Voltage Sources in Parallel

Electrical Circuit Analysis > Parallel DC Circuits

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in Jan 2026

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Introduction

In electrical and electronic circuits, voltage sources are used to provide electrical energy to loads. In many practical applications, more than one voltage source is connected together to improve reliability or current capacity. One such configuration is voltage sources connected in parallel. Understanding how voltage sources behave when connected in parallel is important for circuit design, power systems, and electronics.

What Are Voltage Sources in Parallel?

Voltage sources are said to be connected in parallel when their positive terminals are connected together and their negative terminals are connected together, supplying power to a common load.

In this arrangement:

The voltage across the load remains the same as the voltage of each source.

The current supplied to the load is shared among the sources.

Ideal Voltage Sources in Parallel

Important Rule

Ideal voltage sources with different voltages must never be connected in parallel.

If two ideal voltage sources with unequal voltages are connected in parallel, it creates a short circuit between the sources, resulting in infinite current (theoretically), which is impossible and dangerous in real circuits.

voltage sources can be placed in parallel only if they have the same voltage.

Condition for Parallel Connection

For ideal voltage sources to be connected in parallel:

All sources must have exactly the same voltage magnitude and polarity.

Practical Voltage Sources in Parallel

In real-life circuits, voltage sources are not ideal. Each practical voltage source has a small internal resistance (r). When voltage sources with internal resistance are connected in parallel:

The internal resistances limit the circulating current.

The load current is shared according to the internal resistances.

This makes parallel connection possible and safe in practical systems such as batteries and power supplies.

Equivalent Voltage Source in Parallel

For n identical voltage sources connected in parallel: Equivalent Voltage (Vₑq) $$ V_{eq} = V $$ Equivalent Internal Resistance (rₑq) $$ \frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} + \cdots + \frac{1}{r_n} $$ If all internal resistances are equal: $$ r_{eq} = \frac{r}{n} $$ This shows that connecting voltage sources in parallel reduces internal resistance, allowing the system to deliver more current.

Current Sharing in Parallel Voltage Sources

The total load current is divided among the sources based on their internal resistances.

For two sources: $$ I_1 = \frac{r_2}{r_1 + r_2} I_{total} $$ $$ I_2 = \frac{r_1}{r_1 + r_2} I_{total} $$ Where: ($I_1$, $I_2$) = currents from each source ($r_1$, $r_2$) = internal resistances A source with lower internal resistance supplies more current.

Example 1: Two Batteries in Parallel

Given:

Solution:

Equivalent internal resistance: $$ r_{eq} = \frac{0.2}{2} = 0.1Ω $$ Total circuit resistance: $$ R_{total} = R_L + r_{eq} = 6.1Ω $$ Load current: $$ I = \frac{12}{6.1} \approx 1.97A $$ Each battery supplies: $$ I_1 = I_2 = \frac{1.97}{2} \approx 0.985A $$

Advantages of Voltage Sources in Parallel

✔ Increased current capacity
✔ Improved reliability (backup operation)
✔ Reduced internal resistance
✔ Longer battery life in battery systems

Disadvantages

✖ Circulating currents if voltages are mismatched
✖ Unequal current sharing if internal resistances differ
✖ Requires careful design and protection

Practical Applications

Voltage sources in parallel are widely used in:

Battery banks (UPS systems, solar systems)
DC power supplies
Electric vehicles
Telecommunication power systems
Redundant power systems in industries

Conclusion

Voltage sources connected in parallel provide the same output voltage while increasing the available current capacity. This configuration is widely used in practical power systems where reliability and higher current delivery are required. However, proper design is essential to avoid circulating currents and ensure safe operation.

Why we need Parallel Voltage Sources

The primary reason for placing two or more batteries or supplies in parallel is to increase the current rating above that of a single supply.

For example, in Fig. 1, two ideal batteries of 12 V have been placed in parallel. The total source current using Kirchhoff's current law is now the sum of the rated currents of each supply. The resulting power available will be twice that of a single supply if the rated supply current of each is the same. That is,
with $I_1 = I_2 = I$ then

$$P_T = E(I_1 + I_2)$$ $$= E(I + I) = E(2I) $$ $$= 2(EI) = 2P$$

Fig. 1: Examining the impact of placing two lead-acid batteries of different terminal voltages in parallel.

If for some reason two batteries of different voltages are placed in parallel, both will become ineffective or damaged because the battery with the larger voltage will rapidly discharge through the battery with the smaller terminal voltage. For example, consider two lead acid batteries of different terminal voltages placed in parallel as shown in Fig. 2. It makes no sense to talk about placing an ideal 12 V battery in parallel with a 6 V battery because Kirchhoff's voltage law would be violated. However, we can examine the effects if we include the internal resistance levels as shown in Fig. 2.

Fig. 2: Examining the impact of placing two lead-acid batteries of different terminal voltages in parallel.

The only current-limiting resistors in the network are the internal resistances, resulting in a very high discharge current for the battery with the larger supply voltage. The resulting current for the case would be

$$ I = {E_1 - E_2 \over R_{int1} + R_{int2}}$$

$$= {12 V - 6 V \over 0.03 Ω + 0.02 Ω}$$

$$ = {6 V \over 0.05 Ω} = 120 A$$

This value far exceeds the rated drain current of the $12 V$ battery, resulting in rapid discharge of $E_1$ and a destructive impact on the smaller supply due to the excessive currents. This type of situation did arise on occasion when some cars still had $6 V$ batteries. Some people thought, If I have a $6 V$ battery, a $12 V$ battery will work twice as well "not true"!

In general,

it is always recommended that when you are replacing batteries in series or parallel, replace all the batteries.

A fresh battery placed in parallel with an older battery probably has a higher terminal voltage and immediately starts discharging through the older battery. In addition, the available current is less for the older battery,resulting in a higher-than-rated current drain from the newer battery when a load is applied.

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