Glossary

21.6 DC Circuits Containing Resistors and Capacitors

3 min read•june 18, 2024

RC circuits blend and , creating fascinating electrical behavior. The , τ = RC, is key, determining how quickly capacitors or . This concept is crucial for understanding changes and applications in technology.

RC circuits find use in everyday devices like camera flashes and touchscreens. They exhibit transient and steady-state behaviors, with the playing a vital role in energy storage and release. These principles are essential for grasping DC circuit dynamics.

DC Circuits with Resistors and Capacitors

Time constant calculation for RC circuits

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  • Measure of how quickly a charges or discharges in an
  • Calculated by multiplying the resistance RR and the capacitance CC: τ=RC\tau = RC
    • Measured in seconds (s)
  • Represents the time for the voltage to reach approximately 63.2% of its final value when charging or discharging
    • After one time constant, the capacitor voltage is VC=Vf(1−e−1)≈0.632VfV_C = V_f(1 - e^{-1}) \approx 0.632V_f, where VfV_f is the final voltage
    • After five time constants, the capacitor is considered fully charged or discharged (99.3% of final value)
  • Examples:
    • In an with a 100 kΩ and a 10 μF capacitor, the time constant is τ=(100×103Ω)(10×10−6F)=1\tau = (100 \times 10^3 \Omega)(10 \times 10^{-6} F) = 1 s
    • If a 220 kΩ is used with a 47 nF capacitor, the time constant is τ=(220×103Ω)(47×10−9F)≈10.3\tau = (220 \times 10^3 \Omega)(47 \times 10^{-9} F) \approx 10.3 ms

Voltage changes during capacitor charge/discharge

  • Charging in an RC circuit:
    1. Capacitor voltage increases exponentially from zero to the final voltage VfV_f
    2. Charging equation: VC=Vf(1−e−t/τ)V_C = V_f(1 - e^{-t/\tau}), where VCV_C is the capacitor voltage at time tt
  • Discharging in an RC circuit:
    1. Capacitor voltage decreases exponentially from the initial voltage V0V_0 to zero
    2. Discharging equation: VC=V0e−t/τV_C = V_0e^{-t/\tau}, where VCV_C is the capacitor voltage at time tt
  • Rate of voltage change depends on the time constant τ\tau
    • Smaller τ\tau results in faster charging or discharging
    • Larger τ\tau results in slower charging or discharging
  • Examples:
    • If τ=1\tau = 1 s and Vf=5V_f = 5 V, after 1 s of charging, VC≈3.16V_C \approx 3.16 V (63.2% of VfV_f)
    • For τ=10\tau = 10 ms and V0=3.3V_0 = 3.3 V, after 20 ms of discharging, VC≈0.41V_C \approx 0.41 V (12.4% of V0V_0)
  • The charging and discharging processes follow an pattern

RC circuit applications in technology

  • Timing circuits in various applications:
    • Camera flashes: RC circuit controls flash duration by determining capacitor discharge time through the flash tube
    • Touchscreens: RC circuits detect and locate touch position by measuring capacitance change caused by a finger
    • Pacemakers: RC circuits generate timing pulses to stimulate heart muscle contraction at a steady rate
      • Time constant determines the pacing rate
    • Defibrillators: RC circuits control charging and discharging of high-voltage capacitors for delivering therapeutic electrical shocks to the heart during cardiac arrest
  • Other applications:
    • Debouncing switches: RC circuits filter out rapid voltage fluctuations caused by mechanical switch contacts
    • Smoothing power supply ripples: RC low-pass filters reduce voltage fluctuations in DC power supplies
    • Audio equalizers: RC high-pass and low-pass filters adjust the balance between high and low frequencies in audio signals

Transient and Steady-State Behavior in RC Circuits

  • : The initial period when the circuit is adjusting to a change in voltage or
    • Characterized by rapid changes in voltage and current
    • Duration depends on the time constant of the circuit
  • : The condition reached after the transient response has died out
    • Voltage and current remain constant or follow a repeating pattern
    • In DC circuits, capacitors act as open circuits in steady state
  • Electric field: Stores energy in the capacitor during charging and releases it during discharging
    • Strength of the electric field is proportional to the charge stored on the capacitor plates

Key Terms to Review (30)

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.
Capacitance: Capacitance is a measure of the amount of electric charge that can be stored in an electrical component or system. It is a fundamental concept in the study of electrostatics and the behavior of electric circuits. Capacitance is a crucial factor in understanding the storage and release of electrical energy, as well as the behavior of electrical components like capacitors.
Capacitor: A capacitor is an electrical component that stores energy in the form of an electric field, created by a pair of conductors separated by an insulating material. The ability to store charge is measured in farads (F).
Capacitor: A capacitor is a passive electronic component that stores electrical energy in an electric field. It consists of two conductors separated by an insulator, and it is used in various electronic circuits and devices to store and release electrical charge.
Characteristic time constant: The characteristic time constant in an RL circuit, denoted as $\tau$, is the time it takes for the current to reach approximately 63% of its final value after a sudden change in voltage. It is calculated as the ratio of inductance $L$ to resistance $R$, i.e., $\tau = \frac{L}{R}$.
Charge: Charge is a fundamental property of matter that is the source of all electrical phenomena. It is the basic unit of electrical charge that can be either positive or negative and is responsible for the creation of electric fields and the flow of electric current.
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.
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.
Discharge: Discharge refers to the process by which a capacitor releases or transfers the stored electric charge in a circuit. It is a fundamental concept in the study of DC circuits containing resistors and capacitors.
Electric charge: Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. Charges are either positive or negative and are measured in coulombs (C).
Electric Field: The electric field is a vector field that describes the force experienced by a stationary, positive test charge at any given point in space. It represents the strength and direction of the electric force exerted on a charged particle by other charges in the vicinity, and is a fundamental concept in the study of electromagnetism and the behavior of charged particles.
Exponential Decay: Exponential decay is a mathematical model that describes the gradual reduction of a quantity over time. It is characterized by an initial value that decreases at a rate proportional to its current value, resulting in a smooth, continuous decline.
Farad: The farad (symbol: F) is the unit of electrical capacitance in the International System of Units (SI). It measures the amount of electric charge that a capacitor can store for a given potential difference across its terminals.
Faraday cage: A Faraday cage is an enclosure made of conductive material that blocks external static and non-static electric fields by channeling electricity along and around the exterior. This effect is used to protect sensitive electronic equipment from electromagnetic interference.
Kirchhoff's Rules: Kirchhoff's Rules are two fundamental principles that describe the behavior of electric currents in a closed circuit. These rules provide a framework for analyzing the flow of electric charge and the distribution of potential differences in complex electrical networks.
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.
RC circuit: An RC circuit consists of a resistor (R) and a capacitor (C) connected in series or parallel. It is fundamental in analyzing transient responses in circuits.
RC Circuit: An RC circuit, or resistor-capacitor circuit, is a fundamental electrical circuit that combines a resistor and a capacitor. It is a type of first-order linear circuit that exhibits specific behaviors when subjected to direct current (DC) or alternating current (AC) inputs, making it a crucial component in various electronic applications.
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: 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.
Steady State: Steady state refers to a condition in which a system or process has reached a stable, unchanging state, where the input and output values remain constant over time. This concept is particularly relevant in the context of electrical circuits, where it describes the point at which the circuit has reached a stable, predictable behavior.
Time Constant: The time constant is a fundamental concept in the study of electrical circuits, particularly those involving resistors and capacitors (RC circuits) or inductors and resistors (RL circuits). It represents the time required for a circuit to reach a specific percentage of its final value when subjected to a step change in input.
Transient Response: The transient response refers to the temporary, initial behavior of a system when it is subjected to a change in input or initial conditions. It describes the system's dynamic response before it settles into a steady-state or equilibrium condition.
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).
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