V = V1 + V2). Since V = Q/C, we get Q/C_total = Q/C1 + Q/C2, which simplifies to the reciprocal formula.

This formula is fundamental for anyone working with electronic circuits, from hobbyists to electrical engineers, as it allows for the calculation of the effective capacitance, which is crucial for circuit design and analysis, especially in filters, timing circuits, and energy storage applications.

Who should use it?

• Electronics students and hobbyists learning about circuits.
• Electrical engineers designing or analyzing circuits.
• Technicians troubleshooting electronic equipment.

Common Misconceptions

A common misconception is that adding capacitors in series increases the total capacitance, similar to resistors in parallel. However, it’s the opposite: total capacitance decreases when capacitors are added in series. Another is that the voltage across each capacitor is the same; in series, the voltage divides inversely proportionally to the capacitance (smaller capacitance gets higher voltage).

Two Capacitors in Series Formula and Mathematical Explanation

When two capacitors, C1 and C2, are connected in series, the total capacitance (C_total) is given by the formula:

1/C_total = 1/C1 + 1/C2

To find C_total, we can rearrange this formula:

C_total = (C1 * C2) / (C1 + C2)

Step-by-step Derivation:

1. For capacitors in series, the charge (Q) on each capacitor is the same: Q1 = Q2 = Q.
2. The total voltage (V_total) across the series combination is the sum of the individual voltages: V_total = V1 + V2.
3. We know that voltage across a capacitor is V = Q/C. So, V1 = Q/C1 and V2 = Q/C2.
4. Substituting into the voltage equation: V_total = Q/C1 + Q/C2.
5. The total voltage across the equivalent capacitance C_total is V_total = Q/C_total.
6. Therefore, Q/C_total = Q/C1 + Q/C2.
7. Dividing by Q (assuming Q is not zero), we get: 1/C_total = 1/C1 + 1/C2.
8. To solve for C_total: 1/C_total = (C2 + C1) / (C1 * C2), so C_total = (C1 * C2) / (C1 + C2).

Variables Table

Variable	Meaning	Unit	Typical Range
C1	Capacitance of the first capacitor	Farads (F), µF, nF, pF	pF to thousands of µF
C2	Capacitance of the second capacitor	Farads (F), µF, nF, pF	pF to thousands of µF
Ctotal	Total or equivalent capacitance	Farads (F), µF, nF, pF	Less than min(C1, C2)

Table 1: Variables in the two capacitors in series formula.

Practical Examples (Real-World Use Cases)

Example 1: Combining Standard Capacitors

Suppose you have two capacitors, C1 = 10 µF and C2 = 22 µF, and you connect them in series. What is the total capacitance?

Using the two capacitors in series formula:

1/C_total = 1/10 µF + 1/22 µF = 0.1 + 0.04545 = 0.14545 (1/µF)

C_total = 1 / 0.14545 ≈ 6.875 µF

Alternatively: C_total = (10 * 22) / (10 + 22) = 220 / 32 = 6.875 µF.

The total capacitance (6.875 µF) is less than the smallest individual capacitance (10 µF).

Example 2: Voltage Divider Application

Two capacitors, C1 = 100 nF and C2 = 47 nF, are connected in series across a 12V AC source. The total capacitance helps determine the impedance and how the voltage divides at a given frequency.

First, find the total capacitance:

C_total = (100 nF * 47 nF) / (100 nF + 47 nF) = 4700 / 147 ≈ 31.97 nF

The voltage across each capacitor will be inversely proportional to its capacitance. C2 is smaller, so it will have a larger voltage drop across it. This is a basic principle for capacitive voltage dividers in AC circuits. You can find more about this in our capacitive voltage divider guide.

How to Use This Two Capacitors in Series Formula Calculator

1. Enter Capacitor 1 Value: Input the capacitance value of the first capacitor (C1) into the “Capacitor 1 (C1)” field.
2. Select C1 Unit: Choose the unit (µF, nF, pF, or F) for C1 from the dropdown menu next to its value.
3. Enter Capacitor 2 Value: Input the capacitance value of the second capacitor (C2) into the “Capacitor 2 (C2)” field.
4. Select C2 Unit: Choose the unit for C2 from its dropdown menu.
5. Select Output Unit: Choose the desired unit for the total capacitance result from the “Desired Output Unit” dropdown.
6. Calculate: The calculator automatically updates the results as you input values. You can also click the “Calculate” button.
7. Read Results: The “Total Capacitance (C_total)” will be displayed prominently, along with intermediate values like 1/C1, 1/C2, and the sum of reciprocals. The formula used is also shown.
8. Reset: Click “Reset” to clear inputs and results to default values.
9. Copy Results: Click “Copy Results” to copy the main result, intermediates, and formula to your clipboard.

The calculator applies the two capacitors in series formula C_total = (C1 * C2) / (C1 + C2) after converting both C1 and C2 to Farads for calculation, and then converts the result to your chosen output unit.

Capacitance Comparison Chart

Chart 1: Comparison of C1, C2, and Total Capacitance (values shown in µF for chart display).

Key Factors That Affect Total Series Capacitance

• Individual Capacitance Values (C1, C2): The total capacitance is directly dependent on the values of the individual capacitors. Smaller individual capacitances lead to a smaller total capacitance. The total capacitance is always less than the smallest individual capacitance.
• Number of Capacitors: Although this calculator is for two, if more capacitors are added in series, the total capacitance further decreases (1/C_total = 1/C1 + 1/C2 + 1/C3 + …).
• Tolerance of Capacitors: Real-world capacitors have a tolerance (e.g., ±10%). The actual total capacitance will vary within a range determined by the tolerances of C1 and C2.
• Frequency (in AC circuits): While the capacitance value itself doesn’t change with frequency, the capacitive reactance (X_C = 1/(2πfC)) does. The behavior of the series combination in an AC circuit depends on the frequency.
• Dielectric Material: The capacitance of each capacitor depends on its dielectric material. If you are choosing capacitors to combine, their material might affect their stability and frequency response.
• Voltage Rating: When capacitors are in series, the total voltage divides between them. The voltage across each capacitor (V1 = V_total * C_total / C1, V2 = V_total * C_total / C2) must not exceed its voltage rating. The smaller capacitor will experience a larger voltage drop. The total voltage rating of the series combination is NOT simply the sum of individual ratings unless the capacitances are identical and leakage currents are managed. It is often limited by the voltage division.

Understanding these factors is crucial when using the two capacitors in series formula for practical circuit design. Consider exploring more about capacitor basics.

Frequently Asked Questions (FAQ)

What is the formula for two capacitors in series?

The formula is 1/C_total = 1/C1 + 1/C2, or equivalently, C_total = (C1 * C2) / (C1 + C2).

Why does total capacitance decrease in series?

In a series connection, it’s like increasing the thickness of the dielectric or the distance between the outermost plates, effectively reducing the overall ability to store charge for a given voltage. The smallest capacitor limits the charge storage for the entire series.

Is the voltage the same across capacitors in series?

No. The charge stored on each capacitor is the same (Q). The voltage across each capacitor (V=Q/C) is inversely proportional to its capacitance. The smaller capacitor will have a larger voltage drop across it.

What happens if the capacitors have different voltage ratings?

You must ensure the voltage across each individual capacitor does not exceed its rating. The voltage divides inversely to capacitance, so the smaller capacitor will see more voltage. The effective voltage rating of the series combination is complex and not simply the sum, especially with DC or if leakage currents differ. Adding balancing resistors in parallel with each capacitor is sometimes done in high-voltage DC applications.

How do you calculate total capacitance for more than two capacitors in series?

The formula extends: 1/C_total = 1/C1 + 1/C2 + 1/C3 + … + 1/Cn.

Can I use this formula for AC circuits?

Yes, the capacitance value C_total is the same for AC and DC. However, in AC circuits, you would often be more interested in the total capacitive reactance X_C,total = X_C1 + X_C2, where X_C = 1/(2πfC).

What is the unit of capacitance?

The base unit is the Farad (F). More common units are microfarad (µF = 10^-6 F), nanofarad (nF = 10^-9 F), and picofarad (pF = 10^-12 F).

How does the two capacitors in series formula relate to impedance?

The impedance of a capacitor is Zc = 1/(jωC) or -j/(2πfC), where j is the imaginary unit and ω=2πf is the angular frequency. For series capacitors, total impedance Z_total = Zc1 + Zc2 = 1/(jωC1) + 1/(jωC2) = 1/(jω) * (1/C1 + 1/C2) = 1/(jωC_total).

For more circuit calculations, check our RC circuit calculator.

• Series and Parallel Capacitor Calculator: Calculate combinations of more than two capacitors in series or parallel.
• Capacitor Basics Explained: A guide to understanding how capacitors work.
• Capacitive Voltage Divider Calculator: Calculate voltage division in AC circuits with series capacitors.
• RC Circuit Time Constant Calculator: Analyze resistor-capacitor circuits.
• Capacitor Energy Storage Calculator: Calculate the energy stored in a capacitor.
• Parallel Capacitors Calculator: Calculate the total capacitance of capacitors in parallel.