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15.3 RLC Series Circuits with AC

4 min read•june 24, 2024

AC circuits with resistors, capacitors, and inductors create fascinating electrical behavior. Current flows through all components equally, but voltage across each part differs. This interplay leads to complex relationships between current and voltage.

RLC circuits showcase how different components affect electricity flow. Resistors keep current and voltage in sync, while capacitors and inductors cause phase shifts. Understanding these interactions is key to grasping AC circuit dynamics.

RLC Series Circuits with AC

Current changes in AC series circuits

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  • In an with an , the current remains the same through all components since they are connected in series
    • The current is , oscillating back and forth, and has the same frequency as the AC voltage source (60 Hz in the US)
    • The current amplitude, or maximum value, is determined by dividing the voltage amplitude by the total of the circuit (Ohm's Law)
  • Resistors in an RLC series circuit with AC
    • The voltage across a resistor is always in phase with the current, meaning they reach their maximum, minimum, and zero values at the same time
    • The voltage amplitude across a resistor is calculated using Ohm's Law: VR=IRV_R = IR, where II is the current amplitude and RR is the resistance
  • Capacitors in an RLC series circuit with AC
    • The voltage across a capacitor lags the current by 90 degrees, meaning the current reaches its maximum value a quarter cycle before the voltage does
    • The voltage amplitude across a capacitor is given by VC=IXCV_C = I X_C, where II is the current amplitude and XC=12πfCX_C = \frac{1}{2\pi fC} is the , which depends on the frequency ff and capacitance CC
  • Inductors in an RLC series circuit with AC
    • The voltage across an inductor leads the current by 90 degrees, meaning the voltage reaches its maximum value a quarter cycle before the current does
    • The voltage amplitude across an inductor is given by VL=IXLV_L = I X_L, where II is the current amplitude and XL=2πfLX_L = 2\pi fL is the , which depends on the frequency ff and inductance LL

Phase angles in RLC circuits

  • visually represent sinusoidal quantities, like voltage and current, as rotating vectors on a
    • The length of the vector represents the amplitude of the sinusoidal quantity (voltage or current)
    • The angle between the vector and the positive x-axis represents the , which indicates the timing difference between the voltage and current
    • are used to mathematically represent these phasors, combining magnitude and phase information
  • In an RLC series circuit, the voltage phasors for each component (resistor, capacitor, and inductor) can be added vectorially to find the total voltage phasor
    • The voltage phasor for a resistor is always in phase with the current phasor, pointing in the same direction
    • The voltage phasor for a capacitor lags the current phasor by 90 degrees, pointing 90 degrees clockwise from the current phasor
    • The voltage phasor for an inductor leads the current phasor by 90 degrees, pointing 90 degrees counterclockwise from the current phasor
  • The between the total voltage phasor and the current phasor represents the overall phase angle of the RLC series circuit
    • If the circuit is predominantly capacitive (XC>XLX_C > X_L), the current leads the voltage, and the phase angle is negative
    • If the circuit is predominantly inductive (XL>XCX_L > X_C), the current lags the voltage, and the phase angle is positive

Total impedance of RLC series

  • , measured in ohms (Ω\Omega) and represented by the symbol ZZ, is the total opposition to current flow in an AC circuit, considering both resistance and
    • Impedance is a complex quantity, consisting of a real part (resistance) and an imaginary part (reactance)
    • The real part represents the resistance, while the imaginary part represents the combined effect of capacitive and inductive reactances
  • In an RLC series circuit, the total impedance is the vector sum of the resistive impedance (ZRZ_R), capacitive impedance (ZCZ_C), and inductive impedance (ZLZ_L)
    • The resistive impedance is equal to the resistance: ZR=RZ_R = R, and has no imaginary component
    • The capacitive impedance is given by ZC=−jXCZ_C = -jX_C, where jj is the imaginary unit and XC=12πfCX_C = \frac{1}{2\pi fC} is the , which is always negative
    • The inductive impedance is given by ZL=jXLZ_L = jX_L, where XL=2πfLX_L = 2\pi fL is the , which is always positive
  • The total impedance of an RLC series circuit is calculated using the formula: Z=ZR+ZC+ZL=R−jXC+jXLZ = Z_R + Z_C + Z_L = R - jX_C + jX_L
    • The magnitude of the total impedance, ∣Z∣|Z|, is found using the Pythagorean theorem: ∣Z∣=R2+(XL−XC)2|Z| = \sqrt{R^2 + (X_L - X_C)^2}
    • The phase angle of the total impedance, ϕ\phi, is found using the arctangent function: ϕ=arctan⁡(XL−XCR)\phi = \arctan\left(\frac{X_L - X_C}{R}\right), which gives the angle in radians
    • The of an RLC circuit is a special case when the inductive and capacitive reactances are equal, resulting in a purely resistive impedance

Power and Measurements in RLC Circuits

  • RMS (Root Mean Square) values are commonly used to describe AC voltages and currents, representing the equivalent DC value that would produce the same power dissipation
  • The in an RLC circuit is the cosine of the phase angle between voltage and current, indicating the fraction of apparent power that is real power
  • are often used to express power ratios or signal levels in AC circuits, particularly in audio and communication systems

Key Terms to Review (35)

AC Source: An AC source, or alternating current source, is an electrical circuit component that provides an alternating current (AC) signal, where the flow of electric charge periodically reverses direction. This is in contrast to a direct current (DC) source, which provides a unidirectional flow of electric charge.
Bandwidth: Bandwidth in an AC circuit refers to the range of frequencies over which the circuit can effectively operate. It is typically defined as the difference between the upper and lower frequency limits where the circuit's performance remains within acceptable levels.
Bandwidth: Bandwidth refers to the range or capacity of frequencies that can be transmitted or processed within a given communication system or electronic circuit. It is a fundamental concept in various fields, including electronics, telecommunications, and signal processing.
Capacitive reactance: Capacitive reactance is the opposition that a capacitor offers to alternating current (AC), due to the phase difference between voltage and current. It is inversely proportional to both the frequency of the AC signal and the capacitance.
Capacitive Reactance: Capacitive reactance is the opposition to the flow of alternating current (AC) in a capacitive circuit, which is caused by the capacitance of the circuit elements. It is a measure of the capacitor's resistance to changes in voltage and is an important concept in the analysis of AC circuits.
Characteristic Impedance: Characteristic impedance is a property of transmission lines that defines the relationship between voltage and current as a wave travels along the line. It is represented by the symbol Z₀ and is crucial for understanding how alternating current (AC) signals behave in RLC series circuits, as it helps determine the efficiency of power transfer and the reflection of signals at discontinuities.
Complex Numbers: Complex numbers are a mathematical concept that extend the real number system to include imaginary numbers. They are represented in the form $a + bi$, where $a$ is the real part and $bi$ is the imaginary part, with $b$ being a real number and $i$ representing the imaginary unit (the square root of -1).
Complex Plane: The complex plane is a two-dimensional coordinate system used to represent and analyze complex numbers. It provides a visual representation of complex numbers, where the real and imaginary parts are plotted on perpendicular axes, allowing for a deeper understanding of their properties and behavior in the context of RLC series circuits with AC.
Decibels: Decibels (dB) is a logarithmic unit used to measure the ratio between two quantities, typically related to sound or electrical power. It is commonly used to quantify the intensity or loudness of a sound, as well as the gain or loss in an electrical signal.
Farad: A farad (F) is the SI unit of capacitance, defined as one coulomb of electric charge stored per one volt of potential difference. It quantifies a capacitor's ability to store electrical energy.
Farad: The farad (symbol: F) is the SI unit of electrical capacitance, which is the ability of a body or system to store an electrical charge. It is a fundamental unit that is essential in understanding the behavior of capacitors, which are key components in electrical circuits and devices.
Henry: The henry (H) is the unit of measurement for the physical quantity of inductance, which is a measure of the magnetic field created by an electric current passing through a coil or other inductor. It is a fundamental unit in the study of electromagnetism and is essential in understanding the behavior of circuits involving inductors.
Impedance: Impedance is the measure of opposition that a circuit presents to the flow of alternating current (AC) at a particular frequency. It combines resistance, inductive reactance, and capacitive reactance into a single value represented as a complex number.
Impedance: Impedance is a measure of the opposition to the flow of alternating current (AC) in an electrical circuit. It encompasses the combined effects of resistance, capacitance, and inductance, and determines the overall behavior of the circuit under AC conditions.
Inductive reactance: Inductive reactance is the opposition that an inductor presents to alternating current due to its inductance. It is proportional to both the frequency of the AC signal and the inductance.
Inductive Reactance: Inductive reactance is the opposition to the flow of alternating current (AC) in an inductor, caused by the inductor's self-induced magnetic field. It is a measure of an inductor's resistance to changes in the current flowing through it, and it plays a crucial role in the behavior of AC circuits.
Inductive time constant: The inductive time constant, denoted as $\tau_L$, is the time required for the current in an RL circuit to change significantly (about 63.2%) towards its final value after a change in voltage. It is calculated as $\tau_L = \frac{L}{R}$, where $L$ is the inductance and $R$ is the resistance.
Magnetic resonance imaging: Magnetic Resonance Imaging (MRI) is a medical imaging technique used to visualize internal structures of the body using magnetic fields and radio waves. It relies on the principles of nuclear magnetic resonance to generate detailed images of organs and tissues.
Phase angle: Phase angle is the measure of the phase difference between the voltage and current in an AC circuit, usually expressed in degrees. It indicates whether the current leads or lags behind the voltage.
Phase Angle: The phase angle is the difference in the timing or displacement between two periodic signals, such as voltage and current, in an alternating current (AC) circuit. It represents the angular difference between the peak values of these signals and is a crucial parameter in understanding the behavior of AC circuits.
Phasor Diagram: A phasor diagram is a graphical representation of the magnitude and phase relationship between voltage, current, and impedance in an AC circuit. It provides a visual tool to understand the behavior of alternating current (AC) circuits, particularly those involving resistors, inductors, and capacitors.
Phasor diagrams: Phasor diagrams are graphical representations of the magnitudes and phase relationships between sinusoidal functions in AC circuits. They simplify the analysis of alternating currents and voltages by converting complex numbers into rotating vectors.
Power factor: Power factor is the ratio of real power (P) to apparent power (S) in an AC circuit. It indicates how effectively the electrical power is being converted into useful work output.
Power Factor: Power factor is a dimensionless quantity that describes the relationship between the real power and apparent power in an alternating current (AC) electrical system. It is a measure of how efficiently the load converts supplied AC power into useful work, with a value ranging from 0 to 1.
Quality factor: Quality factor, often denoted as Q, measures the sharpness of resonance in an AC circuit. It is defined as the ratio of the resonant frequency to the bandwidth over which the circuit can efficiently resonate.
Quality Factor: The quality factor, also known as the Q-factor, is a dimensionless parameter that characterizes the resonance properties of a system. It quantifies the sharpness or selectivity of a resonant circuit, indicating how well the circuit can store and dissipate energy during oscillations.
Reactance: Reactance is a measure of the opposition to the flow of alternating current (AC) in an electrical circuit, caused by the presence of inductors and capacitors. It represents the reactive component of impedance, which is distinct from the resistive component that dissipates energy as heat.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a force that matches the system's natural frequency of oscillation, resulting in a dramatic increase in the amplitude of the system's response. This concept is fundamental in understanding the behavior of various physical systems, including electrical circuits, mechanical vibrations, and acoustic waves.
Resonant frequency: Resonant frequency is the natural frequency at which an AC circuit's impedance is minimized, and current is maximized. It occurs when the inductive reactance equals the capacitive reactance, leading to a purely resistive impedance.
Resonant Frequency: Resonant frequency is the natural frequency at which a system oscillates or vibrates with the greatest amplitude when subjected to an external driving force. This concept is fundamental to understanding the behavior of RLC series circuits with AC and the phenomenon of resonance in AC circuits.
RLC Series Circuit: An RLC series circuit is an electrical circuit that contains a resistor (R), an inductor (L), and a capacitor (C) connected in series. This combination of circuit elements is fundamental in understanding the behavior of alternating current (AC) circuits.
RMS Values: RMS (Root Mean Square) values are a way to measure the effective or equivalent value of an alternating current (AC) signal. It represents the magnitude of a sinusoidal waveform and is used to quantify the power or energy content of the signal, which is important in the analysis of AC circuits.
Sinusoidal: Sinusoidal refers to a waveform that follows a sine function, characterized by a smooth, repeating, and oscillating pattern. This term is particularly relevant in the context of RLC series circuits with AC, as the voltage and current in such circuits often exhibit a sinusoidal behavior.
Skin Effect: The skin effect is a phenomenon that occurs in conductors carrying alternating current (AC), where the current tends to flow primarily near the surface of the conductor rather than being evenly distributed throughout its cross-section. This effect is caused by the interaction between the magnetic field and the conductive material, and it has important implications in various electrical and electronic applications.
Time Constant: The time constant is a fundamental concept that describes the rate of change in various electrical and physical systems. It represents the time required for a system to reach approximately 63% of its final value when undergoing a step change in input.
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15.4 Power in an AC Circuit