Electrical Circuit Introduction

Electrical circuit principles are foundational for understanding and designing systems that integrate sensors, actuators, and controls. Key concepts include voltage, current, and capacitance with fundamental principles like Ohm's law and Kirchhoff's law. Basic electrical components such as resistors, capacitors, and inductors are also explored, including the mathematical models.

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Voltage, Current, and Capacitance

  • Voltage (V): The electrical potential difference between two points in a circuit, measured in volts (V). It drives the flow of electric current.
  • Current (I): The rate of flow of electric charge, measured in amperes (A). Defined as \( I = \frac{dQ}{dt} \), where \( Q \) is the charge.
  • Capacitance (C): The ability of a component to store electrical energy in the form of an electric field, measured in farads (F). Relates the charge stored \( Q \) to the voltage \( V \): \( Q = CV \).

Ohm's Law

  • Relates voltage, current, and resistance in a circuit Where V is Voltage (volts), I is Current (amperes), and R is Resistance (ohms, `\Omega`).

$$V = IR$$

Kirchhoff's Laws

1. Kirchhoff's Current Law (KCL): The total current entering a junction equals the total current leaving the junction.

$$\sum I_\text{in} = \sum I_\text{out}$$

2. Kirchhoff's Voltage Law (KVL): The total voltage around any closed loop in a circuit is zero.

$$\sum V = 0$$

Basic Electrical Components

  • Resistors: Components that impede the flow of current and dissipate energy as heat. Resistance is constant in an ideal resistor and follows Ohm's law.
  • Capacitors: Components that store electrical energy. The voltage across a capacitor is related to the current by the differential equation:

$$I = C \frac{dV}{dt}$$

  • Inductors: Components that store energy in a magnetic field when current flows through them. The voltage across an inductor is related to the current by:

$$V = L \frac{dI}{dt}$$

where L is the inductance, measured in henries (H).
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