Introduction

Electricity plays a crucial role in our everyday existence, and a grasp of the fundamental principles governing electrical circuits is indispensable for individuals engaged in electrical engineering or electronics. Electrical circuits can be broadly categorized into two primary types: series circuits and parallel circuits. This content offers an extensive overview of the foundational theory behind series and parallel circuits, encompassing their characteristics, practical applications, and mathematical equations.

Series-parallel circuits are a fundamental class of circuits that are widely used in electrical engineering and electronics. By understanding the properties and applications of series-parallel circuits, one can effectively design and analyze complex circuits for a wide range of applications. There are a variety of circuit analysis techniques and design considerations that must be taken into account when designing series-parallel circuits, and it is important to have a thorough understanding of these concepts in order to succeed in the field of electrical engineering and electronics.

When designing series-parallel circuits, there are several important considerations to keep in mind, including:

• Component Selection : The process of picking suitable elements for a series-parallel circuit holds paramount significance in guaranteeing the circuit's dependable and effective performance. Optimal component choice involves the careful consideration of voltage and current ratings that align with the circuit's requirements, in addition to ensuring compatibility with other constituent parts of the circuit.
• Wiring and Layout : The wiring and layout of a series-parallel circuit can have a significant impact on its performance and reliability. It is important to ensure that the wiring is done correctly and that the components are laid out in a way that minimizes interference and maximizes efficiency.
• Power Dissipation : Power dissipation is a critical consideration in series-parallel circuit design, as excessive power dissipation can lead to component failure and reduced circuit efficiency. It is important to carefully calculate the power dissipation of each component in the circuit and to select components with appropriate power ratings.
• Protection and Safety : Protecting the circuit and ensuring safety is also a crucial consideration in series-parallel circuit design. It is important to include appropriate fuses, circuit breakers, and other safety devices in the circuit to protect against over-current, over-voltage, and other potential hazards.

Nodes and Current Flow

An electrical node is a point in an electrical circuit where two or more electrical components are connected. It is the point of connection where the current flowing through one component is shared by the other components connected to the same node. The nodes represent the wire or circuit path between components.

In simpler terms, an electrical node is like a junction point or a meeting point for electrical components, where electrical current flows in and out. Nodes are important in analyzing and designing electrical circuits as they help to determine the behavior of the circuit and its various components. The number of nodes in a circuit determines the complexity of the circuit and the number of equations required to solve it.

Figure 1. Example schematic with four uniquely colored nodes.

Electric current materializes when charged particles, notably electrons, traverse a conductor or an electrical circuit. In the majority of electrical circuits, this current flow originates from a disparity in electric potential, often referred to as voltage, existing between two locations within the circuit. Electrons migrate from the site of higher potential (positive) towards the site of lower potential (negative), engendering the formation of an electric current.

The intensity of the current is ascertained by both the voltage disparity across the two circuit points and the resistance of the conductor or the connected load. The standard unit of measurement for electrical current is the ampere (A), defined as the passage of one coulomb of charge every second.

There are different types of currents, including direct current (DC) and alternating current (AC). In a DC circuit, the current flows in one direction only, while in an AC circuit, the direction of the current changes periodically.

Using the above circuit in figure 1 as an example, figure 2 shows how current would flow as it runs from the battery's positive terminal to the negative:

Figure 2. Current (indicated by the blue, orange, and pink lines) flowing through the same example circuit as above. Different currents are indicated by different colors.

Figure 2 reveals that in specific nodes, such as the one situated between R1 and R2, the incoming current equals the outgoing current. Conversely, at distinct nodes like the tri-junction involving R2, R3, and R4, the primary (blue) current bifurcates into two separate streams. This delineates the manner in which current behaves within series and parallel circuits.

Definition of Series Circuits

A series connection embodies an electrical circuit arrangement in which the components are linked in a consecutive manner, facilitating the passage of the identical current through each component in succession. In a series circuit, the components are structured in a way that the output of one component serves as the input for the subsequent component, thus constructing a singular pathway for the current to traverse.

Within a series connection, the voltage drop across each component is directly proportionate to its respective resistance or impedance. The cumulative resistance of the series circuit corresponds to the summation of the individual resistances of each component. Furthermore, the total voltage in the series circuit amounts to the summation of the voltages present across each individual component.

Figure 3 shows and example of a series circuit containing 3 resistors (R1, R2, and R3). The current through each resistor is the same, as there is only one path for the current to flow through. If one component in a series circuit fails or is removed, the circuit is broken, and the current stops flowing. This is the reason for series circuits often used in situations where it is important to control the current flow, such as in lighting or heating applications.

Figure 3. An example of a series circuit containing 3 resistors and a voltage source.

Calculating Equivalent Resistances in Series Circuits

The equivalent resistance of a series circuit is important to know because it allows us to calculate the total current flowing in the circuit and the total power consumed by the circuit. Additionally, it allows us to replace the series circuit with a single resistor that has an equivalent resistance, simplifying circuit analysis and design.

The equivalent resistance in a series circuit signifies the combined resistance of all the resistors connected sequentially within the circuit. This value is determined by simply summing up the individual resistances of each resistor in the series configuration.

The formula to calculate the equivalent resistance of a series circuit is:

where Rtot is the equivalent resistance, R1, R2, R3, and so on, are the individual resistances of each resistor in the circuit, and n is the total number of resistors in the circuit.

Within a series circuit, an identical current courses through each resistor owing to the presence of a solitary path for current flow. In contrast, the voltage across each resistor is directly proportional to its individual resistance. The cumulative voltage in the series circuit is achieved by summing up the voltages present across each individual resistor.

Figure 4

Consider a circuit shown in figure 4 where a voltage source is connected to a single 10kΩ resistor. Note that the current flows can be calculated by dividing the voltage by the total resistance in the circuit. In the case above, the total resistance is equal to 10kΩ since there is only one resistor in the circuit.

In figure 5 below, we then insert another 10kΩ resistor in series with the first resistor and leave the voltage source unchanged. Note that the current is reduced to 50% because the resistance is doubled (Rtot = 10k + 10k = 20k).

Figure 5

Rules of Thumb for Series Resistors

• The summation of individual resistances yields the total resistance: Rtot = R1 + R2 + R3 + ...
• Uniform current flows through each resistor.
• Voltage drop across each resistor varies directly with its resistance: V1 = IR1, V2 = IR2, V3 = IR3, and so forth.
• The overall voltage drop equals the cumulative voltage drops across individual resistors: Vtot = V1 + V2 + V3 + ...
• Series-connected resistors can form a voltage divider, dividing an input voltage into a lower output voltage.
• An LED's current can be controlled and safeguarded from burning out by incorporating a resistor in series. This approach is also applied in power supplies to limit the output current to a safe level.
• Impedance matching: In audio circuits, resistors in series can be used to match the output impedance of one device to the input impedance of another device, ensuring maximum power transfer and minimizing signal distortion.
• When choosing resistors for a series circuit, make sure they are rated to handle the total power dissipated by the circuit. This can be calculated using P = I2 * R, where I is the current and R is the total resistance.
• Be mindful of the temperature coefficient of resistance (TCR) when selecting resistors for a series circuit. If the TCR of the resistors is different, the total resistance can change with temperature, leading to unpredictable behavior.

Definition of Parallel Circuits

A parallel circuit is a specific electrical circuit arrangement in which the components are interconnected in a manner that enables multiple pathways for the current to travel. In contrast to a series circuit where components are connected end-to-end, parallel circuit components are linked side by side.

In a parallel connection, the voltage across each component is the same, while the current through each component is determined by its resistance or impedance. The total resistance of a parallel circuit is calculated differently than in a series circuit, as it involves taking the reciprocal of each component's resistance and then adding them together to find the total resistance. The total current of the parallel circuit is the sum of the currents through each component.

One of the key properties of a parallel circuit is that if one component fails or is removed, the other components continue to operate as normal. Parallel circuits are often used in situations where it is important to maintain a constant voltage across each component, such as in electrical power distribution or electronic circuits that require different components to operate at different voltages.

In a parallel circuit, multiple components share two common nodes, creating multiple pathways for the current to flow. Figure 6 illustrates a parallel circuit comprising three resistors connected in parallel with a voltage source. Starting from the positive battery terminal, the current follows paths to R1, R2, and R3. The node that links the battery to R1 is also interconnected with the other resistors. At the opposite ends of these resistors, they are similarly joined together and ultimately connected back to the negative terminal of the battery. This configuration provides three distinct routes that the current can take before returning to the battery, defining these resistors as being connected in parallel.

Other examples of parallel circuits include the wiring in most homes, where electrical outlets and appliances are connected in parallel, and the internal circuitry of most electronic devices, where components such as resistors, capacitors, and transistors are typically connected in parallel.

Figure 6. An example of a parallel configuration of three resistors connected to a voltage source.

Calculating Equivalent Resistances in Parallel Circuits

The equivalent resistance of a parallel circuit is the total resistance of all the resistors connected in parallel in the circuit. It is calculated by taking the reciprocal of each resistor and adding them together, then taking the reciprocal of the sum.

The formula to calculate the equivalent resistance of a parallel circuit is:

 

where Rtot is the equivalent resistance, R1, R2, R3, and so on, are the individual resistances of each resistor in the circuit, and n is the total number of resistors in the circuit.

In a parallel circuit, the voltage across each resistor is the same, while the current through each resistor is determined by its resistance. The total current of the parallel circuit is the sum of the currents through each resistor. The power consumed by each resistor is calculated as the voltage across the resistor multiplied by the current flowing through it.

The equivalent resistance of a parallel circuit is important to know because it allows us to calculate the total current flowing in the circuit and the total power consumed by the circuit. Additionally, it allows us to replace the parallel circuit with a single resistor that has an equivalent resistance, simplifying circuit analysis and design.

In the previous example, as illustrated in Figure 4, we had a circuit with a 10V power supply and a 10kΩ resistor. In this scenario, we introduced an additional 10kΩ resistor in parallel, as shown in Figure 6. This arrangement creates two separate pathways for the electric current to follow. Since the supply voltage remains constant at 10V, Ohm's Law dictates that the initial resistor will continue to draw 1mA of current. Simultaneously, the newly added resistor will also draw 1mA, resulting in a total current of 2mA drawn from the power source, effectively doubling the original 1mA. This indicates that the total resistance in the circuit has been halved, as evidenced by this change in current behavior.

Figure 7

While we can say that 10kΩ || 10kΩ = 1/ (1/10 + 1/10) = 1/ (2/10) = 10/2 = 5kΩ

Note that “||” sign means “in parallel with.

Instead of reciprocals, we can also use a method called “product over sum” when we have two resistors in parallel:

Rtot = (10 x 10) / (10 + 10) = 100 / 20 = 5kOhm

Nonetheless, it's worth noting that while this method is suitable for calculating the combined resistance of two resistors in parallel (R1 || R2), when dealing with more than two resistors, it becomes more complex. In such cases, you would need to calculate the value of the first two resistors in parallel, then take that result and calculate it in parallel with a third resistor (using the product-over-sum method again). While this approach can handle multiple resistors in parallel, the reciprocal method (taking the reciprocals of individual resistances, summing them, and then reciprocating the total) often proves to be less labor-intensive and more straightforward.

Rules of Thumb for Parallel Resistors

• The reciprocal of the total resistance is equal to the sum of the reciprocals of the individual resistances:    1/Rt = 1/R1 + 1/R2 + 1/R3 + ...
• The voltage drop across each resistor is the same.
• The current through each resistor is proportional to its conductance (the inverse of its resistance):              I1 = V/R1, I2 = V/R2, I3 = V/R3, ...
• The total current is equal to the sum of the currents through each resistor: It = I1 + I2 + I3 + ...
• Use parallel resistors when you want to create a current divider or increase the overall current-carrying capacity of a circuit.
• Within a parallel circuit, it's important to note that the total resistance is invariably lower than the smallest individual resistance. Consequently, the inclusion of additional resistors in parallel leads to a reduction in the overall resistance of the circuit.
• When choosing resistors for a parallel circuit, make sure they are rated to handle the total current flowing through the circuit. This can be calculated using I = V/R, where V is the voltage and R is the total resistance.
• Heat dissipation: Multiple resistors in parallel can be used to dissipate more heat than a single resistor of the same value. This is commonly used in power resistors and electronic components that generate a lot of heat, such as voltage regulators and audio amplifiers.
• Power distribution: In high-power applications, resistors in parallel can be used to distribute power among multiple components or circuits, ensuring that each one receives the required amount of power.
• Be aware of the tolerance of the resistors when using them in parallel. If the resistors have significantly different values, the current may not be divided equally, leading to unpredictable behavior.

Calculating Resistance, Current, Voltage Drop, and Power Dissipation in a Series Circuit

Calculating Resistance, Current, Voltage Drop, and Power Dissipation in a Parallel Circuit

Combination of Series and Parallel Circuits

A series-parallel combination circuit is a type of electrical circuit that combines both series and parallel connections. In this type of circuit, some components are connected in series, while others are connected in parallel. The purpose of combining these two types of connections is to achieve a desired electrical characteristic or functionality.

Analysis of Series-Parallel Circuits

Calculating equivalent resistance for combination series and parallel resistors

Combinations of series and parallel resistors are often used to create more complex connections. Such combinations are frequently encountered, particularly when the resistance of the wire is taken into account. In such cases, the wire resistance is typically in series with other resistances that are arranged in parallel.

The technique illustrated in Figure 11 can be used to simplify combinations of series and parallel resistors into a single equivalent resistance. By identifying the parts as either series or parallel and reducing them to their equivalent values, the resistors can be further reduced until a single resistance is obtained. Although the process may be time-consuming, it is not particularly difficult.

Figure 11a. Replacing R2, R3 and R4 with Rp, and replacing R5 and R6 with Rp’

 

Figure 11b. Replacing Rp and Rp’ with Rs

 

Figure 11c. Replacing R7 and Rs with Rp”

 

Figure 11d. Replacing R1 and Rp’’ with Rs’

 

Figure 11e. The simplified circuit with the equivalent resistance