By S. Hussain Ather, Updated Mar 24, 2022
Image credit: Kbarzycki/iStock/GettyImages
Direct Current (DC) flows in a single, constant direction. Alternating Current (AC) periodically reverses direction, producing a sinusoidal waveform that can be described by its frequency.
In a DC circuit, the relationship between voltage (V), current (I), and resistance (R) is described by Ohm’s Law: V = I R. The same law applies to AC, but the resistance is replaced by impedance (Z), which captures both resistive and reactive effects.
Impedance can be expressed as a complex number: Z = R + jX, where R is the real resistance and X is the reactance contributed by inductors and capacitors.
For sinusoidal sources, the instantaneous current is I = I_m sin(ωt + θ), where I_m is the peak current, ω = 2πf is the angular frequency, and θ is the phase shift. The corresponding voltage is V = V_m sin(ωt).
Dividing voltage by current gives the impedance:
Z = V_m sin(ωt) / (I_m sin(ωt + θ))
Reactances are defined as:
Inductive reactance: X_L = 2πfL (in ohms, where L is inductance in henries).
Capacitive reactance: X_C = 1 / (2πfC) (in ohms, where C is capacitance in farads).
The magnitude of total impedance for common configurations is:
In a parallel RLC circuit, the total current from the source equals the vector sum of the currents through each branch:
I_S² = I_R² + (I_L – I_C)²
Each branch’s impedance can be calculated from its voltage: R = V / I_R, X_L = V / I_L, X_C = V / I_C. The overall admittance (Y = 1/Z) is then:
Y = √[(1/R)² + (1/X_L – 1/X_C)²]
In a series RLC, the same current flows through all components. Voltage drops across each element add algebraically to the supply voltage:
V_S – V_R – V_L – V_C = 0
Where V_R = I R, V_L = I X_L sin(ωt + 90°), and V_C = I X_C sin(ωt – 90°). The phase relationships highlight that inductors lead the voltage by 90°, while capacitors lag by 90°.
Understanding the distinction between DC resistance and AC impedance is essential for designing power distribution systems, selecting components for electronic circuits, and troubleshooting electrical problems. Accurate impedance calculations enable engineers to predict voltage drops, power losses, and phase shifts in real-world applications.
While the underlying physics of electric circuits remains the same, the presence of alternating signals introduces reactive elements that alter how we calculate resistance. By applying Ohm’s Law to impedance and using the correct formulas for inductive and capacitive reactance, engineers can precisely model both DC and AC circuits for optimal performance.
