Source: University Physics: Elec & Mag, UIUC
Tags: resonance, resonant frequency, Q factor, average power, RMS, root mean square, power line, transformer, primary coil, secondary coil, turns ratio
Difficulty: Intermediate to Advanced Prerequisites: AC circuit analysis (impedance, reactance, phasor diagrams), series RLC circuits.
Big picture: Resonance is the payoff of everything in the AC circuits sequence. When you drive a series RLC circuit at exactly the right frequency, impedance drops to its minimum (just R), current is maximised, and the generator voltage and current snap into phase. This has enormous practical importance: it is how radios select a station, how MRI machines excite nuclei, and how wireless chargers transfer energy. This section also covers AC power (why we use RMS values and why power lines run at high voltage) and transformers (how to step voltage up or down).
Resonance occurs in a series RLC circuit when X_L = X_C, which happens at ω₀ = 1/√(LC). At resonance, Z = R and current is at its peak. Average AC power is P = I_rms² R, where I_rms = I_peak/√2. Transformers change voltage using the turns ratio: V_S/V_P = N_S/N_P.
Resonant frequency (ω₀)
The angular frequency at which X_L = X_C, minimising impedance: ω₀ = 1/√(LC). In simple terms, this is the "sweet spot" frequency where the inductor and capacitor cancel each other out and the circuit responds most strongly.
Q factor (quality factor)
Q² = L/(R²C). Alternatively, Q can be thought of as the ratio of resonant frequency to bandwidth. A higher Q means a sharper, narrower resonance peak: the circuit is very selective about which frequency it responds to. A lower Q gives a broader, less selective peak.
RMS (root mean square)
The effective steady-state equivalent of a time-varying quantity. For a sinusoidal current, I_rms = I_peak / √2. RMS values are what household voltage ratings refer to: 120 V (or 230 V) is an RMS figure, not a peak figure.
Average power (P_avg)
The time-averaged power dissipated in a resistor in an AC circuit: P_avg = I_rms² R = ½ I_peak² R. Only the resistor dissipates power; ideal capacitors and inductors store and return energy each cycle.
Transformer
A device with two coils (primary and secondary) wound around a shared core. It changes voltage and current levels according to V_P/N_P = V_S/N_S, where N is the number of turns.
From the impedance formula: Z = √[R² + (X_L − X_C)²].
To maximise I_max = E_max / Z, we need to minimise Z.
Z is minimised when X_L = X_C, i.e. when (X_L − X_C) = 0 and Z = R.
Setting X_L = X_C:
ωL = 1/(ωC)
ω² = 1/(LC)
ω₀ = 1/√(LC), the resonant frequency.
At resonance:
Z = R (minimum possible impedance).
I_max = E_max / R (maximum possible current).
φ = 0 (current and generator voltage are in phase).
V_L and V_C are equal in magnitude but opposite in phase, so they cancel.
Q² = L/(R²C), or equivalently one can define x = ω/ω₀ and write:
I_max = (E_max / R) × 1/√[1 + Q²(x² − 1)²/x²]
A higher Q factor produces a taller, narrower resonance peak: the circuit is more selective.
A lower Q factor produces a shorter, broader peak: less selective, but more tolerant of frequency variation.
Graphically, a Q = 2.7 circuit has a broad hump; a Q = 9.5 circuit has a sharp spike at ω₀.
Instantaneous power: P(t) = I(t)² R. Since I varies sinusoidally, P oscillates between zero and I_peak² R.
Average power:
P_avg = ½ I_peak² R
RMS values simplify this:
I_peak = I_rms √2, so I_rms = I_peak / √2
P_avg = I_rms² R
Only the resistor dissipates power. The inductor and capacitor alternately absorb and release energy each quarter-cycle, contributing nothing to the time-averaged power.
A power line has resistance R_line, causing power loss P_lost = I² R_line.
Since power delivered is P = IV, for the same power, increasing V allows a proportional decrease in I.
Halving the current reduces line losses by a factor of four (since loss goes as I²).
This is why power is transmitted at high voltage (hundreds of kV) and then stepped down near the consumer.
A transformer consists of a primary coil (P) and a secondary coil (S) linked by a magnetic core.
The fundamental relation: V_P / N_P = V_S / N_S.
Step-up transformer: N_S > N_P, so V_S > V_P (voltage increases).
Step-down transformer: N_S < N_P, so V_S < V_P (voltage decreases).
For an ideal transformer, power is conserved: V_P I_P = V_S I_S.
Stepping up voltage steps down current, and vice versa.
Quantity | Formula |
|---|---|
Resonant frequency | ω₀ = 1/√(LC) |
Impedance at resonance | Z = R |
Q factor | Q² = L/(R²C) |
RMS current | I_rms = I_peak / √2 |
Average power | P_avg = I_rms² R = ½ I_peak² R |
Power line loss | P_lost = I² R_line |
Transformer voltage ratio | V_P/N_P = V_S/N_S |
Resonance is how a radio selects one station from dozens broadcasting simultaneously: the tuning knob adjusts C (or L) until ω₀ matches the desired station's frequency. MRI scanners exploit nuclear magnetic resonance at very precise frequencies. Power-line engineers choose high transmission voltages specifically to reduce I²R losses over long distances, with transformers at each end stepping voltage up and down. Your phone charger contains a small transformer for exactly this purpose.
Students sometimes think resonance means the circuit "blows up" or current goes to infinity. It does not. At resonance, Z = R, so I_max = E_max/R. The current is large but finite, limited by the resistance.
Confusing peak and RMS values. Household "120 V" is RMS; the peak is 120√2 ≈ 170 V. Forgetting the √2 factor is a common error in power calculations.
Assuming transformers create free energy. They conserve power (ideally). If voltage goes up, current goes down proportionally.
Thinking the Q factor is the same as charge Q. They share the same letter but are entirely different quantities. Context (resonance vs. capacitor charge) tells you which is meant.
⚠️ Be ready to derive the resonant frequency from the condition X_L = X_C. This is a standard derivation question.
⚠️ Power questions almost always require RMS values. If given a peak value, convert before plugging into P = I²R.
⚠️ Transformer problems typically ask for V_S given V_P, N_P, and N_S. Know the ratio and know that power is conserved in an ideal transformer.
⚠️ Expect a question on why high-voltage transmission is more efficient. The answer hinges on P_lost = I²R and the fact that higher V means lower I for the same power.
Fill in the blank: Resonance occurs when ω = ______.
True or false: At resonance, the phase angle between current and source voltage is 90°.
Fill in the blank: I_rms = I_peak / ______.
True or false: Doubling the transmission voltage cuts power-line losses by a factor of four.
Fill in the blank: For an ideal transformer, V_P/N_P = ______.
Answers: 1. 1/√(LC). 2. False (φ = 0). 3. √2. 4. True. 5. V_S/N_S.
Q: A series RLC circuit has R = 50 Ω, L = 20 mH, and C = 5 μF. Find the resonant frequency and the maximum current if E_max = 100 V.
A: ω₀ = 1/√(LC) = 1/√(0.02 × 5×10⁻⁶) = 1/√(10⁻⁷) = 1/(3.162×10⁻⁴) ≈ 3162 rad/s. At resonance, Z = R = 50 Ω, so I_max = E_max/R = 100/50 = 2 A.
Q: An AC circuit has I_peak = 3 A and R = 40 Ω. What is the average power dissipated?
A: P_avg = ½ I_peak² R = ½ (9)(40) = 180 W. Equivalently, I_rms = 3/√2 ≈ 2.12 A, and P = I_rms² R = (4.5)(40) = 180 W.
Q: A transformer has 500 turns on the primary and 50 turns on the secondary. If the primary voltage is 240 V, what is the secondary voltage?
A: V_S = V_P × (N_S/N_P) = 240 × (50/500) = 24 V. This is a step-down transformer.
Q: Explain, in one or two sentences, why power companies transmit electricity at high voltages.
A: For a given amount of power, P = IV, a higher voltage means a lower current. Since power losses in the transmission line scale as I²R, reducing the current dramatically cuts the energy wasted as heat in the wires.
Resonance ties together the full arc of the course from Faraday's law through inductors, capacitors, and impedance. The resonant frequency ω₀ = 1/√(LC) is the same natural frequency from the LC oscillation section, now appearing as the driven frequency that maximises response. Transformers connect back to mutual inductance and Faraday's law (changing flux in one coil induces voltage in another). RMS and average power concepts reappear in any later physics or engineering course involving alternating signals.
Related Terms / Search Tags: resonance, resonant frequency, series RLC, Q factor, quality factor, impedance minimum, RMS, root mean square, average power, power dissipation, power line loss, transformer, turns ratio, step-up, step-down, primary coil, secondary coil, PHY 212, UIUC, midterm 3