Physics Ninja: AC Circuits

Explore three simple ac circuits—resistor, capacitor, and inductor—driven by an ideal ac generator with Vmax cos(omega t) to derive the voltage–current relationship and lay groundwork for analyzing more complex circuits.

Analyze current in an AC resistor circuit by applying Ohm's law to a generator with voltage v_max cos(ωt) and derive I(t) as v_max cos(ωt) divided by R, illustrating phasor representation.

Calculate power in an AC circuit resistor by multiplying instantaneous voltage and current: p(t) = Vmax Imax cos^2(ωt); average power equals Vmax Imax/2 or Vrms^2/R.

Analyze a sinusoidal 48 V peak across a 12 ohm resistor to find i rms, average power, and maximum power (i max = 4 A, p max = 192 W).

Explore capacitors in an AC circuit by relating generator voltage to capacitor voltage, deriving q(t)=qmax cos(ωt) and i(t)=dq/dt, and using phasor diagrams to show current leads voltage by 90 degrees.

Explore how voltage and current are out of phase by pi/2 and define capacitance reactance x_c in ohms as 1/(omega c), showing how it governs maximum current and frequency dependence.

Plot the capacitor voltage as the x-axis projection on phasor diagram, and the current at omega t plus pi/2 rotating counterclockwise, with its x-axis projection giving cos(omega t + pi/2).

Analyze an AC generator with an inductor, derive i(t) by integrating v = L di/dt, and show voltage leads current by 90 degrees, i(t) = Imax cos(ωt − π/2).

Define inductive reactance x_L as ωL and derive V = I x_L in an Ohm's law–style form. Note that x_L has ohms units and that maximum current depends on frequency.

Explore how capacitor and inductor reactances vary with angular frequency, acting as frequency-dependent resistances. Compare their limits to dc and high-frequency cases and to actual resistance.

Explore the phasor diagram for an inductor in an AC circuit, showing VL and I with amplitude Imax, −90° phase shift, and projection onto the positive x axis.

Explain ac circuits with resistor, capacitor, and inductor by showing how current relates to voltage: resistor in phase, capacitor current leads, inductor current lags; use Eli the ice man mnemonic.

Explore a series LRC circuit driven by an AC generator, applying Kirchhoff's loop rule to sum the resistor, capacitor, and inductor drops, while accounting for phase differences in AC.

Define the series ac current, equal through all elements, in phasor form with Imax, omega, t, and phase delta; current equals the vector projection at angle omega t minus delta.

Apply Ohm's law to express the resistor voltage as VR = R I with I = Imax cos(ωt - δ). Show their in-phase relation on a shared phasor diagram.

Learn to determine the capacitor and inductor voltages in an RLC circuit using phasors, relate them to the resistor and driving voltage, and find the max current and phase angle.

Plot a phasor diagram for the resistor, capacitor, and inductor voltages, define delta between Vmax and VR, and solve for Imax and the phase angle with the drive voltage.

Apply phasor analysis to a series RLC circuit, using Eli the Ice Man to relate voltages to current and derive impedance Z = sqrt(R^2 + (X_L - X_C)^2).

Analyze phasor diagrams to identify whether a series RLC circuit is inductively or capacitively dominated, by comparing emf and current phase relationships and recognizing leading or lagging behavior.

Analyze RLC circuit with a 200-ohm resistor, 15 µf capacitor, 230 mH inductor, driven by 36 V at 60 Hz. Compute ω, ω0, Xc, XL, impedance, Imax, and phase angle.

Derive instantaneous power p(t)=v(t)i(t) in a series RLC circuit and show the average power equals (Imax Vmax/2) cos delta, or Vrms Irms cos phi.

Tune the driving frequency to resonance in an RLC circuit, where XL equals XC, maximizing Imax as Vmax over R; at resonance, omega_r = 1/√(LC) and voltage and current align.

Explore how resonance determines average power in an RLC circuit, derive the power factor from impedance, and analyze the frequency dependence with a peak at the resonant frequency.

Explore an lrc series circuit by calculating the phase angle from xl and xc using tan phi = (xl - xc)/r, and compare the drive frequency to resonance.

Compute the full width at half maximum of the power resonance curve for a series LRC circuit, locating the two half maximum frequencies; width scales with R and equals R/L.

Define the quality factor (Q) for a harmonic oscillator and relate it to resonance in LRC circuits via energy stored over energy lost per cycle and delta omega.

Calculate the resonant angular frequency of a series RLC with L=2 H, C=2 μF, and R=20 Ω, and identify Q=50, FWHM=10 rad/s, and Pmax about 250 W.

Remove one reactive component from series lrc circuits to create simple ac filters that tune the output amplitude, using capacitors and inductors as practical filter elements.

Explore how an RC low-pass filter passes low-frequency signals and attenuates high-frequency components by analyzing the capacitor voltage using capacitive reactance Xc = 1/(omega C) and the RC impedance.

Explore the RC high-pass filter, derive the voltage across the resistor and its amplitude as a function of frequency, and show how high frequencies pass while low frequencies are attenuated.

Analyze RL filters by using inductive reactance ωL and impedance sqrt(R^2+(ωL)^2); the inductor voltage produces a high-pass response, while the resistor voltage gives a low-pass result.

Learn how transformers use coils around magnetic material to step up or step down AC power, from power plants to household voltages, guided by Faraday's law.

Explore how transformer cores link primary and secondary windings to transfer voltage via turns ratio, using Faraday's law and ideal assumptions to explain step-up and step-down behavior.

Power in transformers shows how an ideal transformer conserves power from primary to secondary, relates voltages and currents through turns ratio, and contrasts step-up and step-down behavior.

Apply the transformer model to reflect the secondary resistance to the primary, yielding Rp = (Np/ns)^2 Rs and establishing an equivalent resistance on the primary side.

An ideal transformer with 50:10 turns steps 120 V to 24 V; powers a 10 Ω load with 2.4 A secondary current and 0.48 A primary current.

Solve a transformer practice problem with a step-down transformer at 60 Hz; determine the secondary Imax using a 40 V output and the resistor-capacitor impedance, about 0.78 A.

Practice applying concepts with an extensive ac circuits problem set; download the pdf, attempt 10–15 problems at a time, then review video solutions to check your answers.

Solve ac circuit problems by linking period to frequency, rms to peak values, and peak-to-peak voltages. Apply these to compute resistance and power for 120 V rms, 60 Hz.

Explore problem solutions 13–25 for ac circuits, covering rms current, capacitive reactance, capacitor energy and current phase, impedance in rc circuits, and inductor energy at zero current.

Solve ac circuit problems by calculating inductive reactance, capacitive reactance, impedance, and phase angle; determine rms current, power factor, and resonance to derive inductance.

Learn to analyze series RLC circuits: determine resonance where XL=XC, compute L and C from reactances, and evaluate impedance, currents, voltages, and average power using phasor methods.

Doubling both l and c in a lrc circuit halves the resonant frequency, while an ideal diode in an ac circuit allows current to flow only in one direction.