21.1 Resistors in Series and Parallel
3 min read•june 18, 2024
circuits are the building blocks of electrical systems. They come in two main types: and . Each type affects , , and differently, shaping how electricity flows through devices.
Understanding circuits is crucial for designing and troubleshooting electrical systems. helps calculate drops and currents, while calculations differ between and configurations. These principles form the foundation for analyzing complex circuits.
Resistor Circuits
Circuits with resistors
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- Series circuits connect resistors end-to-end allowing to flow through each resistor sequentially ()
- Voltage drops across each resistor in series as current encounters resistance
- Total is the sum of individual voltage drops across each resistor
- Parallel circuits connect resistors side-by-side allowing current to split and flow simultaneously ()
- Voltage across each parallel resistor is the same as the source voltage
- Total current is the sum of currents through each parallel branch
Ohm's law for voltage drops
- Ohm's law relates voltage, current, and resistance: [V = IR](https://www.fiveableKeyTerm:V_=_IR)
- represents voltage measured in volts (V)
- represents current measured in amperes (A)
- represents resistance measured in ohms ()
- In series circuits, current remains constant while voltage drops across each resistor
- Apply Ohm's law to each resistor to calculate individual voltage drops (, , etc.)
- Sum of voltage drops equals the source voltage ()
- In parallel circuits, voltage remains constant across each resistor while current divides
- Apply Ohm's law to each resistor to calculate branch currents (, , etc.)
- Sum of branch currents equals the total current ()
Resistance calculation in series vs parallel
- Series circuits: Total resistance is the sum of individual resistances
- Resistances add because the same current encounters each resistor's opposition
- Equivalent to water flowing through a narrow pipe (more resistance)
- Parallel circuits: equals the sum of reciprocals of individual resistances
- Conductances (1/R) add because the same voltage is applied across each resistor
- Equivalent to water flowing through wider pipes (less resistance)
- The total resistance of a circuit configuration is also known as the
Parallel circuits and total resistance
- Parallel resistors provide multiple paths for current, reducing the overall resistance
- Each additional path acts like opening another lane on a highway for traffic
- More paths for current results in less opposition to current flow
- Mathematically, the reciprocal formula for parallel resistance yields a smaller total resistance
- Example: Two 10Ω resistors in parallel
- Total resistance (5Ω) is less than the individual resistances (10Ω)
- Example: Two 10Ω resistors in parallel
Complex circuit resistance calculations
- Identify series and parallel sections within the complex circuit
- Calculate the total resistance for each series section by adding resistances
- Example:
- Calculate the total resistance for each parallel section using the reciprocal formula
- Example:
- Combine the series and parallel section total resistances
- Treat each section's total resistance as an individual resistor
- Repeat steps 2-4 as needed until the circuit is reduced to a single total resistance
- The final single resistance represents the total resistance of the entire complex circuit
Circuit Analysis Principles
- are fundamental principles used to analyze complex circuits:
- Kirchhoff's Current Law: The sum of currents entering a equals the sum of currents leaving it
- Kirchhoff's Voltage Law: The sum of voltage drops around any closed loop in a circuit equals zero
- in resistors is calculated using P = I²R, where power is measured in watts
- refers to the current flowing through a specific path or component in a
- A junction is a point in a circuit where three or more conductors meet, crucial for applying Kirchhoff's Current Law
Key Terms to Review (46)
Acceleration: Acceleration is the rate of change of velocity over time. It is a vector quantity, meaning it has both magnitude and direction.
Air resistance: Air resistance is a force that opposes the motion of an object through the air. It depends on the object's speed, surface area, and shape.
Ampere: An ampere (A) is the unit of electric current in the International System of Units (SI). It represents the flow of one coulomb of charge per second.
Ampere: The ampere (symbol: A) is the base unit of electric current in the International System of Units (SI). It is defined as the constant flow of one coulomb of electrical charge per second, and it is a fundamental quantity used to describe the movement of electric charge.
Battery: A battery is a device that stores chemical energy and converts it into electrical energy, providing a source of power for various electrical devices and circuits. It is a fundamental component in the understanding of topics such as electric potential energy, Ohm's law, resistors in series and parallel, and electromotive force.
Branch Current: Branch current refers to the current that flows through a specific branch or path within an electrical circuit containing multiple resistors in parallel. It is a crucial concept in understanding the behavior of resistors in parallel configurations.
Christmas Lights: Christmas lights are a popular holiday decoration consisting of a series of electric lights that are typically hung around the home, on a Christmas tree, or in other outdoor areas to create a festive and illuminated display during the Christmas season.
Conductance: Conductance is a measure of a material's ability to allow the flow of electric current. It is the reciprocal of resistance and is an important concept in understanding the behavior of resistors in electrical circuits.
Current: Current is the flow of electric charge through a conductor, typically measured in amperes (A). It is driven by a voltage difference and encounters resistance within the material.
Current: Current is the flow of electric charge in a circuit, typically measured in amperes (A). It represents how much charge passes through a point in the circuit per unit of time, and it plays a crucial role in determining how electrical energy is distributed and consumed in various applications.
Current Divider: A current divider is a circuit configuration that allows the division of an input current into two or more output currents, with the ratio of the output currents determined by the resistor values in the circuit. It is a fundamental concept in the analysis of resistors in series and parallel configurations.
Decay series: A decay series is a sequence of radioactive decays where the product of one decay becomes the parent nuclide for the next. This process continues until a stable nuclide is formed.
Equivalent Resistance: Equivalent resistance is the single resistance value that can replace a network of resistors, producing the same overall effect as the original network. It is a crucial concept in understanding the behavior of resistors connected in series and parallel circuits.
Home Wiring: Home wiring refers to the electrical system that distributes power throughout a residential building, enabling the use of various electrical devices and appliances. It is a crucial component that ensures the safe and efficient operation of a home's electrical infrastructure.
I: The term 'I' refers to the moment of inertia, a fundamental concept in the study of rotational motion, wave intensity, electrical resistance, and circuit analysis. Moment of inertia describes an object's resistance to changes in its rotational motion, while intensity is a measure of the energy carried by a wave, and resistance is a measure of an object's opposition to the flow of electric current. Understanding the role of 'I' in these various contexts is crucial for comprehending the underlying principles of physics.
I_total = I1 + I2 + ... + In: The total current in a circuit is equal to the sum of the individual currents flowing through each component. This principle applies to both series and parallel circuits, and is a fundamental concept in understanding the behavior of electrical circuits.
Joule’s law: Joule's law states that the rate of heat production by an electric current in a conductor is directly proportional to the square of the current, the resistance of the conductor, and the time. Mathematically, it is expressed as $P = I^2R$ where $P$ is power, $I$ is current, and $R$ is resistance.
Junction: A junction is a point where two or more electrical components, such as resistors or conductors, are connected. It is a critical concept in understanding the behavior of electrical circuits, particularly in the context of resistors in series and parallel, as well as Kirchhoff's rules.
Junction rule: The junction rule states that the sum of currents entering a junction equals the sum of currents leaving the junction. It is a consequence of the conservation of electric charge.
Kirchhoff's Laws: Kirchhoff's Laws are a set of fundamental principles that describe the behavior of electric circuits. They provide a framework for understanding the flow of current and the distribution of voltage in both simple and complex electrical networks.
Ohm’s law: Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points. It is mathematically expressed as $V = IR$, where $V$ is voltage, $I$ is current, and $R$ is resistance.
Ohm's Law: Ohm's law is a fundamental principle in electrical engineering that describes the relationship between the voltage, current, and resistance in an electrical circuit. It states that the current flowing through a conductor is directly proportional to the voltage applied across it, and inversely proportional to the resistance of the conductor.
Parallel: Parallel circuits are electrical circuits where components are connected across common points or junctions, providing multiple paths for current to flow. In a parallel configuration, the voltage across each component is the same.
Parallel: Parallel refers to a configuration or arrangement where components or elements are connected side-by-side, with each one operating independently but simultaneously. This concept is fundamental in understanding various electrical and electronic systems.
Parallel Circuit: A parallel circuit is an electrical circuit in which components are connected along multiple paths, allowing current to flow through different branches simultaneously. In a parallel circuit, the voltage across each component is the same, while the current through each component can vary.
Power Dissipation: Power dissipation refers to the amount of energy converted into heat within an electrical or electronic component or system. It is a crucial concept in understanding the performance, efficiency, and thermal management of various electrical devices and circuits.
R: In physics, 'r' typically represents the distance from a rotation axis or center of mass to a point of interest. This distance is crucial in understanding rotational motion, electric circuits, and how different components interact within a system. It plays a vital role in calculations for rotational inertia, resistance, and electromotive force, linking the concepts of linear and angular relationships in physics.
R_total = R1 + R2 + R3: R_total, or the total resistance, is the sum of the individual resistances R1, R2, and R3 in an electrical circuit. This relationship is a fundamental concept in understanding resistors in series and parallel configurations.
Reciprocal of Total Resistance: The reciprocal of the total resistance in an electrical circuit is equal to the sum of the reciprocals of the individual resistances. This relationship is a fundamental principle in understanding the behavior of resistors connected in series and parallel configurations.
Resistance: Resistance is a measure of the opposition to the flow of electric current in a material. It is measured in ohms ($\Omega$).
Resistance: Resistance is a measure of the opposition to the flow of electric current in a circuit. It is a fundamental concept in electrical engineering and physics, as it determines the behavior of electrical systems and the energy dissipation within them.
Resistor: A resistor is an electrical component that limits or regulates the flow of electrical current in a circuit. It provides resistance, measured in ohms ($\Omega$), to control voltage and current levels.
Resistor: A resistor is a passive electronic component that is used to control or limit the flow of electric current in a circuit. It is a fundamental element in electrical and electronic systems, playing a crucial role in various applications such as voltage division, current regulation, and signal processing.
Series: A series circuit is one in which resistors are connected end-to-end so that the same current flows through each resistor. The total resistance in a series circuit is the sum of the individual resistances.
Series: In the context of electrical circuits, the term 'series' refers to the configuration where components are connected one after the other, forming a single continuous path for the flow of electric current.
Series Circuit: A series circuit is an electrical circuit in which the components are connected end-to-end, forming a single continuous path for the flow of electric current. This arrangement ensures that the same current flows through each component in the circuit.
Series-Parallel: Series-parallel is a circuit configuration where components are arranged in both series and parallel connections. This combination allows for more flexible control over the flow of current and distribution of voltage within an electrical circuit.
V: In physics, 'v' represents velocity, which is a vector quantity that indicates the rate of change of an object's position with respect to time, including its direction. Velocity is crucial in understanding motion, as it not only tells how fast an object is moving but also in which direction it is traveling. This concept extends beyond simple motion to fluid dynamics and electrical circuits, highlighting its versatility across different scientific fields.
V = IR: V = IR, also known as Ohm's Law, is a fundamental relationship in electrical circuits that describes the relationship between voltage (V), current (I), and resistance (R). This equation is a critical concept in understanding the behavior of electrical components and circuits.
V_source = V1 + V2 + ... + Vn: The total voltage source, or V_source, in an electrical circuit is equal to the sum of all the individual voltage sources (V1, V2, etc.) that are connected in series. This relationship is a fundamental principle in the analysis of circuits with multiple voltage sources.
Volt: A volt is the unit of electric potential difference, which measures the amount of electric potential energy per unit charge between two points in an electric circuit. It helps us understand how much energy is available to move electric charges through a conductor, connecting it to concepts like power, energy storage, and circuit behavior.
Voltage: Voltage, also known as electric potential difference, is the measure of potential energy between two points in an electric field. It is measured in volts (V) and drives electric current through a circuit.
Voltage: Voltage, also known as potential difference, is the electrical potential energy difference between two points in an electrical circuit. It is the driving force that causes the flow of electric current, and it is measured in units of volts (V).
Voltage Divider: A voltage divider is a simple circuit that produces an output voltage that is a fraction of its input voltage. It is a fundamental concept in electrical engineering and is commonly used to reduce or adjust voltage levels in various electronic circuits.
Voltage drop: Voltage drop is the reduction in voltage across a component or resistor in an electrical circuit. It occurs due to the resistance that opposes the flow of current.
ω: Omega (ω) is a Greek letter that represents angular velocity, a fundamental concept in rotational motion, wave energy, and electrical circuits. It describes the rate of change of angular displacement, the dynamics of rotational inertia, the intensity of waves, and the electromotive force in electrical systems.