This cheat sheet covers alternating current circuits using impedance, phasors, and resonance in resistor, inductor, and capacitor combinations. Students need these tools because AC circuit quantities change with time and cannot always be added like simple numbers. Phasors make sinusoidal voltage and current easier to compare, while impedance shows how resistors, capacitors, and inductors oppose AC.
The reference is designed for quick review during homework, labs, and exam preparation.
The core idea is that resistance, inductive reactance, and capacitive reactance combine into a complex impedance . In a series RLC circuit, the impedance magnitude is and the resonance frequency is . At resonance, , the phase angle is , and current is maximum in a series circuit.
Power depends on both rms values and phase through .
Key Facts
- Inductive reactance is , so an inductor opposes AC more strongly as frequency increases.
- Capacitive reactance is , so a capacitor opposes AC less strongly as frequency increases.
- For a series RLC circuit, the impedance is and the magnitude is .
- Ohm’s law for AC rms values is .
- The phase angle in a series RLC circuit satisfies .
- Series resonance occurs when , giving and .
- Average power in an AC circuit is for a series RLC circuit.
- The quality factor for a series RLC circuit can be written as .
Vocabulary
- Impedance
- Impedance is the total opposition an AC circuit gives to current, including both resistance and reactance.
- Reactance
- Reactance is the frequency-dependent opposition to AC current caused by inductors or capacitors.
- Phasor
- A phasor is a rotating vector used to represent the magnitude and phase of a sinusoidal voltage or current.
- Resonance
- Resonance is the condition in an RLC circuit where and the net reactance is zero.
- Phase Angle
- The phase angle measures how far voltage and current are shifted from each other in an AC circuit.
- Power Factor
- The power factor is , the fraction of apparent power converted into average real power.
Common Mistakes to Avoid
- Adding reactances without signs is wrong because inductive reactance and capacitive reactance act in opposite phasor directions. In a series RLC circuit, use , not , for net reactance.
- Using peak values in rms power formulas is wrong unless the formula is adjusted. For average power, use .
- Forgetting frequency dependence is wrong because and change when changes. Always calculate and at the given frequency.
- Assuming resonance means zero impedance is wrong for a series RLC circuit with resistance. At resonance, , so current is limited by the resistor.
- Confusing lead and lag is wrong because capacitors and inductors shift current in opposite directions. In a capacitive circuit current leads voltage, while in an inductive circuit current lags voltage.
Practice Questions
- 1 A series RLC circuit has , , , and . Find , , and .
- 2 A series circuit has , , and . Find the phase angle and state whether the circuit is inductive or capacitive.
- 3 Find the resonance frequency of a series RLC circuit with and using .
- 4 Explain why the current is maximum at resonance in a series RLC circuit even though the inductor and capacitor may each have large reactance.
Understanding Impedance, Phasors & RLC Resonance Reference
A phasor is best understood as a rotating arrow. Its length represents the size of a sinusoidal quantity, and its angle represents where that quantity is in its cycle. A full rotation matches one complete AC cycle.
When two phasors point in different directions, their voltage peaks happen at different times. This timing difference is phase difference. In a resistor, current and voltage rise and fall together.
In an inductor, current changes more slowly, so current lags voltage. In a capacitor, current responds to changing voltage, so current leads voltage. A phasor diagram turns these time relationships into simple geometry.
Reactance does not waste energy in the same way as resistance. An ideal resistor converts electrical energy into thermal energy. An ideal inductor stores energy temporarily in a magnetic field.
An ideal capacitor stores energy temporarily in an electric field. During each AC cycle, inductors and capacitors take energy from the source, then return energy later. This return of energy creates a phase difference between voltage and current.
Real components are not ideal. Wire resistance, core losses, and internal resistance in capacitors remove some energy as heat. That loss is why a real resonant circuit never has unlimited current or perfectly sharp resonance.
At resonance in a series RLC circuit, energy moves back and forth mainly between the inductor and capacitor. The source mostly replaces energy lost in the resistor. The total opposition to current is then at its smallest, so the current can become large.
The voltage across the inductor or capacitor can be much larger than the supply voltage, even though their effects cancel in the full circuit. This does not break conservation of energy.
Their voltages have opposite phase directions, so they combine to a smaller total. Students should treat these component voltages carefully in calculations and laboratory work.
The quality factor describes how selective a resonant circuit is. A larger quality factor means strong response over a narrow range of frequencies near resonance. A smaller quality factor means a broader, less selective response.
Radio receivers use tuned circuits to select one station frequency while reducing signals from other stations. Filters in audio equipment separate bass, middle, and high frequency signals.
Wireless chargers, metal detectors, sensor circuits, and power electronics use related ideas. The same physics can appear in any system that stores energy in two forms and transfers it repeatedly.
Use rms values when calculating practical AC voltage, current, and power. RMS values connect an AC signal to the heating effect of an equivalent steady DC signal. On circuit diagrams and meters, stated AC voltages are usually rms values unless another label is given.
Keep frequency units consistent, since angular frequency is not the same as ordinary frequency. Draw a phasor sketch before using equations.
Label which quantity leads or lags, then check whether the circuit should act mostly inductive or mostly capacitive. These checks catch many sign errors and help equations describe the actual circuit.