Electricity Equations
1. Ohm’s Law
Ohm’s Law states the relationship between voltage (\(V\)), current (\(I\)), and resistance (\(R\)):
\[ V = IR \]
Example:
If \(I = 2A\) and \(R = 10\Omega\), then: \[ V = (2A)(10\Omega) = 20V \]
2. Kirchhoff’s Laws
Kirchhoff’s Current Law (KCL)
The sum of currents entering a junction is equal to the sum of currents leaving:
\[ \sum I_{in} = \sum I_{out} \]
Kirchhoff’s Voltage Law (KVL)
The sum of voltage drops around a closed loop is zero:
\[ \sum V = 0 \]
Example:
For a closed loop with resistors \(R_1\) and \(R_2\) and voltage source \(V\): \[ V - I R_1 - I R_2 = 0 \]
3. AC & DC Circuits
Direct Current (DC) Circuits
A steady flow of electric charge in one direction.
Alternating Current (AC) Circuits
A flow of electric charge that periodically reverses direction:
\[ V(t) = V_m \sin(\omega t) \]
where: - \(V_m\) = Peak voltage - \(\omega = 2\pi f\) (angular frequency) - \(f\) = Frequency in Hz
4. Capacitance & Inductance
Capacitor:
Stores electrical energy in an electric field: \[ Q = CV \]
where: - \(Q\) = Charge (Coulombs) - \(C\) = Capacitance (Farads) - \(V\) = Voltage (Volts)
Inductor:
Stores energy in a magnetic field: \[ V = L \frac{dI}{dt} \]
where: - \(L\) = Inductance (Henrys) - \(\frac{dI}{dt}\) = Rate of change of current
5. Power in Electrical Circuits
DC Power:
\[ P = VI \]
AC Power:
For sinusoidal voltages and currents: \[ P = VI \cos(\phi) \]
where: - \(\phi\) = Phase angle between voltage and current
Example:
If \(V = 230V\), \(I = 10A\), and \(\phi = 30^\circ\): \[ P = (230)(10) \cos(30^\circ) = 1992W \]
6. Electrical Machines
Transformers:
Used to step up or step down voltage: \[ \frac{V_1}{V_2} = \frac{N_1}{N_2} \]
where: - \(V_1, V_2\) = Primary and secondary voltages - \(N_1, N_2\) = Number of turns in primary and secondary windings
Motors & Generators:
Faraday’s Law of Electromagnetic Induction: \[ \mathcal{E} = -N \frac{d\Phi}{dt} \]
where: - \(\mathcal{E}\) = Induced EMF (Volts) - \(N\) = Number of turns - \(\Phi\) = Magnetic flux
7. Semiconductor Devices
Diodes:
Allows current to flow in one direction: \[ I = I_S \left( e^{\frac{V}{nV_T}} - 1 \right) \]
Transistors:
Used for amplification and switching. For a bipolar junction transistor (BJT): \[ I_C = \beta I_B \]
where: - \(I_C\) = Collector current - \(I_B\) = Base current - \(\beta\) = Current gain
8. Digital Electronics
Logic Gates:
Basic building blocks of digital circuits: - AND: $ Y = A B $ - OR: $ Y = A + B $ - NOT: $ Y = $
Flip-Flops:
Used for memory storage and sequential logic.
9. Control Systems
A system that regulates its own behavior. ### Transfer Function: \[ H(s) = \frac{Y(s)}{X(s)} \]
PID Controller:
\[ u(t) = K_p e(t) + K_i \int e(t) dt + K_d \frac{de(t)}{dt} \]
where: - \(K_p, K_i, K_d\) = Proportional, Integral, and Derivative gains
10. Electrical Safety
- Grounding: Ensures safety by redirecting excess current.
- Circuit Breakers: Protect against overcurrent.
- Insulation Resistance: Prevents current leakage.