Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Electromagnetism → Electric circuits ↓


RC Circuit


In the field of electromagnetism, RC circuits, also known as resistor-capacitor circuits, are of vital importance due to their simplicity and ubiquitous presence in various electronic systems. These circuits consist primarily of resistors (R) and capacitors (C) connected in series or parallel with a power source such as a battery or AC voltage source. Understanding how these components interact is important for many engineering and physics applications, including filters, timers, and more sophisticated circuits in electronics.

Basic concepts of resistors and capacitors

Before going deeper into RC circuits, it is essential to understand its basic components:

  • Resistors: Resistors are components that resist the electric current flowing through them. The resistance offered by a resistor is measured in ohms (Ω), and it determines how much it will oppose the flow of electric current.
  • Capacitors: Capacitors are components that store and release electrical energy in a circuit. They are measured in farads (F), which indicates their ability to store charge.

RC Circuit Structure

RC circuits typically consist of a resistor and a capacitor that are in series or in parallel with each other. When connected to a power source, these components exhibit unique behaviors that depend on time, making them important in timing and filtering applications.

Series RC Circuit

In a series RC circuit, the resistor and capacitor are connected end-to-end, and the same current flows through both components.

Parallel RC Circuit

In a parallel RC circuit, the resistor and capacitor are connected such that they have the same voltage across their terminals, but the currents flowing through them may be different.

Key Concepts in RC Circuits

Charging and Discharging

Perhaps the most interesting aspect of an RC circuit is the process of charging and discharging a capacitor:

Charge

When the voltage source is connected to the RC circuit, the capacitor begins to charge with time t. The voltage across the capacitor as a function of time can be described by the equation:

v(t) = v_0 (1 - e^(-t/rc))

Where:

  • V(t) is the voltage across the capacitor at time t.
  • V_0 is the initial voltage applied to the circuit.
  • e is the Euler number, the base of the natural logarithm (approximately 2.718).
  • R is the resistance in ohms.
  • C is the capacitance in farads.

Discharge

When the power source is removed, the capacitor will discharge its stored energy through the resistor. The voltage across the capacitor during discharging is given by:

v(t) = v_0 e^(-t/rc)

This exponential behavior is the hallmark of RC circuits, where the time constant τ (tau) of an RC circuit is defined as:

τ = rc

Time constant (τ)

The time constant τ is an important parameter in RC circuits. It represents the time it takes for the voltage across the capacitor to reach approximately 63.2% of its final value during charging or discharging. The higher the time constant, the slower the response of the circuit.

The effects of the time constant can be visually demonstrated in an SVG example. Imagine a graph with time on the x-axis and voltage on the y-axis, showing how the voltage changes over time for different values of τ.

τ = rc

Applications of RC Circuits

RC circuits are fundamental in myriad applications due to their ability to filter signals and store energy. Some common applications include:

  • Filters: RC circuits can act as low-pass or high-pass filters. A low-pass filter allows signals with frequencies lower than a certain cutoff frequency to pass through and attenuates signals with frequencies higher than the cutoff.
  • Timing circuits: Because they can introduce delays, RC circuits are widely used in time-delay circuits and timers. In these applications, the time constant τ determines the delay interval.
  • Integrators and Differentiators: In some operational amplifier circuits, RC combinations are used to form integrators and differentiators, which are important in analog signal processing.

Using an RC circuit as a low-pass filter, let's take a text example where the application of filtering in audio equipment is shown. Let's say an audio signal with high frequencies needs to be reduced:

1. Connect the input audio signal to the RC series circuit.

2. The output signal is routed across capacitors. Given that capacitors impede low frequencies less than high frequencies, this configuration helps to reduce unwanted high frequency noise.

Mathematical analysis

Understanding RC circuits also requires a good understanding of the governing equations and methods for solving them, particularly differential equations due to the time-varying nature of capacitors.

Kirchhoff's loop rule

In a closed loop, the sum of the rise and fall of the voltage is zero. This fundamental law helps in deriving the characteristic equations of RC circuits. For a series RC circuit, the loop rule can be expressed as:

V_0 = V_R(t) + V_C(t)

Where:

  • V_0 is the voltage supplied by the battery.
  • V_R(t) is the voltage across the resistor.
  • V_C(t) is the voltage across the capacitor.

Difference equation for RC circuit

To properly analyze the behavior of RC circuits, one must derive and solve the differential equation they follow. This equation expresses the relationship between current, resistor, and capacitor. For a series RC circuit:

V_0 = Ri(t) + Q(t)/C

Where:

  • I(t) is the current at time t.
  • Q(t) is the charge on the capacitor at time t.

Since I(t) = dQ(t)/dt, substituting this into the above equation gives:

V_0 = r(dQ/dt) + Q/C

This represents a first-order linear differential equation, where the solution can describe how the charge on a capacitor, and hence the voltage, changes over time.

Solving the differential equation

Once the differential equation is set up, it can be solved using standard techniques to obtain expressions for current and voltage over time.

The solution for charging is as follows:

q(t) = c * v_0 (1 - e^(-t/rc))

For discharge:

q(t) = q_0 e^(-t/rc)

Both solutions highlight the exponential nature of charging and discharging in RC circuits.

Practical considerations

In practical applications, it is necessary to keep a few things in mind when designing circuits containing resistors and capacitors:

  • Non-ideal components: Real-world components can exhibit non-ideal behavior, such as parasitic inductance in resistors or series resistance in capacitors, which can slightly change the circuit dynamics.
  • Tolerance: Resistors and capacitors are manufactured with certain tolerances, which means that their actual values may differ slightly from their nominal values. This may affect the accuracy of the time constant and other parameters.

Conclusion

RC circuits, with their ability to control time-dependent processes, are an essential part of electronic and electrical engineering. Their applications range from simple tasks, such as creating delays, to more complex activities such as signal filtering. By mastering the concepts of charging and discharging, time constants, and the mathematical analysis of these circuits, one can gain important insights into both the theory and practice of electronics.


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RC Circuit

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