7.1 Series and parallel combinations of resistors
2 min read•august 7, 2024
Resistors in series and parallel are fundamental building blocks of DC circuits. They determine how voltage and current flow through a circuit, shaping its behavior and functionality.
Understanding these combinations is crucial for analyzing complex circuits. Series resistors add up, increasing total resistance, while parallel resistors offer multiple paths, decreasing overall resistance. This knowledge is essential for designing and troubleshooting electrical systems.
Series Circuits
Characteristics and Equivalent Resistance
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- Series circuits consist of components connected end-to-end, forming a single continuous path for current flow
- Current remains constant throughout the entire while voltage drops across each component
- () of a series circuit is the sum of individual resistances:
- Total resistance in a series circuit is always greater than any individual resistance
Voltage Divider and Applications
- refers to the division of voltage across each component in a series circuit
- Voltage across each resistor is proportional to its resistance relative to the total resistance:
- Voltage dividers are used in various applications such as:
- Adjusting voltage levels in electronic circuits (potentiometers)
- Sensing physical quantities (temperature, light, pressure) using variable resistors (thermistors, photoresistors, strain gauges)
- Kirchhoff's voltage law () states that the sum of voltage drops across components in a series circuit equals the source voltage:
Parallel Circuits
Characteristics and Equivalent Resistance
- Parallel circuits have components connected across the same two nodes, providing multiple paths for current flow
- Voltage remains constant across all parallel branches while current divides among the branches
- Equivalent resistance () of a is the reciprocal of the sum of reciprocals of individual resistances:
- Total resistance in a parallel circuit is always less than the smallest individual resistance
Current Divider and Power Dissipation
- refers to the division of current among parallel branches
- Current through each branch is inversely proportional to its resistance:
- Kirchhoff's current law () states that the sum of currents entering a node equals the sum of currents leaving the node:
- in each resistor can be calculated using or
- Total power dissipated in a parallel circuit is the sum of power dissipated by individual resistors:
- Parallel circuits are commonly used in:
- Household wiring (outlets connected in parallel to maintain constant voltage)
- Redundant systems (backup power supplies, safety mechanisms) to ensure continued operation if one component fails
Key Terms to Review (22)
Ampere: An ampere, often abbreviated as 'amp', is the SI unit of electric current, defined as the flow of one coulomb of charge per second. This unit is foundational in understanding how electric currents work and is crucial when discussing current density, resistors, multi-loop circuits, and the magnetic effects of currents in conductors and loops.
Christmas lights in series: Christmas lights in series refer to a configuration where multiple light bulbs are connected end-to-end, forming a single path for the electric current to flow. In this setup, the same current passes through each bulb, and if one bulb burns out or is removed, it breaks the circuit, causing all the lights to go out. This configuration highlights important concepts of voltage, current, and resistance within electrical circuits.
Circuit analysis: Circuit analysis is the process of examining and solving electrical circuits to determine the voltage, current, and resistance values within them. This involves applying various techniques and laws to understand how different components interact within a circuit, which is essential for designing and troubleshooting electrical systems. Understanding circuit analysis allows for the prediction of circuit behavior under various conditions, including the arrangement of components like resistors.
Conservation of charge: Conservation of charge is a fundamental principle stating that the total electric charge in an isolated system remains constant over time. This principle implies that charge can neither be created nor destroyed; it can only be transferred from one part of the system to another. Understanding this concept is crucial for grasping how electric charges behave in various physical situations, such as the flow of current in circuits or interactions between charged particles.
Current Divider: A current divider is a circuit configuration that allows current to be split among multiple parallel paths, enabling the calculation of the current flowing through each branch based on its resistance. This concept is vital for understanding how current behaves in parallel circuits, where the total current entering a junction divides among the available paths according to their resistances. The relationship between the currents and resistances in these branches can be described mathematically, making it an essential tool for analyzing electrical circuits.
Current Division: Current division refers to the process by which the total current entering a junction in a parallel circuit is split among the different branches of that circuit. Each branch receives a portion of the total current that is inversely proportional to its resistance, meaning that lower resistance branches will carry a larger share of the current, while higher resistance branches will carry less. Understanding current division is crucial for analyzing and designing electrical circuits effectively.
Equivalent Resistance: Equivalent resistance is the total resistance that a circuit presents to an external source of voltage when all the individual resistances are combined into a single value. This concept simplifies circuit analysis, allowing one to easily calculate current and voltage across components in both series and parallel configurations.
Equivalent Resistance in Parallel Circuits: The equation $$\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} + ... + \frac{1}{r_n}$$ is used to calculate the equivalent resistance of multiple resistors connected in parallel. This relationship highlights that in a parallel configuration, the total or equivalent resistance decreases as more resistors are added, allowing more current to flow through the circuit. Understanding this equation is essential for analyzing electrical circuits and optimizing their performance.
Household wiring in parallel: Household wiring in parallel is a system where multiple electrical devices are connected across the same voltage source, allowing each device to operate independently. This setup ensures that if one device fails or is turned off, it does not affect the operation of other devices in the circuit, providing reliability and convenience for everyday use.
KCL: KCL, or Kirchhoff's Current Law, states that the total current entering a junction or node in an electrical circuit must equal the total current leaving that junction. This fundamental principle is essential for analyzing complex circuits, particularly when dealing with series and parallel combinations of resistors, as it helps determine how current is distributed among various paths in the circuit.
Kirchhoff's Laws: Kirchhoff's Laws are two fundamental principles that govern the behavior of electrical circuits, specifically relating to the flow of current and the distribution of voltage. These laws help analyze complex circuits by establishing relationships between currents at junctions and the sum of voltages around closed loops. Understanding these laws is crucial for solving problems involving resistors, capacitors, and inductors in different circuit configurations.
KVL: Kirchhoff's Voltage Law (KVL) states that the sum of the electrical potential differences (voltage) around any closed network is zero. This principle is essential in analyzing electrical circuits, particularly when dealing with series and parallel combinations of resistors, as it allows for the calculation of voltage drops across each component in a circuit and helps ensure that energy is conserved.
Ohm: An ohm is the unit of measurement for electrical resistance in a circuit, defined as the resistance between two points when a potential difference of one volt produces a current of one ampere. This unit connects to various concepts including how resistors behave in series and parallel configurations, the relationship between voltage, current, and resistance through Ohm's Law, and how to analyze complex circuits with multiple loops and resistances.
Ohm's Law: Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor. This fundamental principle helps in understanding how voltage, current, and resistance interact within electrical circuits, linking concepts such as electric potential energy and current density with the behavior of resistors in series and parallel configurations.
Parallel Circuit: A parallel circuit is a type of electrical circuit where two or more components are connected across the same voltage source, providing multiple pathways for current to flow. In this configuration, the voltage across each component remains the same while the total current is divided among them. This allows for components to operate independently; if one fails, the others continue to function, making parallel circuits vital in both household and industrial applications.
Power Dissipation: Power dissipation refers to the process by which electrical energy is converted into heat energy in a circuit, typically due to resistance. This phenomenon occurs when current flows through resistive components, leading to energy loss that can affect the efficiency of the circuit. Understanding power dissipation is crucial for analyzing how energy is consumed in different configurations of resistors and in complex multi-loop circuits, where the distribution of power across components can significantly impact performance and reliability.
R_eq = r_1 + r_2 + ... + r_n: The equation $$r_{eq} = r_1 + r_2 + ... + r_n$$ defines the equivalent resistance in a series circuit, where the total resistance is the sum of all individual resistances connected in series. This concept illustrates how resistors behave in a circuit when they are arranged one after the other, impacting the overall current flow. Understanding this equation is crucial for analyzing series circuits and designing electrical systems effectively.
Schematic Diagram: A schematic diagram is a visual representation that illustrates the components of an electrical circuit and their connections using standardized symbols. These diagrams simplify complex circuits by focusing on the essential elements, making it easier to understand how electrical components, such as resistors, capacitors, and power sources, are interconnected in various configurations like series and parallel combinations.
Series Circuit: A series circuit is an electrical circuit where components are connected end-to-end in a single path for the current to flow. In this setup, the same current flows through each component, and the total voltage across the circuit is the sum of the voltages across each individual component. This type of circuit can be analyzed using laws that describe the behavior of current and voltage, which are essential for understanding how resistors and other components interact within a circuit.
Volt: A volt is the unit of electric potential difference or electric potential energy per unit charge. It represents the energy required to move a unit charge between two points in an electric field, establishing a critical connection between energy and charge in electrical systems. Understanding volts is essential for analyzing electric circuits, potential energy changes, and the behavior of charged particles in various configurations.
Voltage Divider: A voltage divider is a simple circuit configuration that allows for the division of an input voltage into smaller output voltages. This is typically achieved using two or more resistors in series, where the output voltage is taken across one of the resistors. Voltage dividers are essential in many electronic applications, particularly when a specific voltage level is required from a higher voltage source.
Voltage drop: Voltage drop refers to the reduction in voltage across a component or conductor in an electrical circuit, caused by the resistance of the material through which the current is flowing. This concept is crucial for understanding how electrical energy is distributed and consumed in various circuit configurations, affecting the overall efficiency and functionality of electrical systems.