Electric circuits are the backbone of modern electronics. This section dives into the key components—resistors, inductors, and capacitors—and how they interact in RLC circuits. We'll explore their behavior using differential equations to model current and voltage.

Understanding electric circuits is crucial for analyzing real-world applications. We'll learn about transient and steady-state responses, which help us predict how circuits behave when turned on or off, and during normal operation.

Circuit Components

Electrical Properties

• Inductance ($L$) represents the ability of a circuit element to store energy in a magnetic field when current flows through it
  • Measured in henries (H)
  • Opposes changes in current ($v_L = L \frac{di}{dt}$)
  • Examples: coils, transformers, motors
• Capacitance ($C$) represents the ability of a circuit element to store energy in an electric field
  • Measured in farads (F)
  • Opposes changes in voltage ($i_C = C \frac{dv}{dt}$)
  • Examples: parallel plate capacitors, dielectric materials
• Resistance ($R$) represents the opposition to the flow of electric current in a circuit
  • Measured in ohms ($\Omega$)
  • Relates voltage and current linearly (Ohm's law: $v = iR$)
  • Examples: resistors, wires, light bulbs

Complex Impedance

• Impedance ($Z$) is a complex quantity that represents the total opposition to current flow in an AC circuit
  • Measured in ohms ($\Omega$)
  • Consists of resistance ($R$), inductive reactance ($X_L$), and capacitive reactance ($X_C$)
  • Expressed as $Z = R + j(X_L - X_C)$, where $j$ is the imaginary unit
  • Magnitude of impedance: $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$
  • Phase angle of impedance: $\theta = \tan^{-1}(\frac{X_L - X_C}{R})$

Circuit Fundamentals

Basic Quantities

• Current ($i$) is the rate of flow of electric charge through a circuit
  • Measured in amperes (A)
  • Conventional current flows from positive to negative
• Voltage ($v$) is the potential difference between two points in a circuit
  • Measured in volts (V)
  • Represents the work done per unit charge to move a positive test charge from one point to another
• Charge ($q$) is a fundamental property of matter that determines its electromagnetic interactions
  • Measured in coulombs (C)
  • Relates to current by $i = \frac{dq}{dt}$

RLC Circuits and Kirchhoff's Laws

• RLC circuit is a combination of resistors, inductors, and capacitors connected in series or parallel
  • Used to model various electrical systems and analyze their behavior
  • Example: a simple series RLC circuit consists of a resistor, an inductor, and a capacitor connected in series with a voltage source
• Kirchhoff's laws describe the conservation of current and voltage in electrical circuits
  • Kirchhoff's current law (KCL): the sum of currents entering a node equals the sum of currents leaving the node ($\sum i_{in} = \sum i_{out}$)
  • Kirchhoff's voltage law (KVL): the sum of voltage drops around any closed loop in a circuit equals zero ($\sum v = 0$)

Circuit Analysis

Transient Response

• Transient response refers to the behavior of a circuit immediately after a change in the input (e.g., switching a power supply on or off)
  • Characterized by the presence of exponential terms in the solution
  • Involves the natural response (homogeneous solution) and the forced response (particular solution)
  • Example: in an RLC series circuit, the transient response of the current after switching on a DC voltage source is given by $i(t) = \frac{V}{R}(1 - e^{-\alpha t})\cos(\omega_d t + \phi)$, where $\alpha$ and $\omega_d$ depend on the circuit parameters
• The natural response is determined by the initial conditions and the circuit's time constants
  • For an inductor: $\tau_L = \frac{L}{R}$
  • For a capacitor: $\tau_C = RC$
  • These time constants determine how quickly the transient response decays

Steady-State Response

• Steady-state response refers to the behavior of a circuit long after any transients have died out
  • Characterized by the absence of exponential terms in the solution
  • For AC circuits, the steady-state response is sinusoidal with the same frequency as the input
  • Example: in an RLC series circuit driven by a sinusoidal voltage source $v(t) = V_m \cos(\omega t)$, the steady-state current is given by $i(t) = I_m \cos(\omega t - \theta)$, where $I_m = \frac{V_m}{|Z|}$ and $\theta$ is the phase angle of the impedance
• Phasor analysis is a powerful tool for analyzing the steady-state response of AC circuits
  • Represents sinusoidal quantities as complex numbers (phasors)
  • Simplifies circuit analysis by converting differential equations into algebraic equations