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.
Voltage, Current, and Capacitance ⚡
- Voltage (V): The electrical potential difference between two points. Think of it as the "pressure" that pushes charges through a circuit.
Example: A typical AA battery has a voltage of 1.5 V.
- Current (I): The flow of electric charge, similar to water flow in a pipe.
Formula: `I = \frac{dQ}{dt}`
Example: 10 coulombs (C) of charge pass a point in 2 seconds
The current is `I = \frac{10}{2} = 5 \text{A}`.
- Capacitance (C): The ability of a capacitor to store electrical energy.
Formula: `Q = CV`
Example: Capacitor with `C = 10 \mu\text{F}` and `V = 5 \text{V}`.
Stores `Q = 10 \times 10^{-6} \times 5 = 50 \mu\text{C}`.
Ohm's Law 🔧
Formula: `V = IR`
Example: A resistor `R = 10 \Omega` has a current `I = 2 \text{A}` flowing through it.
The voltage across the resistor is `V = 10 \times 2 = 20 \text{V}`.
Kirchhoff's Laws 🔄
- Kirchhoff's Current Law (KCL): At any node, `\sum I_\text{in} = \sum I_\text{out}`.
Example: If `I_1 = 3 \text{A}`, `I_2 = 2 \text{A}`, and `I_\text{out} = 5 \text{A}`
The law is satisfied: 3 + 2 = 5.
- Kirchhoff's Voltage Law (KVL): Around any loop, `\sum V = 0`.
Example: In a loop with a battery `V = 12 \text{V}`
Resistors `R_1 = 2 \Omega`, `R_2 = 4 \Omega`:
Current `I = \frac{12}{2+4} = 2 \text{A}`.
Voltage drop across `R_1`: `2 \times 2 = 4 \text{V}`.
Voltage drop across `R_2`: `2 \times 4 = 8 \text{V}`.
Total: 12 - (4 + 8) = 0.
Basic Electrical Components 📟
- Resistors 🛑
Impede the flow of current and convert energy to heat.
Example: A 10 `\Omega` resistor connected to a 5V battery.
Current: `I = \frac{V}{R} = \frac{5}{10} = 0.5 \text{A}`.
- Capacitors 🌀
Store electrical energy temporarily.
Example: A 10 `\mu\text{F}` capacitor charged with 10V.
Charge Stored: Q = CV = `10 \times 10^{-6} \times 10 = 100 \mu\text{C}`.
- Inductors 🔄
Store energy in a magnetic field when current flows through them.
Example: An inductor with L = 5H and changing current `\frac{dI}{dt} = 2\text{A/s}`.
Voltage Across Inductor: `V = L \frac{dI}{dt} = 5 \times 2 = 10 \text{V}`.
Breadboards and Building Simple Circuits 🤖
- Breadboards: A platform to connect electrical components without soldering.
Rows are electrically connected; vertical rails are typically power (+) and ground (-).
- Resistor Example:
Place a resistor in series with an LED. Connect one leg to a 5V power supply, the other to the LED. Add a ground connection.
- Capacitor Example:
Place a capacitor in parallel with a power supply to stabilize voltage fluctuations. Connect one leg to + and the other to -.
- Inductor Example:
Use an inductor in series with a resistor to form an RL circuit. Observe the time it takes for the current to stabilize.
Simulating Circuits with Python 📊
import numpy as np
import matplotlib.pyplot as plt
# RC Circuit parameters
R = 1000 # ohms
C = 1e-6 # farads
V_in = 5 # volts
# Time points
t = np.linspace(0, 0.01, 1000) # 10 ms
V_out = V_in * (1 - np.exp(-t / (R * C)))
# Plot results
plt.figure(figsize=(5,2.5))
plt.plot(t, V_out)
plt.xlabel('Time (s)')
plt.ylabel('Voltage (V)')
plt.grid(); plt.tight_layout()
plt.show()
import matplotlib.pyplot as plt
# RC Circuit parameters
R = 1000 # ohms
C = 1e-6 # farads
V_in = 5 # volts
# Time points
t = np.linspace(0, 0.01, 1000) # 10 ms
V_out = V_in * (1 - np.exp(-t / (R * C)))
# Plot results
plt.figure(figsize=(5,2.5))
plt.plot(t, V_out)
plt.xlabel('Time (s)')
plt.ylabel('Voltage (V)')
plt.grid(); plt.tight_layout()
plt.show()
By understanding these principles and experimenting with simple circuits, you can design and analyze more complex systems involving sensors and controls.