RCL Circuits

Articles 3, 4 and 5 met the resistor, the capacitor and the coil one at a time. Article 9 put parts together on DC, and article 10 did the same on AC. This article combines all three: how they shape signals over time and frequency, how they resonate, where that is useful and why it can be dangerous.

This site calls them RCL circuits. Most textbooks write RLC,[1]​[2] and the instrument that measures these parts is sold as an LCR meter.[3] All three names mean the same thing: a circuit built from resistance (R), inductance (L) and capacitance (C). The order of the letters does not matter.

About the analogies. Everyday comparisons appear only in shaded "Analogy" boxes beside the real physics, and each box says where the comparison stops working.

Resonance can multiply voltage. A resonant circuit fed from a few volts can build up far higher voltages across its coil and capacitor. Section 12 shows how much.

1. A short recap

2. RC circuits

Charging and discharging: τ = RC

Connect a capacitor to a DC source through a resistor. At first the whole supply sits across the resistor and the current is largest. As charge builds, the capacitor's voltage rises, the current falls and charging slows, so the voltage climbs along an exponential curve (Figure 1). The pace is set by the time constant, τ = RC, in seconds when R is in ohms and C in farads.[5] After one τ the voltage has reached 63.2 percent of its final value.[6]​[5] After five time constants it is within 1 percent and counts as finished. Discharging through a resistor is the mirror image: the voltage falls to 36.8 percent in each τ. (Recap: charging through a resistor, with a worked 1-second example, and transients in DC circuits.)

Capacitor charging and discharging curves Two graphs against time measured in time constants from 0 to 5. Left, charging: the capacitor voltage rises steeply at first and levels off, reaching 63.2 percent of the supply at one time constant and 99.3 percent at five. Right, discharging: the voltage falls from 100 percent to 36.8 percent at one time constant and 0.7 percent at five. Charging 0 1τ 2τ 3τ 4τ 5τ 0% 50% 100% Time, in time constants (τ = RC) % of supply voltage 63.2% at 1τ 99.3% at 5τ Discharging 0 1τ 2τ 3τ 4τ 5τ 0% 50% 100% Time, in time constants (τ = RC) % of starting voltage 36.8% at 1τ 0.7% at 5τ
Figure 1. Charging (left) and discharging (right) a capacitor through a resistor, with time measured in time constants. Values are the exact exponential curves (calculated); the 63.2 percent rule is from OpenStax and Kuphaldt.

RC filters and the cutoff frequency

A filter is a circuit that lets some frequencies through and holds others back.[7] Two parts are enough to make one (Figure 2):

The dividing line is the cutoff frequency, fc, where the output has fallen to 70.7 percent of the input. For a simple RC filter that is where the capacitor's reactance equals the resistance,[7] which gives

fc = 1 / (2πRC)

At 70.7 percent of the voltage, the power is half, so fc is also called the half-power point.[8]

Worked example (calculated). Take R = 1 kΩ and C = 100 nF. Then fc = 1 / (2π × 1,000 × 0.0000001) ≈ 1,592 Hz, about 1.6 kHz, and the time constant is 100 µs. As a low-pass filter it passes 99.5 percent of a 159 Hz signal, 70.7 percent at 1.6 kHz and only about 10 percent at 15.9 kHz. As a high-pass filter the same parts do the reverse. These figures assume nothing is connected to the output; a load changes the response.[7]

Four simple RC and RL filters Four schematics, each with input terminals on the left and output terminals on the right. RC low-pass: a resistor in series, a capacitor across the output. RC high-pass: a capacitor in series, a resistor across the output. RL low-pass: a coil in series, a resistor across the output. RL high-pass: a resistor in series, a coil across the output. RC low-pass R C in out RC high-pass C R in out RL low-pass L R in out RL high-pass R L in out
Figure 2. Two-part filters. Top: RC low-pass and high-pass. Bottom: RL low-pass and high-pass. In each, the part across the output terminals decides which frequencies are shorted away. Source: Kuphaldt.

3. RL circuits

A coil and a resistor behave as the mirror image of an RC circuit. Switched onto a DC source, the coil's current cannot jump; it rises toward V/R along the same kind of exponential curve, with the time constant τ = L/R.[9]​[6] Here a larger resistance makes the circuit faster. (Recap: the RL time constant, with a worked example.)

RL filters follow the same logic as RC ones. A coil in series makes a low-pass filter, because its reactance rises with frequency, and a coil across the output makes a high-pass filter (Figure 2).[7] The cutoff is again where reactance equals resistance, which for a coil gives fc = R / (2πL). For example, 10 mH and 100 Ω give τ = 100 µs and fc ≈ 1.6 kHz, the same as the RC example (calculated). In a power supply, where added series resistance is unwanted, a series coil is the better low-pass filter; where light weight and small size matter more, a capacitor filter may make more sense.[7] (Recap: filters and chokes.)

4. Frequency response

A frequency-response graph plots how much of the input reaches the output at each frequency, often with the frequency axis in steps of ten so a wide range fits. Filters come in four general types, shown in Figure 3:[8]​[7]

Band-pass and band-stop filters can be made by combining low-pass and high-pass sections, or more neatly with a coil and capacitor tuned to resonance (Section 6).[7]​[8]

Frequency response of the four filter types Four graphs of output as a percentage of input against frequency on a scale where each step is ten times the last. Low-pass: near 100 percent at low frequencies, crossing 70.7 percent at the cutoff frequency and falling away above it. High-pass: the mirror image, rising through 70.7 percent at the cutoff. Band-pass: a peak of 100 percent at the resonant frequency, low on both sides. Band-stop: near 100 percent on both sides with a notch to zero at the resonant frequency. Low-pass 0.01 0.1 1 10 100 Frequency ÷ fc 0% 50% 100% High-pass 0.01 0.1 1 10 100 Frequency ÷ fc 0% 50% 100% Band-pass 0.01 0.1 1 10 100 Frequency ÷ f0 0% 50% 100% Band-stop 0.01 0.1 1 10 100 Frequency ÷ f0 0% 50% 100% Vertical axes: output as % of input. Each step along the bottom is ten times the last. Dashed line: 70.7%.
Figure 3. The four filter types. Each graph plots output as a percentage of input against frequency, in steps of ten. The dashed line marks 70.7 percent, the half-power level. Curves calculated for simple first-order (low-pass and high-pass) and resonant (band-pass and band-stop) circuits; the four types are from NEETS and Kuphaldt.

5. Impedance in a series RCL circuit

Now put a resistor, a coil and a capacitor in series across an AC source. The same current flows through all three, but their voltages are out of step, so resistance and reactance cannot simply be added. Article 10 showed this for a resistor and a coil; here is the general case.[1] (Recap: series and parallel circuits.)

Phasors: rotating arrows

A phasor is an arrow whose length stands for the size of a sine wave and whose angle stands for its phase. Picture it spinning counterclockwise once per cycle; its shadow on the vertical axis traces the sine wave.[10] Because every voltage in the circuit has the same frequency, all the arrows spin together, and you can freeze them at one instant to compare their angles.[11]

Draw the current arrow pointing right (Figure 4). The resistor's voltage, in step with it, points the same way. The coil's voltage points straight up, 90° ahead of the current. The capacitor's voltage points straight down, 90° behind.[11] These two partly cancel. Only their difference, set by the net reactance XL − XC, adds to the resistor's voltage at right angles. By Pythagoras's theorem:[1]​[11]

Z = √(R2 + (XL − XC)2)

The source voltage is the sum of the arrows, and it sits at the phase angle φ (phi) from the current, where cos φ = R/Z.[1] If XL is bigger, the circuit acts like a coil and the current lags. If XC is bigger, it acts like a capacitor and the current leads.[2]

Phasor diagram of a series RCL circuit Arrows drawn from a common origin. The current direction points right as the reference. The resistor voltage arrow points right, in step with the current. The coil voltage arrow points straight up, 90 degrees ahead; the capacitor voltage arrow points straight down, 90 degrees behind, and is shorter. A dashed line from the tip of the resistor arrow shows the difference between the coil and capacitor voltages. The source voltage arrow runs from the origin to that point, at phase angle phi above the current. A curved arrow notes that all the arrows turn together, counterclockwise, once per cycle. I (reference) VR VL VC VL − VC V (source) φ All arrows turn together, counter- clockwise, once per cycle VR: in step with I VL: 90° ahead of I VC: 90° behind I V: the sum, at angle φ Here the coil's voltage is bigger than the capacitor's, so V leads I and the circuit acts inductive.
Figure 4. Phasor diagram for a series RCL circuit at one instant. The current I is the reference. VR is in step with it, VL leads by 90° and VC lags by 90°. VL and VC partly cancel, and the source voltage V is the sum, at phase angle φ. All the arrows rotate together. Sources: OpenStax.

Optional: why engineers use complex numbers. A phasor can be written as a pair of numbers: how far it reaches to the right, and how far up. Engineers mark the "up" part with the letter j and call it the "imaginary" part; the name just labels the vertical direction. A series RCL impedance becomes Z = R + j(XL − XC), and complex arithmetic handles the arrow geometry automatically.[12] You do not need it for this article.

Worked example (calculated). Take a 10 Ω resistor, a 10 mH coil and a 1 µF capacitor in series on a 1 V RMS source at 1 kHz. Then XL = 2π × 1,000 × 0.01 ≈ 62.8 Ω and XC = 1/(2π × 1,000 × 0.000001) ≈ 159.2 Ω, so the net reactance is about −96.3 Ω. The impedance is Z = √(102 + 96.32) ≈ 96.8 Ω, and the current is 1/96.8 ≈ 10.3 mA. The phase angle is about 84°, with the current leading, because at 1 kHz the capacitor dominates.

6. Series resonance

Because XL rises with frequency and XC falls, at exactly one frequency they are equal. That is the resonant frequency, f0:[1]​[8]

f0 = 1 / (2π√(LC))

At f0 the coil and capacitor voltages cancel completely. The impedance falls to its minimum, just R, so the current reaches its maximum, and voltage and current are in phase.[1]​[2] Below f0 the circuit acts capacitive; above it, inductive.[2] The same L and C give the same f0 whether they are in series or in parallel.[8] (Recap: resonance in a radio tuner.)

Worked example (calculated). Use the same 10 Ω, 10 mH and 1 µF parts. Then f0 = 1 / (2π√(0.01 × 0.000001)) ≈ 1,592 Hz. There XL = XC = 100 Ω, so Z = 10 Ω, and 1 V RMS drives 100 mA, ten times the current at 1 kHz. Notice the voltage across the coil: 100 mA × 100 Ω = 10 V, and the same across the capacitor. That is ten times the source voltage. The two cancel, so the source never sees them, but each part does (Section 12).

Series resonance: reactance, impedance and current Left graph, ohms against frequency from 100 hertz to 10 kilohertz. The coil reactance rises with frequency and the capacitor reactance falls. They cross at 100 ohms at about 1.59 kilohertz, where the impedance curve dips to a minimum of 10 ohms. Right graph, current from a 1 volt source against the same frequencies: a sharp peak of 100 milliamps at about 1.59 kilohertz, falling away on both sides. Reactance and impedance 100 Hz 1 kHz 10 kHz 0 Ω 100 Ω 200 Ω 300 Ω 400 Ω XL XC Z At f0 ≈ 1.59 kHz, the reactances cross at 100 Ω and Z dips to 10 Ω (dots). Frequency Current from a 1 V source 100 Hz 1 kHz 10 kHz 0 mA 50 mA 100 mA 100 mA peak at f0 Frequency
Figure 5. Series resonance for 10 Ω, 10 mH and 1 µF. Left: XL rises and XC falls with frequency; they cross at 100 Ω at about 1.59 kHz, where the impedance Z dips to 10 Ω. Right: the current from a 1 V source peaks at 100 mA there. Values calculated; the relationships are from OpenStax.

7. Parallel resonance: the tank circuit

Connect a coil and capacitor in parallel and you get a tank circuit, so called because it stores energy and passes it back and forth.[2] Here is the mechanism, step by step (Figure 6):[13]​[2]

  1. The capacitor starts charged. All the energy is in its electric field, and no current flows yet.
  2. The capacitor discharges through the coil, which opposes the change, so the current and its magnetic field build gradually.
  3. When the capacitor is empty, the current is at its maximum and all the energy is in the coil's magnetic field.
  4. The coil now resists the fall in current, so its collapsing field keeps the current flowing in the same direction. That current charges the capacitor the opposite way.
  5. The process reverses, and one cycle later it is back where it started.

The exchange happens at f0, however much energy is stored.[2] Resistance turns a little energy into heat each cycle, so a tank left alone rings down; an amplifier that tops it up keeps it going. The Navy's training manuals call this the flywheel effect.[8]

Energy sloshing in a tank circuit Four snapshots of a capacitor connected in a loop with a coil. 1: the capacitor is charged, top plate positive, with an electric field between its plates and no current; the energy is in the electric field. 2: the capacitor is empty and the current around the loop is at its maximum, with a magnetic field around the coil holding the energy. 3: the capacitor is charged the other way, top plate negative, and the current is zero again. 4: the capacitor is empty and the current is at its maximum in the reverse direction, with the energy back in the magnetic field. Then the cycle repeats from 1. 1 + − E C full, no current Energy in the electric field 2 B current C empty, current at maximum. Energy in the magnetic field 3 − + E C full the other way, no current. Energy in the electric field 4 B current C empty, current at maximum, reversed. Energy in the magnetic field
Figure 6. One cycle of a tank circuit. Energy moves from the capacitor's electric field to the coil's magnetic field and back, with the current reversing each half cycle. Sources: OpenStax; Kuphaldt.

Analogy: a pendulum (Kuphaldt's own comparison).[2] A swinging pendulum trades height for speed and back again. At the top of each swing it is still, like the charged capacitor with no current. At the bottom it is fastest, like the coil at maximum current. Air friction damps it, as resistance damps the tank.

Where it breaks down: a pendulum's rate depends on its length and gravity, and is steady only for small swings; a tank's depends only on L and C. And nothing in a tank swings: the energy is in fields, and the electrons in the wire drift only slowly.

What the source sees. At f0 a practical tank's impedance is at a maximum, the opposite of series resonance, so it draws little current from the source,[8] while a much larger current circulates between the coil and capacitor. In a parallel resonant circuit, the current through L or C is Q times the current from the source.[2] (Q is explained next.) Depending on where they are placed in a circuit, series and parallel resonant circuits can act as band-pass or band-stop filters.[8]​[2]

8. Q factor and bandwidth

The quality factor, Q, measures how sharp a resonance is. For a series circuit it is the reactance of either part at resonance divided by the resistance:[2]​[14]

Q = XL / R

The coil's wire resistance usually limits Q, because capacitors have far lower losses.[8] The bandwidth is the range of frequencies between the two half-power points, where the response falls to 70.7 percent of its peak:[8]​[14]

bandwidth = f0 / Q[2]

High Q gives a narrow, tall peak; low Q gives a broad, low one (Figure 7).[8] In the worked example, Q = 100 Ω / 10 Ω = 10, so the bandwidth is about 1,592 / 10 ≈ 159 Hz, from about 1,514 to 1,673 Hz (calculated). With only 1 Ω of resistance, Q would be 100 and the bandwidth about 16 Hz.

Resonance curves for three values of Q Graph of current as a percentage of its own peak against frequency from 800 to 3,200 hertz, for 10 millihenries and 1 microfarad with three resistances. All three peak at about 1,592 hertz. With 10 ohms (Q of 10) the peak is narrow; with 20 ohms (Q of 5) it is wider; with 50 ohms (Q of 2) it is broad. A dashed line marks 70.7 percent, and a bracket shows the 159 hertz bandwidth of the Q of 10 curve. 1,000 Hz 1,500 Hz 2,000 Hz 2,500 Hz 3,000 Hz 0% 50% 100% Current, % of its peak Frequency 70.7% 10 Ω, Q = 10 20 Ω, Q = 5 50 Ω, Q = 2 Bandwidth at Q = 10: about 159 Hz, between the 70.7% points
Figure 7. Resonance curves for 10 mH and 1 µF with three series resistances, each scaled to its own peak: 50 Ω (Q = 2), 20 Ω (Q = 5) and 10 Ω (Q = 10). The bracket marks the 159 Hz bandwidth at Q = 10. Values calculated; the definitions are from NEETS and OpenStax.

Selectivity. This sharpness is what lets a radio pick one station out of many. You tune the circuit by adjusting L or C until f0 matches the station. With a high Q, stations even slightly off frequency are rejected.[14]​[8]

Tuned circuits in radio and radar receivers

Radar frequencies are too high to amplify easily, so a radar receiver is a superheterodyne. A mixer combines the incoming echo with a local oscillator's signal to produce a fixed, lower intermediate frequency (IF), usually 30 or 60 MHz according to the Navy's radar manual. Several IF amplifier stages then provide most of the gain and often set the overall bandwidth. The detector that follows uses a tuned circuit resonant at the IF.[15] Even the transmitter's magnetron uses tuned circuits, in the form of cavities in its plate.[15]

Bandwidth in a radar is a trade-off: receiver noise is proportional to bandwidth, but too narrow a receiver does not amplify and process the echoes properly. In practice a pulse radar's receiver bandwidth is close to the reciprocal of its pulse length: about 1 MHz for 1 µs pulses.[16] At a 60 MHz IF, a 1 MHz bandwidth corresponds to an overall Q of about 60 (calculated, from bandwidth = f0/Q).

9. Damping: ringing, critical and overdamped

Switch a DC voltage onto a series RCL circuit, or kick a tank, and what follows depends on the resistance. OpenStax describes three cases (Figure 8), the same three that a damped spring shows:[17]​[18]

The boundary comes from OpenStax's condition for oscillation, and works out to R = 2√(L/C).[17] For 10 mH and 1 µF that is 2 × √10,000 = 200 Ω (calculated). With the 10 Ω of the worked example the circuit is strongly underdamped. It rings at about 1.59 kHz, and the swings shrink by 63 percent every 2L/R = 2 ms (calculated from OpenStax's decay factor, e−Rt/2L). Figure 8 uses 20 Ω, so its swings shrink faster and fit on the graph.

Underdamped, critically damped and overdamped responses Graph of capacitor voltage against time from 0 to 5 milliseconds after a 1 volt step is applied to 10 millihenries and 1 microfarad in series. With 20 ohms the voltage overshoots to about 1.7 volts and rings above and below 1 volt, the swings shrinking over a few milliseconds. With 200 ohms it rises smoothly to 1 volt without overshoot. With 1,000 ohms it rises more slowly, still well short of 1 volt at the start and creeping up over the 5 milliseconds. 0 ms 1 ms 2 ms 3 ms 4 ms 5 ms 0 V 0.5 V 1 V 1.5 V 2 V Time after the switch closes Capacitor voltage 20 Ω: underdamped, overshoots and rings 200 Ω: critically damped, no overshoot 1 kΩ: overdamped, slower final value, 1 V
Figure 8. Capacitor voltage after a 1 V step is switched onto 10 mH and 1 µF in series, for three resistances: 20 Ω (underdamped, rings), 200 Ω (critically damped) and 1 kΩ (overdamped). Curves calculated; the three cases are from OpenStax.

Analogy: a weight on a spring (OpenStax relates the RCL circuit to a damped spring).[17] Pull a weight on a spring and let go. With little friction it bounces many times before settling (underdamped). In a thick fluid it oozes back without a bounce (overdamped). OpenStax gives a car's shock absorbers as an example of critical damping: the car settles as quickly as possible without bouncing.[18]

Where it breaks down: the spring's damping comes from moving through a fluid, and the weight really moves. In the circuit the "damping" is resistance turning current into heat, and the "bounce" is energy trading between two fields. The coil plays the part of the weight's inertia and the capacitor the spring, but no part of the circuit moves.

Damping matters in practice. TI's EMI-filter design guide warns that an LC filter at a switching power supply's input can make the supply unstable, and adds a damping capacitor to prevent it.[19]

10. Where RCL circuits are used

11. Measuring and experimenting

The LCR meter

An LCR meter measures inductance, capacitance and resistance by applying a sine wave at a chosen test frequency. From the resulting current and its phase it works out the impedance, and most also show Z, phase angle, Q and ESR (equivalent series resistance).[3] B&K Precision advises testing a part at or near the frequency where it will be used.[3] Never connect an LCR meter to a powered circuit, and discharge capacitors first.[3] (Recap: stored charge.)

A beginner experiment: find a resonance

A function generator (a bench source of adjustable sine waves) and an oscilloscope can do this by hand, with a small series resistor whose voltage shows the current.[3] You need:

Wire the parts in series with the generator, resistor at the ground end (Figure 9). Generators and scopes usually have one side tied to ground,[3] so this lets both probe grounds share one point. Connect channel 1 across the generator and channel 2 across the resistor. (Recap: why a scope ground clip goes only to ground.)

  1. Sweep the generator from 1 kHz to 20 kHz. Channel 2 should peak near the calculated f0 = 1/(2π√(0.01 × 0.0000001)) ≈ 5.0 kHz (calculated). At the peak, the two traces line up in phase.
  2. Find the two frequencies where channel 2 falls to 70.7 percent of its peak. Their difference is the bandwidth, and f0 divided by it is your Q.
  3. Swap the 10 Ω resistor for 100 Ω and repeat. The peak gets lower and broader.

Expect your numbers to differ from the calculation. Part tolerances shift f0, and the coil's wire resistance and the generator's own output resistance (stated in its manual) add to R, lowering Q. At resonance the coil's reactance is about 316 Ω, so Q is 316 Ω divided by the total resistance. Even with only the 10 Ω, Q would be about 32, putting about 32 V RMS across the capacitor (calculated). That is why it is rated for 100 V.

Test circuit for finding series resonance Schematic. A function generator set to a 1 volt RMS sine wave drives a loop: a 10 millihenry coil, then a 100 nanofarad capacitor, then a 10 ohm resistor down to ground, and back to the generator. Scope channel 1 connects to the generator output. Scope channel 2 connects to the top of the 10 ohm resistor, so it shows a voltage proportional to the current. The resistor sits at the ground end, so the generator ground and both scope ground clips share one ground point. Function generator, 1 V RMS sine L = 10 mH C = 100 nF R = 10 Ω Ground: the generator and both scope ground clips CH1 source CH2 current (as V across 10 Ω)
Figure 9. Finding resonance. A function generator drives a 10 mH coil, a 100 nF capacitor and a 10 Ω sense resistor in series. The resistor sits at the ground end, so both scope channels share one ground: channel 1 shows the source and channel 2 shows the current. Method from B&K Precision; values calculated.

12. Safety: voltage magnification and stored energy

A low-voltage source does not guarantee low voltages. At series resonance the voltage across the coil, and across the capacitor, is Q times the source voltage. Work out Q × V before you switch on.

Kuphaldt warns that the high current at series resonance can produce dangerously high voltages across the capacitor and inductor.[2] That voltage is Q times the applied voltage,[2] which the Navy calls the "gain" of a series-resonant circuit.[8] TI's wireless-charger guide warns that tank voltages "can swing very high" and that capacitor voltage ratings must be observed.[23]

Worked example (calculated). Feed the 10 mH and 1 µF pair from a 12 V RMS AC source, such as a small transformer's secondary, through only 3.3 Ω. Then:

Stored energy. A capacitor stores ½CV2[24] and a coil stores ½LI2.[25] In the example, the capacitor at 514 V peak holds ½ × 0.000001 × 5142 ≈ 0.13 J. A quarter cycle later the coil holds the same 0.13 J at its 5.1 A peak (calculated). Switch off and the capacitor may keep its charge (recap). Interrupt the coil's current and it kicks back a voltage spike (recap).

Habits for experimenting:

The whole story in brief

Looking back over the Learn path

This article completes the first stretch of the Learn path. Articles 1 and 2 started with electrons and materials. Articles 3 to 8 introduced the parts: resistors, capacitors, coils, diodes, transistors and transformers. Article 9 and article 10 put them into DC and AC circuits, and this one showed how R, C and L together filter, tune, ring and resonate. The articles that follow cover practical components one at a time, and you can put the ideas to work in the Microcontrollers pages.

Coming up in this section: 12 Switches and Relays.

Sources

All links were checked in October 2026. External links open in a new tab.

  1. OpenStax, College Physics 2e, section 23.12 "RLC Series AC Circuits": https://openstax.org/books/college-physics-2e/pages/23-12-rlc-series-ac-circuits
  2. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume II (AC), chapter 6, "Resonance": https://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_6.html
  3. B&K Precision, "LCR Meter Guide" (PDF): https://bkpmedia.s3.amazonaws.com/downloads/guides/en-us/lcr_meter_guide.pdf
  4. OpenStax, College Physics 2e, section 23.11 "Reactance, Inductive and Capacitive": https://openstax.org/books/college-physics-2e/pages/23-11-reactance-inductive-and-capacitive
  5. OpenStax, University Physics Volume 2, section 10.5 "RC Circuits": https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits
  6. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume I (DC), chapter 16, "RC and L/R Time Constants": https://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_16.html
  7. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume II (AC), chapter 8, "Filters": https://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_8.html
  8. U.S. Navy, NEETS Module 9: Introduction to Wave-Generation and Wave-Shaping, chapter 1, "Tuned Circuits", and chapter 2, "Oscillators" (PDF): https://legacy.maritime.org/doc/neets/mod09.pdf
  9. OpenStax, University Physics Volume 2, section 14.4 "RL Circuits": https://openstax.org/books/university-physics-volume-2/pages/14-4-rl-circuits
  10. OpenStax, University Physics Volume 2, section 15.2 "Simple AC Circuits": https://openstax.org/books/university-physics-volume-2/pages/15-2-simple-ac-circuits
  11. OpenStax, University Physics Volume 2, section 15.3 "RLC Series Circuits with AC": https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac
  12. Tony R. Kuphaldt, Lessons In Electric Circuits, Volume II (AC), chapter 2, "Complex Numbers": https://www.ibiblio.org/kuphaldt/electricCircuits/AC/AC_2.html
  13. OpenStax, University Physics Volume 2, section 14.5 "Oscillations in an LC Circuit": https://openstax.org/books/university-physics-volume-2/pages/14-5-oscillations-in-an-lc-circuit
  14. OpenStax, University Physics Volume 2, section 15.5 "Resonance in an AC Circuit": https://openstax.org/books/university-physics-volume-2/pages/15-5-resonance-in-an-ac-circuit
  15. U.S. Navy, NEETS Module 18: Radar Principles, chapter 2, "Radar Subsystems" (PDF): https://legacy.maritime.org/doc/neets/mod18.pdf
  16. Christian Wolff, Radartutorial.eu, "Radar Receivers": https://www.radartutorial.eu/09.receivers/en/rx04.en.html
  17. OpenStax, University Physics Volume 2, section 14.6 "RLC Series Circuits": https://openstax.org/books/university-physics-volume-2/pages/14-6-rlc-series-circuits
  18. OpenStax, University Physics Volume 1, section 15.5 "Damped Oscillations": https://openstax.org/books/university-physics-volume-1/pages/15-5-damped-oscillations
  19. Texas Instruments, AN-2162, "Simple Success With Conducted EMI From DC-DC Converters" (PDF): https://www.ti.com/lit/an/snva489c/snva489c.pdf
  20. U.S. Navy, NEETS Module 7: Introduction to Solid-State Devices and Power Supplies, chapter 4, "Solid-State Power Supplies" (PDF): https://legacy.maritime.org/doc/neets/mod07.pdf
  21. Carl W. Moreland, "BFO Theory", Geotech (PDF, 1999): https://geotech1.com/pages/metdet/info/bfotheory/bfo.pdf
  22. Wireless Power Consortium, "Principle of Inductive Power": https://www.wirelesspowerconsortium.com/knowledge-base/magnetic-induction/principle-of-inductive-power/
  23. Texas Instruments, SLUA635A, "Building a Wireless Power Transmitter" (PDF): https://www.ti.com/lit/an/slua635a/slua635a.pdf
  24. OpenStax, College Physics 2e, section 19.7 "Energy Stored in Capacitors": https://openstax.org/books/college-physics-2e/pages/19-7-energy-stored-in-capacitors
  25. OpenStax, University Physics Volume 2, section 14.3 "Energy in a Magnetic Field": https://openstax.org/books/university-physics-volume-2/pages/14-3-energy-in-a-magnetic-field