Series Battery Circuit Calculations Electric Current, Terminal Voltage, And Lost Voltage

This article explores the principles behind calculating electrical characteristics in a series battery circuit. Specifically, we will analyze a scenario involving five 2-volt batteries, each possessing an internal resistance of 5 ohms, connected in series to a 20-ohm resistor. We will delve into the calculations required to determine the electric current flowing through the circuit, the terminal voltage across the battery combination, and the voltage lost due to the internal resistance of the batteries. Mastering these calculations is crucial for anyone working with battery-powered systems, from electronics hobbyists to electrical engineers. Understanding how batteries behave in series circuits is fundamental to designing efficient and reliable power supplies. This analysis will provide a clear and comprehensive guide to tackling such circuit problems.

Circuit Configuration and Key Concepts

In this series circuit configuration, the positive terminal of one battery is connected to the negative terminal of the next, and so on. This arrangement effectively increases the total voltage of the battery combination. However, it also increases the total internal resistance. Each battery's internal resistance acts as a small resistor in series with the battery's ideal voltage source. This internal resistance is an inherent property of batteries and affects their performance, especially under load. When a current flows through the circuit, a voltage drop occurs across each battery's internal resistance, reducing the voltage available at the terminals. This voltage drop is known as the lost voltage. To accurately analyze the circuit, we need to consider both the total voltage provided by the batteries and the total resistance in the circuit, including the external resistor and the internal resistances of the batteries. Understanding these concepts is essential for calculating the circuit's current, terminal voltage, and lost voltage. In the following sections, we will break down the calculations step-by-step, providing a clear understanding of how each parameter is determined.

a) Calculating the Electric Current

The first step in analyzing this circuit is to calculate the electric current. To do this, we need to determine the total voltage supplied by the batteries and the total resistance in the circuit. Since the five 2-volt batteries are connected in series, their voltages add up. The total voltage (Vtotal) is calculated as follows:

$$V_{total} = 5 imes 2V = 10V$$

Next, we need to calculate the total internal resistance (Rinternal) of the batteries. Since the batteries are in series, their internal resistances also add up. Given that each battery has an internal resistance of 5 ohms:

$$R_{internal} = 5 imes 5 ext{ }\[Omega] = 25\[Omega]$$

Now, we need to add the external resistance (Rexternal), which is 20 ohms, to the total internal resistance to find the total resistance (Rtotal) in the circuit:

$$R_{total} = R_{internal} + R_{external} = 25\[Omega] + 20\[Omega] = 45\[Omega]$$

Using Ohm's Law, which states that current (I) is equal to voltage (V) divided by resistance (R), we can calculate the electric current:

$$I = \frac{V_{total}}{R_{total}} = \frac{10V}{45\[Omega]} = 0.22 ext{ }A$$

Therefore, the electric current flowing through the circuit is approximately 0.22 Amperes. This calculation demonstrates the fundamental principle of Ohm's Law in action and highlights the importance of considering both internal and external resistances in circuit analysis. In the subsequent sections, we will use this current value to calculate the terminal voltage and lost voltage in the circuit.

b) Calculating the Terminal Voltage

The terminal voltage is the voltage available at the output terminals of the battery combination when the circuit is under load (i.e., when current is flowing). Due to the internal resistance of the batteries, the terminal voltage is less than the total voltage supplied by the batteries. To calculate the terminal voltage (Vterminal), we need to subtract the voltage drop across the internal resistance from the total voltage. The voltage drop across the internal resistance (Vinternal) can be calculated using Ohm's Law:

$$V_{internal} = I imes R_{internal}$$

We have already calculated the current (I) as 0.22 A and the total internal resistance (Rinternal) as 25 ohms. Substituting these values, we get:

$$V_{internal} = 0.22 ext{ }A imes 25\[Omega] = 5.5V$$

Now, we can calculate the terminal voltage by subtracting the voltage drop across the internal resistance from the total voltage:

$$V_{terminal} = V_{total} - V_{internal} = 10V - 5.5V = 4.5V$$

Therefore, the terminal voltage of the battery combination is 4.5 Volts. This result illustrates the impact of internal resistance on the available voltage in a real-world circuit. The difference between the total voltage and the terminal voltage represents the energy lost within the batteries themselves. Understanding how to calculate terminal voltage is crucial for designing circuits that operate efficiently and reliably. In the next section, we will focus on calculating the lost voltage, which is directly related to the voltage drop across the internal resistance.

c) Calculating the Lost Voltage

The lost voltage represents the voltage drop across the internal resistance of the batteries. As we discussed in the previous section, this voltage drop reduces the terminal voltage available to the external circuit. The lost voltage (Vlost) is essentially the same as the voltage drop across the internal resistance (Vinternal), which we calculated in part (b). Using the values we previously determined:

$$V_{lost} = V_{internal} = I imes R_{internal}$$

Substituting the values for current (I = 0.22 A) and total internal resistance (Rinternal = 25 ohms), we get:

$$V_{lost} = 0.22 ext{ }A imes 25\[Omega] = 5.5V$$

Therefore, the lost voltage in the circuit is 5.5 Volts. This means that out of the total 10 Volts supplied by the batteries, 5.5 Volts are lost due to the internal resistance, leaving only 4.5 Volts as the terminal voltage. The lost voltage represents energy dissipated as heat within the batteries, which is an undesirable effect. Minimizing lost voltage is essential for maximizing the efficiency of a battery-powered system. Factors that influence lost voltage include the internal resistance of the batteries and the current flowing through the circuit. Batteries with lower internal resistance will exhibit lower lost voltage and deliver more power to the load. Understanding and calculating lost voltage is crucial for optimizing battery performance and selecting the appropriate batteries for a given application. In the concluding section, we will summarize the key findings and discuss the implications of these calculations.

Conclusion and Implications

In summary, we have analyzed a series circuit consisting of five 2-volt batteries, each with an internal resistance of 5 ohms, connected to a 20-ohm resistor. Through our calculations, we determined the electric current flowing through the circuit to be 0.22 Amperes, the terminal voltage to be 4.5 Volts, and the lost voltage to be 5.5 Volts.

These results highlight several important considerations for working with series battery circuits. Firstly, the internal resistance of batteries significantly impacts the performance of the circuit. The voltage drop across the internal resistance reduces the terminal voltage available to the load and dissipates energy as heat. This effect becomes more pronounced as the current draw increases. Secondly, connecting batteries in series increases the total voltage but also increases the total internal resistance. Therefore, careful selection of batteries with low internal resistance is crucial for minimizing voltage losses and maximizing efficiency. Thirdly, understanding Ohm's Law and its application in circuit analysis is essential for accurately predicting circuit behavior. By calculating the current, terminal voltage, and lost voltage, we can gain valuable insights into the performance of a battery-powered system.

The principles discussed in this article are applicable to a wide range of applications, from portable electronic devices to electric vehicles. By understanding the behavior of batteries in series circuits and the impact of internal resistance, engineers and hobbyists can design more efficient, reliable, and long-lasting battery-powered systems. Further exploration of parallel battery connections and more complex circuit configurations can build upon this foundation, providing a comprehensive understanding of battery management and power delivery.