AC Circuits and Resonance

01 / UNDERSTAND & APPLY

1. Alternating values need a meaningful average

A sinusoidal voltage can be written v(t) = V₀ sinωt, where V₀ is peak voltage, ω is angular frequency and f = ω/(2π). Its average over a full cycle is zero, but it can still heat a resistor because heating depends on the square of current. The ordinary average is therefore not the quantity used to compare AC heating with DC heating.

Root-mean-square voltage is the square root of the time average of v². For a sine wave, the mean of sin² is one half, so Vᵣₘₛ = V₀/√2. Likewise Iᵣₘₛ = I₀/√2 for sinusoidal current. A DC current equal to Iᵣₘₛ produces the same average heating in a fixed resistor. The √2 relation is waveform-specific; it is not valid for every alternating signal.

Keep peak, rms and instantaneous values separate. If a source is stated as 230 V rms, its sinusoidal peak is about 325 V. Multiplying a peak voltage by an rms current introduces a spurious factor. Use one convention consistently throughout a phasor or power calculation, normally rms for circuit ratings.

Phase describes timing relative to another sinusoid of the same frequency. Two quantities can have the same rms magnitude yet reach their maxima at different times. A phase difference is dimensionless when expressed in radians, and a time delay Δt corresponds to phase ωΔt. Phase, not just magnitude, is essential when adding AC voltage drops.

02 / UNDERSTAND & APPLY

2. Resistors, inductors and capacitors have different timing

For an ideal resistor, vR = Ri, so voltage and current are in phase. For an ideal inductor, vL = L di/dt. Differentiating a sine shifts it ahead by a quarter cycle, so inductor voltage leads current by 90°. For a capacitor, i = C dvC/dt, so current leads capacitor voltage by 90°.

Reactance relates sinusoidal voltage and current amplitudes without implying dissipation. Inductive reactance is XL = ωL and increases with frequency. Capacitive reactance magnitude is XC = 1/(ωC) and decreases with frequency. At very low frequency, an ideal inductor offers little reactance while a capacitor offers much; at high frequency the trend reverses.

An ideal inductor or capacitor has zero average real power over a cycle because it alternately stores and returns energy. Instantaneous power can be positive or negative. Real coils and capacitors include losses; an inductor’s winding resistance is not eliminated by calling it an inductive component.

A phasor represents the amplitude and phase of a sinusoid as a rotating vector or complex number. Only sinusoids of the same angular frequency combine in a single steady-state phasor diagram. Phasors suppress the common time dependence; they do not mean that a circuit voltage is literally a spatial vector.

03 / UNDERSTAND & APPLY

3. Add series voltage drops as phasors

In a series RLC circuit, the same current flows through all elements. Choose current as the horizontal reference phasor. The resistor voltage lies along it, the inductor voltage points 90° ahead and capacitor voltage 90° behind. Therefore the reactive voltage is the difference VL − VC, not their sum of magnitudes.

The source voltage magnitude obeys V² = (IR)² + [I(XL − XC)]². Impedance magnitude is Z = √[R² + (XL − XC)²] and I = V/Z, using either rms or peak consistently. The source voltage phase relative to current satisfies tan φ = (XL − XC)/R for R > 0. Positive φ means current lags a net-inductive source voltage; negative φ means current leads.

At series resonance XL = XC, giving ω₀ = 1/√(LC). The net reactive drop vanishes, impedance becomes R and current is maximal for fixed source voltage and fixed R. The inductor and capacitor voltages need not vanish; they are equal and opposite phasors and can each exceed the source voltage.

The familiar current maximum at ω₀ refers to a series circuit with ideal L and C and frequency-independent R. Other quantities, such as capacitor voltage in a damped circuit, can peak at slightly different frequencies. Parallel resonance has a different impedance behaviour and cannot be inferred by copying the series result.

04 / UNDERSTAND & APPLY

4. Power factor and resonance sharpness answer different questions

For sinusoidal source voltage and current with phase difference φ, average real power is P = Vᵣₘₛ × Iᵣₘₛ × cos φ. Here P is power averaged over a full cycle. Vᵣₘₛ and Iᵣₘₛ are the root mean square (rms), or effective, voltage and current; cos φ is the power factor. In a fixed resistor, an rms current produces the same average heating as a DC current of that value. In a series RLC circuit cos φ = R/Z and P = Iᵣₘₛ² × R. Apparent power Vᵣₘₛ × Iᵣₘₛ is measured in volt-amperes and is not automatically the dissipated power. An ideal pure reactance has zero average real power despite nonzero current.

At resonance, φ = 0 and power factor is unity in the ideal series model. Away from resonance, some energy oscillates between source and reactive elements each cycle, raising current requirements for a given real-power delivery. Power-factor correction changes the reactive demand; it does not create additional real energy.

The quality factor Q = ω₀L/R = 1/(ω₀CR) measures resonance sharpness in the series model. Large Q corresponds to relatively weak damping and a narrow current resonance. At resonance, VL/Vsource = VC/Vsource = Q for ideal components. Large internal voltages can therefore coexist with a modest applied source.

The half-power frequencies occur when current magnitude is 1/√2 of its resonant value, or |XL − XC| = R. Their angular-frequency separation is Δω = R/L for the ideal series model, yielding Q = ω₀/Δω. Avoid confusing the quality factor Q with charge q or thermal heat Q; symbols inherit meaning from context.

05 / UNDERSTAND & APPLY

5. An isolated LC circuit exchanges electric and magnetic energy

An ideal charged capacitor connected to an ideal inductor can oscillate without an external AC source. Let q denote signed capacitor charge and choose the loop current consistently. The loop equation leads to L d²q/dt² + q/C = 0, the same mathematical form as a mass–spring oscillator. The natural angular frequency is 1/√(LC).

If initially q = Q₀ and current is zero, a convenient solution is q = Q₀ cosω₀t. The current magnitude peaks a quarter period later when the capacitor charge is zero. The sign of current depends on its chosen direction; using i = dq/dt gives i = −ω₀Q₀ sinω₀t. Stating the sign convention prevents a meaningless dispute over a minus sign.

Total energy is q²/(2C) + Li²/2 = Q₀²/(2C) in the ideal lossless model. At maximum charge it is all electric; at zero charge it is all magnetic. Thus Imax = Q₀/√(LC) = V₀√(C/L) when the initial capacitor voltage is V₀. Charge and current are a quarter cycle out of phase, while their energy contributions vary twice as fast.

Real resistance dissipates energy and damps the oscillation. For a series RLC free response, underdamped oscillation requires R < 2√(L/C), and damped angular frequency is √[1/(LC) − (R/2L)²]. A circuit with stronger damping may return without oscillating. Lossless LC motion and forced steady-state resonance are related but distinct situations.

06 / UNDERSTAND & APPLY

6. A transformer changes voltage by sharing alternating flux

An ideal transformer has two windings linking the same changing core flux. Faraday’s law gives induced emf per turn equal in both, so voltage ratio Vs/Vp = Ns/Np in magnitude. With negligible losses and ideal coupling, input power equals output power; for the simple matched power-factor idealisation, current ratio Is/Ip = Np/Ns.

A step-up transformer has more secondary than primary turns and produces higher voltage with lower load current for the same transferred power. A step-down transformer does the reverse. The turns ratio does not multiply power. Increasing secondary voltage and current simultaneously without another energy source would violate conservation.

A transformer needs changing flux. A steady DC input does not produce a sustained secondary emf after the transient; a real winding can draw excessive current and overheat. The ideal transformer equations also omit magnetizing current, winding resistance, leakage flux and core losses. They describe an AC operating regime, not every instant after arbitrary switching.

For a resistive secondary load RL, the resistance reflected to the primary of an ideal transformer is Rin = (Np/Ns)²RL. Derive this from Vp/Ip using both voltage and current ratios. A transformer can therefore match impedances as well as change voltage. In general AC loads the same turns-ratio-squared relation applies to complex impedance in the ideal model.

07 / UNDERSTAND & APPLY

7. Transmission losses show why voltage matters

For a fixed real power delivered through a line, current is approximately P/V in a DC or unity-power-factor AC model. Resistive line loss is I²Rline, so raising voltage reduces loss quadratically for the same transmitted power and line resistance. Transformers make voltage conversion efficient in conventional AC systems.

If power factor is below one, rms current for real power P is P/(Vᵣₘₛ × cos φ), so line heating is larger than a unity-power-factor estimate. A problem specifying power, voltage and line resistance may assume unity power factor; state that assumption rather than silently using the DC relation for every AC load.

Real transformers lose energy through winding resistance, eddy currents, hysteresis and imperfect magnetic coupling. Laminated cores reduce eddy currents; suitable magnetic materials reduce hysteresis loss. Efficiency is Pout/Pin and must be less than or equal to one. If a problem gives efficiency, use it explicitly rather than imposing ideal equal power.

High transmission voltage introduces insulation, clearance and equipment constraints. The physics lesson is the current-squared loss advantage, not a recommendation to alter an electrical installation. For quantitative questions, distinguish total round-trip line resistance from one-conductor resistance and include whichever the problem specifies.

08 / UNDERSTAND & APPLY

8. Original practice: phase, resonance and conversion

All AC magnitudes are rms unless stated otherwise. Assume ideal elements and a sinusoidal steady state for driven-circuit questions.

09 / UNDERSTAND & APPLY

9. Choose the state before choosing the formula

First distinguish driven sinusoidal steady state from free LC oscillation or a switching transient. Phasor impedance belongs to the steady sinusoidal case; initial conditions are essential for free motion. Then mark peak versus rms values and choose a phase reference. Add voltage phasors, not their unsigned magnitudes.

For transformers, apply turns ratio to voltage and conservation to power before inferring current. For transmission, state power factor and line resistance conventions. For resonance, say which quantity is resonating and whether the circuit is series or parallel. These small statements keep a compact formula from being applied to a different physical problem.

• Rms is a heating-equivalent quantity, not the ordinary sinusoidal mean.
• L and C store and return energy with opposite phase shifts.
• Series resonance cancels reactances but not individual reactive voltages.
• Lossless LC oscillation conserves electric plus magnetic energy.
• An ideal transformer changes voltage/current ratio, not total power.
• Transmission loss scales with rms current squared.