Electricity and Circuits for the FE Exam
Electrical concepts appear on every FE exam discipline, though they are most heavily weighted on the FE Electrical and Computer exam. Even if you are taking the Civil or Mechanical FE, you will encounter basic circuit analysis and electrical fundamentals questions. This review covers the essential concepts you need.
Fundamental Quantities
- Charge (Q): Measured in coulombs (C). The fundamental unit of electrical charge.
- Current (I): The rate of charge flow, measured in amperes (A). I = dQ/dt. One ampere equals one coulomb per second.
- Voltage (V): The electric potential difference, measured in volts (V). It represents energy per unit charge.
- Resistance (R): The opposition to current flow, measured in ohms (omega). R = V / I (Ohm's law).
- Power (P): The rate of energy transfer, measured in watts (W). P = V * I.
Ohm's Law
Ohm's law is the most fundamental circuit equation: V = I * R. It relates the voltage across a resistor to the current through it and its resistance. From this, we can derive:
- I = V / R (current given voltage and resistance)
- R = V / I (resistance given voltage and current)
- P = V * I = I^2 * R = V^2 / R (power dissipated by a resistor)
Kirchhoff's Laws
Kirchhoff's Current Law (KCL)
The sum of currents entering a node equals the sum of currents leaving the node. This is based on conservation of charge. At any junction in a circuit, the total current flowing in equals the total current flowing out.
Kirchhoff's Voltage Law (KVL)
The sum of all voltages around any closed loop in a circuit equals zero. This is based on conservation of energy. As you traverse a closed path in a circuit, the total voltage rise equals the total voltage drop.
Series and Parallel Circuits
Series Resistors
- Same current flows through all resistors
- Total resistance: R_total = R1 + R2 + R3 + ...
- Voltage divides among resistors proportionally to their resistance
- Voltage divider: V_k = V_total * R_k / R_total
Parallel Resistors
- Same voltage across all resistors
- Total resistance: 1/R_total = 1/R1 + 1/R2 + 1/R3 + ...
- For two parallel resistors: R_total = R1 * R2 / (R1 + R2)
- Current divides inversely proportional to resistance
- Current divider: I_k = I_total * R_total / R_k
Capacitors
Capacitors store energy in an electric field:
- Capacitance: C = Q / V, measured in farads (F)
- Energy stored: E = (1/2) * C * V^2
- Series capacitors: 1/C_total = 1/C1 + 1/C2 + ... (opposite of resistors)
- Parallel capacitors: C_total = C1 + C2 + ... (opposite of resistors)
- In DC steady state, a capacitor acts as an open circuit
Inductors
Inductors store energy in a magnetic field:
- Inductance: V = L * dI/dt, measured in henries (H)
- Energy stored: E = (1/2) * L * I^2
- Series inductors: L_total = L1 + L2 + ... (same as resistors)
- Parallel inductors: 1/L_total = 1/L1 + 1/L2 + ... (same as resistors)
- In DC steady state, an inductor acts as a short circuit
AC Circuit Fundamentals
Alternating current circuits involve sinusoidal voltages and currents:
- v(t) = V_max * sin(omega*t + phi), where omega = 2*pi*f is the angular frequency
- RMS (root mean square) values: V_rms = V_max / sqrt(2). RMS values are what meters measure and what most AC calculations use.
- Impedance (Z): The AC equivalent of resistance, including resistance, capacitive reactance, and inductive reactance.
Impedance
- Resistor: Z_R = R (purely real)
- Capacitor: Z_C = 1 / (j*omega*C) = -j / (omega*C) (purely imaginary, negative)
- Inductor: Z_L = j*omega*L (purely imaginary, positive)
Combine impedances the same way you combine resistances: in series, add them; in parallel, use the reciprocal formula. The total impedance has both magnitude and phase angle.
AC Power
- Real power (P): The actual power consumed, measured in watts (W). P = V_rms * I_rms * cos(phi), where phi is the phase angle between voltage and current.
- Reactive power (Q): Power that oscillates between source and load, measured in volt-amperes reactive (VAR). Q = V_rms * I_rms * sin(phi).
- Apparent power (S): The product of RMS voltage and RMS current, measured in volt-amperes (VA). S = V_rms * I_rms.
- Power factor: cos(phi) = P / S. A power factor of 1 means all power is real (resistive load).
Basic Circuit Analysis Methods
- Node voltage method: Write KCL equations at each node using voltage variables. Solve the resulting system of equations.
- Mesh current method: Write KVL equations around each mesh using current variables. Solve the resulting system of equations.
- Thevenin equivalent: Replace a complex circuit with a voltage source in series with a resistance. Useful for analyzing one part of a larger circuit.
- Norton equivalent: Replace a complex circuit with a current source in parallel with a resistance.
Electrical concepts are foundational knowledge for all engineering disciplines. Even if your primary focus is civil or mechanical engineering, mastering these basics ensures you do not lose easy points on the FE exam.