In alternating-current circuits, coils (inductors) and capacitors, as well as resistors, limit the current. Understanding impedance and phase difference is central.
Basic Knowledge
Alternating voltage: v = V₀ sinωt（V₀: maximum value, ω = 2πf: angular frequency）
RMS value: V_eff = V₀/√2 ≈ 0.707V₀
Alternating-current response of each element:
① Resistor R: Z_R = R, current and voltage are in phase
② Coil (inductor) L: X_L = ωL（inductive reactance）, current lags behind voltage by π/2
③ Capacitor C: X_C = 1/(ωC)（capacitive reactance）, current leads voltage by π/2
Impedance of a series RLC circuit:Z = √(R² + (X_L − X_C)²)
Phase difference tanφ = (X_L − X_C)/R
Resonance condition（X_L = X_C）:ω₀ = 1/√(LC)、at this time Z = R and the current is maximum.
RMS value（the alternating-current value that produces the same power as direct current. V_eff = V₀/√2）
Impedance Z（the total opposition to current in an alternating-current circuit. Unit: Ω）
Reactance（coil X_L = ωL, capacitor X_C = 1/(ωC)）
Phase difference（the time difference between current and voltage. Current lags in a coil and leads in a capacitor）
Resonance（when X_L = X_C. Z is minimum and current is maximum）
Further Exploration (Background and Significance)
When impedance is expressed as a complex number (phasor), it is Z = R + j(X_L − X_C), and Ohm’s law V = ZI holds in complex form. This is a powerful tool for analyzing alternating-current circuits.
In series resonance, the current is maximum, and voltages V_L or V_C may become greater than the source voltage (voltage resonance). Radio tuning uses this principle: by changing the resonant frequency of an LC circuit, only a particular radio wave is selected.
Active power (the power actually consumed) is P = V_eff I_eff cosφ. cosφ is called the power factor. In a pure L or C, φ = ±π/2, so the power factor = 0 (no power consumption).
- RMS value: V_eff = V₀/√2
- X_L = ωL（larger at higher frequency）
- X_C = 1/(ωC)（larger at lower frequency）
- Z = √(R² + (X_L − X_C)²)
- Resonance: ω₀ = 1/√(LC), Z = R minimum
- Coil: current lags; capacitor: current leads
- Active power: P = V_eff I_eff cosφ
Notes (Easily Confused Points)
① X_L is larger at higher frequency (a coil makes it difficult for high-frequency current to pass), while X_C is larger at lower frequency (a capacitor makes it difficult for low-frequency current to pass). ② At resonance, Z = R (the minimum value) and current is maximum. ③ Sign of the phase difference: when X_L > X_C, voltage leads current (inductive). ④ Do not confuse the relationship between RMS and maximum values, √2.
Practice
- Find the impedance of a series circuit with R = 30 Ω, X_L = 60 Ω, and X_C = 20 Ω.
- Find the RMS value of an alternating voltage with a maximum value of 141 V（√2 ≈ 1.41）.
- Find the resonant angular frequency when L = 1.0 mH and C = 10 μF in an LC circuit.