Given: L = 1 mH = 10⁻³ H, C = 0·1 μF = 10⁻⁷ F, R = 1 kΩ (R doesn't affect resonance frequency)
Resonance frequency formula:
f꜀ = 1 / (2π√LC)
f꜀ = 1 / (2π × √(10⁻³ × 10⁻⁷))
f꜀ = 1 / (2π × √10⁻¹⁰)
f꜀ = 1 / (2π × 10⁻⁵)
f꜀ = 10⁵ / (2π) Hz
f꜀ = 10⁵ / (2 × 3·14159) = 10⁵ / 6·2832 ≈ 15,915 Hz ≈ 15·9 kHz
Note: 50/π kHz = 50/3·14159 ≈ 15·92 kHz ✓
f꜀ = 1/(2π√LC) = 10⁵/2π Hz ≈ 15·9 kHz
R does NOT affect resonance frequency
In a series RLC circuit, the total opposition to AC current is called impedance Z. It combines resistance R, inductive reactance X_L = ωL, and capacitive reactance X_C = 1/(ωC). The net reactance is X = X_L − X_C. Total impedance: Z = √(R² + (X_L − X_C)²). The phase angle between voltage and current: tan φ = (X_L − X_C)/R. Current lags voltage when X_L > X_C (inductive circuit), leads when X_C > X_L (capacitive circuit).
Resonance occurs when X_L = X_C, i.e., ωL = 1/(ωC). This gives the resonant angular frequency ω꜀ = 1/√(LC) and resonant frequency f꜀ = ω꜀/2π = 1/(2π√LC). At resonance: impedance is minimum and equal to R alone (Z_min = R), current is maximum (I_max = V/R), the circuit is purely resistive (power factor = 1, φ = 0), and the voltages across L and C are equal and opposite — they cancel each other out, but individually can be much larger than the source voltage (voltage magnification).
Q-factor measures the sharpness of resonance — how selective the circuit is. Q = ω꜀L/R = 1/(ω꜀CR) = (1/R)√(L/C). A high Q means sharp resonance (narrow bandwidth, selective) — used in radio tuning. A low Q means broad resonance (less selective). Bandwidth Δω = ω꜀/Q = R/L. The frequencies at which power falls to half the maximum (half-power points) are: ω₁ = ω꜀ − R/(2L) and ω₂ = ω꜀ + R/(2L). In this problem: Q = ω꜀L/R = (10⁵ × 10⁻³)/(1000) = 100/1000 = 0·1 (low Q, broad resonance).
📌 V_R = IR (in phase with current) — equals source voltage V at resonance
📌 V_L = IX_L = IωL (leads current by 90°) — can exceed source voltage!
📌 V_C = IX_C = I/(ωC) (lags current by 90°) — can exceed source voltage!
📌 V_L = V_C at resonance, so they cancel → net reactive voltage = 0
📌 V_L/V = V_C/V = Q (Q-factor = voltage magnification ratio)
📌 This is why capacitors/inductors can have high voltages even with small source voltage!
Instantaneous power P = V·I varies with time in AC. Average (real) power = V_rms × I_rms × cos φ, where cos φ is the power factor. At resonance: φ = 0, cos φ = 1, all power is dissipated in R (maximum power transfer). Purely inductive (φ = 90°) or purely capacitive (φ = −90°) circuits: cos φ = 0, no real power consumed — only reactive power (stored and returned each cycle). Wattless current: the component of current 90° out of phase with voltage does no work.
Series resonance is used in radio and TV tuning circuits — by varying C (or L), the resonant frequency is tuned to match the desired station's frequency, giving maximum current (and hence signal) only at that frequency. The selectivity (ability to distinguish stations) depends on Q-factor. Parallel resonance (anti-resonance) is used in filter circuits and oscillators. MRI (Magnetic Resonance Imaging) uses nuclear magnetic resonance — a completely different type but same concept.
📌 Series resonance: Z minimum = R, current maximum, voltage magnification = Q
📌 Parallel resonance: Z maximum = L/(CR), current minimum, current magnification = Q
📌 Both: resonant frequency f꜀ = 1/(2π√LC) (same formula)
📌 Series used for: current amplification, tuning, band-pass filters
📌 Parallel used for: impedance matching, oscillators, band-stop filters
A transformer works on the principle of mutual inductance — changing current in one coil (primary) induces EMF in another coil (secondary). V_s/V_p = N_s/N_p = I_p/I_s (for ideal transformer). Step-up transformer: N_s > N_p → increases voltage, decreases current. Step-down: N_s < N_p → decreases voltage, increases current. Power is conserved in ideal transformers: V_p I_p = V_s I_s. Real transformers have losses: copper loss (I²R in windings), iron loss (eddy currents + hysteresis in core). Laminating the core reduces eddy current losses.