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body { font-family: Arial, sans-serif; line-height: 1.6; color: #333; } h2 { border-bottom: 2px solid #3598db; padding-bottom: 10px; margin-top: 30px; } h3 { color: #2c3e50; margin-top: 25px; } img { max-width: 100%; height: auto; } .note { background-color: #f9f9f9; border-left: 5px solid #3598db; padding: 15px; margin: 20px 0; } .faq-item { margin-bottom: 20px; } .question { font-weight: bold; font-size: 16px; color: #236fa1; }IntroductionThe Q factor (quality factor) serves as a critical metric telling us how close a real-world inductor is to an ideal inductor. Inductors are ubiquitous components in power electronics converters, filter networks, and communication systems, where they are frequently used in resonant networks. While theoretical studies often treat inductors as having pure inductance, in reality, they possess inherent resistance and parasitic elements. The Q factor is defined as the ratio of the inductive reactance of the coil to its effective resistance.While the most obvious constituent of this resistance is the standard DC resistance (DCR) of the wire, high-frequency AC losses often play a more significant role. So, what is the true relationship between resistance and quality factor? Why is the Q factor so vital?Video: What is Q-Factor?Ⅰ Why is Q Factor Important for an Inductor?When selecting components, engineers must rely on the manufacturer's datasheet and product line cards after calculating the required inductance for the specific application. However, the Inductance value alone is not enough. It is crucial to consider the Quality Factor (Q) of the inductor, particularly for RF (Radio Frequency) circuits and precision analog applications.A high Q factor indicates that the inductor has low energy losses relative to the energy it stores. In resonant circuits, a high Q leads to a sharper resonance peak and narrower bandwidth, which is essential for selectivity in radio tuners. In power applications, a higher Q generally implies lower power dissipation (heat), leading to higher overall system efficiency.Ⅱ Inductor Q Factor Analysis2.1 There Are No Ideal InductorsIn practice, a "perfect" component does not exist. Inductors are constructed by winding conductive coils around cores made of various magnetic materials (ferrite, iron powder, air, etc.). The actual inductance value depends on physical parameters: the number of turns, the permeability of the core material, flux density, and the core's cross-sectional area.Furthermore, in real-world operation, the effective inductance and performance can fluctuate based on the applied current (saturation), signal frequency, aging, and operating temperature. To ensure consistent output accuracy across a wide range of frequencies and environmental conditions, specific parameters must be quantified. The Q-Factor is the primary parameter used to measure the "purity" and consistency of the coil's performance.2.2 What is Q-Factor?Figure 1. Q Factor in InductorsIdeally, an inductor would only exhibit inductance. However, a functional inductor includes fixed DC resistance, variable AC resistance, and parasitic capacitance. These parasitic elements reduce the inductor's efficiency. The Quality Factor (Q) is a dimensionless figure of merit that quantifies the inductor's performance regarding its losses. It is essentially the ratio of Energy Stored to Energy Dissipated per cycle.Let's explore the parasitic resistances that lower the Q Factor in depth:(DCR or RDC) DC ResistanceThe wire used to wind the coil has internal resistance, known as "DC resistance." This value is usually found in the "DCR" or "RDC" column of a datasheet. DCR depends on the total length of the wire and its cross-sectional area (gauge). To achieve a higher inductance, more turns are required, which increases wire length and, consequently, DCR. Designers often have to balance wire thickness and physical size. Larger diameter wires (lower gauge number) yield lower DC resistance but increase the component's size.Note: How to calculate the resistance of copper wire?Engineers often use the standard resistivity formula:Where:R is the resistance in Ohms (Ω)l is the length of the conductor in metersρ is the electrical resistivity of the material (e.g., Copper)A is the cross-sectional area in square millimeters (derived from wire diameter)Skin Effect Due to AC Resistance (Rac)When the frequency increases (roughly above 50 kHz for standard copper wire, though the effect starts earlier), AC resistance (Rac) becomes dominant over DCR. This is due to the "Skin Effect."At higher frequencies, alternating current tends to flow only near the surface (or "skin") of the conductor rather than through the entire cross-section. This effectively reduces the usable cross-sectional area of the wire, significantly increasing resistance. To mitigate this in high-Q applications, engineers often use Litz wire (multistrand insulated wire) to increase surface area.Core Hysteresis Losses (Modeled as Resistance)In magnetic cores, the magnetic domains must align and realign with the changing magnetic field (H). This realignment is not frictionless; energy is lost as heat during each cycle. This is known as Hysteresis Loss. Ideally, the B-H curve (Magnetic Flux Density vs. Magnetic Field Intensity) would be linear. In reality, it forms a loop. The area inside this loop represents energy lost per cycle.As frequency increases, these losses occur more often per second, increasing the effective resistance. This loss appears in the equivalent circuit as a resistor in series (or parallel, depending on the model) with the inductor, lowering the Q factor.Figure 2. BH Curve and Hysteresis LoopDielectric Losses (Rd)Inductors use insulation on the wire (enamel) and sometimes between layers. The core material itself is also a dielectric. These materials have finite resistance and dielectric constants. While often modeled as a parallel resistance (leakage), dielectric absorption causes losses that add to the total system energy loss, further reducing the Q factor at very high frequencies.Calculating Total Resistance and QThe total effective series resistance (ESR) in a functional inductor is the sum of these components:The Quality Factor (Q) is calculated as the ratio of Inductive Reactance ($X_L$) to this Total Resistance ($R_{total}$):Where $ omega = 2pi f $ (frequency).The Q factor can also be expressed in terms of power:Conclusion: If DCR, Skin Effect, or Core Losses increase, the denominator ($R$) increases, causing the Q-Factor to drop. A lower Q means higher power loss and broader bandwidth. Conversely, a high Q value implies that the inductor behaves more like an ideal reactance with minimal energy loss.Ⅲ What is the Role of Q Factor in a Circuit?The Q factor plays a dominant role in the **Filter Bandwidth** of practical circuits.Narrow Bandwidth (High Q): For Radio Frequency (RF) applications—such as police wireless communication or distinct radio channels—filters must be highly selective. They need to accept a specific frequency while rejecting everything else. An inductor with a High Q factor (Red line in theoretical plots) produces a sharp resonant peak, allowing for a narrow bandwidth.Wide Bandwidth (Low Q): Other applications may require a wider frequency range to pass through. An inductor with a lower Q factor (Orange line) produces a flatter, broader curve with less voltage gain at the peak but a wider passband.Additionally, designers must remember the Self-Resonant Frequency (SRF). Every inductor has parasitic capacitance between its windings. At a certain high frequency (SRF), the inductor resonates with its own capacitance and acts as a resistor. Beyond this frequency, it behaves like a capacitor, and the Q factor concept as an inductance metric becomes invalid.Frequently Asked Questions about Q Factor in Inductors1. How do you find the Q factor of an inductor?The quality factor Q of the inductor is defined by the formula $Q = frac{omega L}{R}$, where $omega$ is the angular frequency ($2pi f$), $L$ is the inductance, and $R$ is the effective series resistance (ESR). Since $R$ changes with frequency (due to skin effect and core losses), Q is frequency-dependent. It is usually measured using an LCR meter or an Impedance Analyzer at the specific operating frequency of the circuit.2. How is Q factor calculated from a bandwidth perspective?In a resonant circuit, the Q factor can be determined by the frequency spectrum. It is defined as $Q = frac{f_r}{Delta f}$, where $f_r$ is the resonant frequency (where impedance is maximum or minimum depending on circuit topology) and $Delta f$ is the -3dB bandwidth (the width of the peak at half-power). A narrower peak indicates a higher Q.3. How do I lower the Q factor of a circuit?Sometimes a high Q is undesirable because it causes ringing or oscillation. To lower the Q factor (dampen the circuit), you can add resistance to the circuit. Adding a resistor in series with the inductor increases the denominator in the $Q = frac{omega L}{R}$ equation, thereby reducing Q. Alternatively, placing a resistor in parallel with the inductor can also widen the bandwidth and lower the Q.4. Does a higher Q factor always matter?It depends on the application. Yes: In RF tuning, oscillators, and filter circuits, a high Q is essential for sharp selectivity and frequency stability. No (or less so): In some power supply chokes or wideband filtering, a moderate Q is acceptable. In fact, if the Q is too high in a switching power supply filter, it might cause transient ringing spikes that damage components. In these cases, designers might intentionally choose a lower Q or add damping.5. What is the physical meaning of Q factor?In physics and engineering, the quality factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. A higher Q indicates a lower rate of energy loss relative to the stored energy of the resonator. In simple terms, a high-Q pendulum would swing for a long time (low friction), while a low-Q pendulum would stop quickly (high friction).

IntroductionRLC circuit is a circuit structure composed of resistance (R), inductance (L), and capacitance (C). The LC circuit is a simple example. RLC circuits are also called second-order circuits. The voltage or current in the circuit is the solution of a second-order differential equation, and its coefficients are determined by the circuit structure.If the circuit components are regarded as linear components, an RLC circuit can be regarded as an electronic harmonic oscillator.The natural frequency of this circuit is generally expressed as: (unit: Hz)RLC circuits are often used as band-pass filters or band-stop filters, and the Q factor can be obtained by the following formula:There are generally two types of RLC circuit composition: series and parallel.The animation above demonstrates the operation of the LC circuit (RLC circuit without resistors). The charge is transferred back and forth between the capacitor plate and the inductor. The energy oscillates back and forth between the electric field (E) of the capacitor and the magnetic field (B) of the inductor. The RLC circuit works similarly. The difference is that the oscillating current decays to zero over time due to the resistance in the circuit.CatalogIntroductionCatalogI RLC Series Circuit 1.1 What is Series RLC Circuit? 1.2 What is Transient Response of RLC Circuit? 1.3 Laplacian Domain 1.4 RLC Series Resonance Formula   1.5 Phasor Diagram of RLC Series CircuitII RLC Parallel CircuitIII The Difference Between Series Resonant Circuit and Parallel Resonant Circuit 3.1 Series Resonance 3.2 Parallel ResonanceIV Application of RLC Circuit Resonance 4.1 Application of Series Resonance Circuit 4.2 Application of Parallel Resonance CircuitV Frequently Asked Questions about RLC CircuitI RLC Series Circuit1.1 What is Series RLC Circuit?Figure1. RLC Series CircuitV-supply voltageI-circuit currentR-resistanceL-InductanceC-capacitanceIn this circuit, all three elements are connected in series with the voltage. The main differential equations can be obtained by substituting the constitutive equations of the three elements into Kirchhoff's voltage law (KVL). From Kirchhoff's voltage law:are the voltages across R, L, and C respectively, and V(t) is the voltage of the power supply that changes with time. Substituting the constitutive equation to get:In the case of a constant supply voltage, take the derivative of the above formula and divide by L to obtain the following second-order differential equation:This equation can be written in a more common form:α is called "attenuation", which is used to measure the attenuation rate of the transient response of this circuit when the external input is removed. ω0 is the angular resonance frequency. These two coefficients are given by:The damping coefficient ζ is another commonly used parameter, defined as the ratio of α to ω0:1.2 What is Transient Response of RLC Circuit?Figure2. Transient ResponseThe figure shows the underdamped and overdamped responses of the series RLC circuit. The critical damping is drawn with a thick red curve. These drawings are unified when L = 1, C = 1 and ω0=1. According to the value of different damping coefficient ζ, the solution of the differential equation has three different situations, namely: under-damping (ζ<1), over-damping (ζ>1), and critical damping (ζ=1).The characteristic equation of the differential equation is:The roots of this equation are:The general solution of this differential equation is the linear superposition of two exponential functions:The coefficients A1 and A2 are given by the boundary conditions of the specific problem.The following video introduces how to analyze RLC circuits by way of second order differential equations. Both parallel and series RLC configurations are discussed in it, looking primarily at Natural Response, but also touching on Step Response.RLC Circuit Response Explanation1.2.1 Over-damped responseThe over-damped response (ζ>1) is:Overdamping response is a transient current without oscillation attenuation.1.2.2 Underdamped responseThe underdamped response (ζ<1) is:Through the trigonometric identities, these two trigonometric functions can be expressed by a phased sine function:The underdamped response is an attenuated oscillation with a frequency of ωd. The rate of oscillation decay is α. The α in the index describes the envelope function of the oscillation. B1 and B2 (or B3 and phase difference φ in the second form) are arbitrary constants and are determined by boundary conditions. The frequency ωd is given by:This is the so-called damped resonance frequency or damped natural frequency. It is the frequency at which the circuit naturally vibrates when driven by no external source. The resonant frequency ω0 is the resonant frequency of the circuit when it is driven by an external source, and is often called the undamped resonant frequency in order to facilitate the distinction.1.2.3 Critical damping responseThe critical damping response (ζ=1) is:1.3 Laplacian DomainThe Laplace transform can be used to analyze the AC transient and steady-state behavior of the RLC series circuit. If the waveform generated by the above voltage source is V(s) after Laplace transform (where s is the complex frequency s=σ+iω), then Kirchhoff’s voltage law is applied in the Laplace domain:Among them, I(s) is the current after Laplace transform. Solve for I(s):After rearranging, the following formula can be obtained:1.3.1 Laplace admittanceSolve for Laplace admittance Y(s):The above formula can be simplified by using the parameters α and ωo defined in the above content, and we can get:1.3.2 Pole and zeroThe zero point of Y(s) is s such that Y(s)=0: s=0 and |s|⟶ ∞; the pole of Y(s) is s such that Y(s)⟶ ∞. Solve the quadratic equation. Get:The poles of Y(s) are the roots s1 and s2 of the characteristic equation of the differential equation mentioned above.1.3.3 Sine steady stateThe sine steady state can be represented by letting s=jω, where j is the imaginary unit. Substitute this into the amplitude of the above equation:The function of the current with ω as the variable isThere is a peak.In this special case, ω in this peak is equal to the undamped natural resonance frequency:1.4 RLC Series Resonance FormulaThe so-called series resonance formula refers to the study of the energy value of the voltage and current of the series circuit to reach the same phase, and the inductance of the inductance in the circuit and the capacitive reactance in the capacitor are equal in value. Therefore, in the study of the resistance characteristics of the circuit, In the case of a given terminal voltage, the maximum current is released, and the active power consumed will also be the maximum.Figure3. RLC series resonance formulaResonance definition: The energy of the L and C_ elements in the circuit are equal. When a reactance element in the circuit releases energy, the other reactance element must absorb the same energy, that is, energy pulsation occurs between the two reactance elements.  When series resonance occurs:Inductive reactance XL = capacitive reactance XCSource voltage U = resistance voltage URInductor voltage UL = Capacitor voltage UCInductor's reactive power QL = Capacitor's reactive power QCTotal circuit impedance Z=resistance value RApparent power S = resistance power PExplanation: When the circuit resonates, it must have two components: inductor L and capacitor C, and the frequency corresponding to resonance is called "resonant frequency" or resonant frequency, generally we use fr to indicate.1.5 Phasor Diagram of RLC Series Circuit(1) Phasor diagram of voltage and currentU&=U&R+U&L+U&CFigure4. Phasor diagram of voltage and currentFigure5. Phasor diagram of voltage and current(2) Voltage triangleThe relationship between the voltage triangle and the impedance triangle: divide the effective value of the voltage triangle by I to get the impedance triangle.Figure6. Voltage triangle● The relationship between the total voltage and the effective value of each part of the voltage:● The effective value relationship between total voltage and total current: U=I|Z|● The phase difference relationship between total voltage and total current:II RLC Parallel CircuitFigure7. RLC Parallel CircuitV-supply voltageI-circuit currentR-resistanceL-InductanceC-capacitanceThe characteristics of the RLC parallel circuit can be handled by the duality (electrical circuits) of the circuit. The RLC parallel circuit is treated as the dual impedance of the RLC series circuit, so it can be analyzed in a similar way to the RLC series circuit.The attenuation α of the RLC parallel circuit can be obtained by the following formula:If the factor of 1/2 is not considered, the damping coefficient of the RLC parallel circuit is exactly the reciprocal of the damping coefficient of the RLC series circuit.Frequency domainAdd the admittance of each element in parallel to obtain the admittance of this circuit:After capacitors, resistors, and inductors are connected in parallel, the impedance at the resonance frequency is the maximum, which is the opposite of the case where capacitors, resistors, and inductors are connected in series. The RLC parallel circuit is an antiresonator.In the figure below, it can be seen that if a constant voltage is used for driving, the frequency response of the current has a minimum value at the resonance frequency ω0=1/√LC. If it is driven by a constant current, the frequency response of the voltage has a maximum value at the resonance frequency, which is similar to the frequency response graph of the current in an RLC series circuit.Figure8. Sinusoidal steady state analysisNormalize with R = 1 ohm, C = 1 Farad, L = 1 Henry, and V = 1.0 VoltIII The Difference Between Series Resonant Circuit and Parallel Resonant CircuitIn an AC circuit containing resistance, inductance and capacitance, the voltage at both ends of the circuit and its current are generally out of phase. If the circuit parameters or the power supply frequency are adjusted to make the current and the power supply voltage in phase, the circuit is resistive, which is called resonance for the working state of the circuit at this time.Resonance is a specific phenomenon of sinusoidal AC circuits. It is widely used in electronics and communication engineering. However, in power systems, resonance may damage the normal operation of the system.Resonance is generally divided into series resonance and parallel resonance. As the name implies, series resonance is the resonance that occurs in a series circuit. Parallel resonance is the resonance that occurs in a parallel circuit.3.1 Series Resonance3.1.1 IntroductionIn a series circuit composed of resistance, inductance and capacitance, when the capacitive reactance XC and the inductive reactance XL are equal, that is, XC=XL, the voltage U and the current I in the circuit have the same phase, and the circuit presents pure resistivity. This phenomenon is called series resonance. When the circuit is in series resonance, the total impedance in the circuit is the smallest, and the current will reach the maximum. 3.1.2 Conditions for the occurrence of series resonanceIn order to resonate in a series circuit, certain conditions must be met.When UL=UC, that is, XL=XC,. Voltage and current are in phase, and series resonance occurs in the circuit. From ωL=1/ωC, ω0=1/√LC can be obtained, and the resonance frequency is f=f0=1/2π√LC. 3.1.3 Characteristics of series resonance circuit● Minimum total impedance● When the power supply voltage is constant, the current is the largest● The circuit is resistive, and the voltage on the capacitor or inductor may be higher than the power supply voltage 3.1.4 Energy changes in the circuit at resonanceThe circuit absorbs Q=0 from the power supply, and the circuit energy exchanges between the electric field and the magnetic field inside the circuit during resonance. The power supply only provides energy to R.High voltage may damage the device. Series resonance should be avoided in the power system. And series resonance is widely used in radio engineering.3.2 Parallel Resonance3.2.1 IntroductionIn a circuit where an inductance and a capacitor are connected in parallel, when the size of the capacitor just makes the voltage and current in the circuit have the same phase, that is, when the power supply is consumed by resistance and becomes a resistance circuit, it is called parallel resonance.Parallel resonance is a complete compensation. The power supply does not need to provide reactive power, only the active power required by the resistance. At resonance, the total current of the circuit is the smallest, and the current of the branch is often greater than the total current of the circuit. Therefore, parallel resonance is also called current resonance.When parallel resonance occurs, a large current flows in the inductance and capacitance components, which will cause the fuse of the circuit to blow or burn the electrical equipment; however, it is often used in radio engineering to select signals and eliminate interference. 3.2.2 Parallel resonance conditionsIn the following two types of circuitsFigure9. Two types of circuitsThe resonant frequency formula of (a) has been discussed above, and (b) is determined by,We can get.Under normal circumstances, the coil resistance R is much smaller than XL, therefore, ignoring R we can getthat is f=f0=1/2π√LC. 3.2.3 Features of parallel resonant circuit● When the voltage is constant, the current is the smallest at resonance● Maximum total impedance● The circuit is resistive, and the branch current may be greater than the total currentIV Application of RLC Circuit Resonance4.1 Application of Series Resonance CircuitThe use of series resonance to generate power frequency high voltage, which is used in high voltage technology to do withstand voltage test for power equipment such as transformers, can effectively find dangerous concentrated defects in the equipment, and is the most effective and direct way to test the insulation strength of electrical equipment Methods. Used in radio engineering, series resonance is often used to obtain a higher voltage.In the radio, the series resonance circuit is often used to select the radio signal. This process is called tuning. The following figure shows a typical circuit.Figure10. A typical circuit for tuningWhen the electric waves of various signals of different frequencies generate electric signals of different frequencies on the antenna, they are induced to the coil 2L through the coil 1L. If the oscillation circuit resonates to a certain signal frequency, the current of the signal in the loop is the largest, and a voltage CU higher than the signal voltage Q times is generated across the capacitor. For other signals of various frequencies, because no resonance occurs, the current in the loop is very small, which is suppressed by the circuit. Therefore, the capacitor C can be changed to change the resonant frequency of the loop to select the desired radio signal.4.2 Application of Parallel Resonance CircuitThe application of LC parallel resonant circuit in communication electronic circuit is determined by its characteristics. Specifically, it mainly includes three categories. One is working in resonance, as a frequency-selective network application. At this time, it appears as a large resistance and outputs a larger voltage under the excitation of current; the second is working in detuning The state, present as inductive or capacitive at this time, together with other inductances and capacitors in the circuit, satisfies the oscillation conditions of the three-point oscillation circuit to form a sine wave oscillator; the third is to work in a detuned state, that is, to work on the amplitude-frequency characteristic curve Or one side of the phase-frequency characteristic curve to realize amplitude-frequency conversion, frequency-amplitude conversion, frequency-phase conversion, and phase-frequency conversion to form an angle modulation and demodulation circuit. (1) LC parallel resonant circuit used as frequency selective matching networkFrequency selection is to select useful frequency components from the input signal and suppress useless frequency components or noise. In communication electronic circuits, the LC parallel resonant circuit is the most commonly used as a frequency selection network. It is widely used in high-frequency small-signal amplifiers, Class C high-frequency power amplifiers, mixers and other circuits. The common feature of these circuits is that the LC resonant circuit is not only a frequency-selective network. Through the connection of the transformer, it also plays the role of impedance transformation, reducing the impact of the amplifier tube or the load on the resonant circuit, and obtaining better selectivity. . (2) The LC parallel resonant circuit of the overtone crystal oscillator as a capacitorUnder the action of the applied alternating voltage, in the mechanical vibration generated by the quartz crystal, in addition to the fundamental frequency mechanical vibration, there are many odd frequency overtones. When a crystal oscillator with a very high operating frequency is required, overtone crystal oscillators are often used. The figure below shows the overtone crystal oscillator.Figure11. Circuit composition and reactance curve of L1C1 circuitIn the above figure, the quartz crystal and the CL branch are inductive. The quartz crystal, C2, and L1C1 loop together form a three-point oscillator. According to the composition principle of the three-point oscillator (shooting the same), the L1C1 resonant circuit should be capacitive. Assuming that the quartz crystal in the figure is working at the 5th overtone frequency, the nominal frequency is 5 MHz. In order to suppress the parasitic oscillation of the fundamental frequency and 3rd overtone, the L1C1 loop should be tuned between the 3rd and 5th overtone frequency, that is, 3~ Between 5 MHz.  From the reactance characteristic curve of the L1C1 resonant circuit shown in Figure (b), it can be seen that for the 5th overtone frequency of 5 MHz, the L1C1 circuit is capacitive, and the circuit meets the three-point oscillation condition and can oscillate. For the fundamental and third harmonics that are less than the resonance frequency of the L1C1 loop, the loop has an inductive characteristic, which does not conform to the principle of different components and cannot produce oscillation. For overtones of 7 times and above, although the L1C1 circuit is also capacitive, the equivalent capacitance at this time is too large, the amplitude starting conditions cannot be met, and the oscillation cannot be generated. (3) LC parallel resonant circuit that realizes the functions of amplitude-frequency conversion and frequency-phase conversionThe phase-frequency characteristic of the impedance of the LC parallel resonant circuit is a monotonous curve with a negative slope. The linear part of the curve can be used to perform a linear conversion between frequency and phase. This is mainly used in the phase frequency discrimination circuit; the same, the LC parallel resonant circuit The linear part of the impedance's amplitude-frequency characteristic curve can also perform the linear conversion between frequency and amplitude, so it has also been applied in the slope frequency discrimination circuit.V Frequently Asked Questions about RLC Circuit1. Is LCR and RLC circuit the same?Yes. An RLC circuit (also known as a resonant circuit, tuned circuit, or LCR circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. This configuration forms a harmonic oscillator. 2. What is the resonant frequency of the RLC circuit?What is Resonance in the RLC circuit? Resonance is the phenomenon in the electrical circuit, where the output of the circuit is maximum at one particular frequency. And that frequency is known as the resonant frequency. At the resonant frequency, The capacitive reactance and inductive reactance are equal. 3. Is the RLC circuit linear?In an RLC circuit, the most fundamental elements of a resistor, inductor and capacitor are connected across a voltage supply. All of these elements are linear and passive in nature. 4. What is the bandwidth of the RLC circuit?The bandwidth of any system is the range of frequencies for which the current or output voltage is equal to 70.7% of its value at the resonant frequency, and it is denoted by BW. 5. What is the second-order circuit?A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements. 6. What is the first-order circuit?A first-order circuit can only contain one. energy storage element (a capacitor or an. inductor). The circuit will also contain. 7. What is the half-power frequency?The frequencies for which current in a series RLC (or a series tuned) circuit is equal to 1/√2 (i.e. 70.71%) of the maximum current (current at resonance)are known as Half Power Frequencies. 8. What is the natural response of the RC circuit?The natural response tells us what the circuit does as its internal stored energy (the initial voltage on the capacitor) is allowed to dissipate. It does this by ignoring the forcing input (the voltage step caused by the switch closing). The "destination" of the natural response is always zero voltage and zero current. 9. What is the difference between first-order and second-order filters?The main difference between a 1st and 2nd order low pass filter is that the stopband roll-off will be twice the 1st order filters at 40dB/decade (12dB/octave) as the operating frequency increases above the cut-off frequency ƒc, point as shown. 10. What is the use of a resonant circuit?One use for resonance is to establish a condition of stable frequency in circuits designed to produce AC signals. Usually, a parallel (tank) circuit is used for this purpose, with the capacitor and inductor directly connected together, exchanging energy between each other. 

Warm hint: The word in this article is about 1000 words and reading time is about 5 minutes     This article introduces you some basic and simple rectifiers and their waveforms and working principle and more.   Catalog I. Introduction 1.1 Composition of DC Stabilized Power Supply 1.2 Basic Concepts II. Detail & Analysis 2.1 Half-Wave Rectifier 2.2 Full-Wave Rectifier 2.3 Bridge Rectifier 2.4 Capacitor Filter 2.5 Inductor Filter FAQ   I. Introduction   1.1 Composition of DC Stabilized Power Supply FIG.1   2. Basic Concepts   AC voltage (current): both amplitude and direction change periodically with time. FIG.2 Sinusoidal Voltage/Current Waveform (AC): alternating voltage/current whose amplitude and direction both change sinusoidally and periodically over time. It is often called AC for short.   Effective value: the direct voltage/current thermally equivalent to the alternating voltage/current is called the effective value of the AC (voltage or current).   Peak value: the maximum instantaneous value of AC (voltage or current).   Frequency: the number of times that AC (voltage or current) changes periodically every second.   Direct voltage/current: voltage/current whose value and direction do not change with time. In fact, the direction can be guaranteed not to change over time, but it is impossible for the value to act the same way all the time. So the alternating voltage/current whose direction is always the same and the numerical value changes over time can be explained as a superposition of DC (voltage or current) and AC (voltage or current) whose amplitude and direction change with time. II. Detail & Analysis   2.1 Half-Wave Rectifier The circuit diagram and the waveforms of half-wave rectifier are shown as the following figures. FIG.3 Average value: FIG.4 Effective value: Turns ratio: According to this, the required output voltage can be obtained by selecting the appropriate N1 and N2.   2.2 Full-Wave Rectifier The following figures are the circuit diagram and the waveform of full-wave rectifier. FIG.5 FIG.6 Average value: Effective value: Transformer: The number of turns of the primary winding is N1, and the number of turns of the secondary winding is N2a=N2b. Turns ratio: According to this, the required output voltage can be obtained by selecting the appropriate N1, N2a and N2b.   2.3 Bridge Rectifier The following figures are the circuit diagram and the waveform of bridge rectifier. FIG.7 FIG.8 Average value: Effective value: Transformer: The number of turns of the primary winding is N1, and the number of turns of the secondary winding is N2. Turns ratio: According to this, the required output voltage can be obtained by selecting the appropriate N1 and N2.   2.4 Capacitor Filter The following figures are the circuit diagram and the waveform of capacitor filter. FIG.9 Filtering principles: a~b: u2=uc=u0, the capacitor C is charged in a sine wave; b~c: u2≈uc = u0, the capacitor C discharges in an exponential curve, but the sinusoidal waves of u2 basically coincide. c~d: u2＜uc=u0, the capacitor C continues to discharge exponentially, and u2 to drop in a sine wave. FIG.10 The effects of RL and C on filtering are shown in the following figure. FIG.11 （1）Basic knowledge of capacitors   Definition:   Basic equations:  Energy equation:  FIG.12 （2）Charging the capacitor Where  It is a time constant, and the initial values of current is  FIG.13 （3）Discharging the capacitor Where It is a time constant, and the initial values of current is FIG.14 （4）Output voltage   After the filtered voltage waveform is linearized, the following approximate waveform is obtained: FIG.15 Based on the relationship of similar triangles, there is And  So we have When There is  FIG.16 Rectifier diode: The current and the conduction angle of the rectifier diode in the capacitor filter circuit are shown in the following figure: FIG.17 Where iD is the current of the rectifier diode when the current is switched on, and io is the current in the load.   2.5 Inductor Filter In heavy current load, if a filter capacitor is used, the capacitance of it and the inrush current of the rectifier both will be very large. But if an industrial-frequency inductor is in series with it for filtering after the rectification, then we can solve these problems very well. The following figure is the circuit diagram of the inductor filter. FIG.18 From the energy point of view, the effects of the inductive filter and the capacitive filter are the same. Therefore, the volt-ampere characteristics of the inductive filter is similar to that of the capacitive filter, see the figure below. FIG.19 The quantitative analysis of inductive filter is more complex, so we should do it with the help of the previous analyzed results for the capacitive filter. If  then we have When the inductor filter is used, the waveform of the terminal voltage and current of the inductor and the conduction angle of the rectifier diode are shown in the following figure. Because the rectifier diode is connected in series with an inductor, the conduction angle of it can reach 180°. Therefore, in situations where harmonics are not demanding, we can use inductive filter to meet PFC (Power Factor Correction) requirements. FIG.20 （1）Basic concepts of inductor FIG.21 Definition:  Basic equations: Energy equation: （2）When inductor stores energy   After the switch is closed, here we have According to the initial conditions, the solution is Where It is a time constant, and the initial voltage is FIG.22 （3）When inductor releases energy FIG.23 After the switch is closed, here we have According to the initial conditions, here we have: Where It is a time constant, and the initial voltage is FIG.24 How Amplifiers Work: Rectifiers and Filter Capacitors   FAQ   1. What are the types of rectifiers? The Different Types of Rectifiers: a. Single Phase & Three Phase Rectifiers. b. Half Wave & Full Wave Rectifiers. c. Bridge Rectifiers. d. Uncontrolled & Controlled Rectifiers.   2. What is Rectifier used for? A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC).   3. What is an example of rectifier? Thyristors are commonly used in place of diodes to create a circuit that can regulate the output voltage. Many devices that provide direct current actually generate three-phase AC. For example, an automobile alternator contains six diodes, which function as a full-wave rectifier for battery charging.   4. Is Zener diode a rectifier? A Zener diode is a special type of rectifying diode that can handle breakdown due to reverse breakdown voltage without failing completely. Here we will discuss the concept of using diodes to regulate voltage drop and how the Zener diode operates in reverse-bias mode to regulate voltage in a circuit.   5. What is the working principle of rectifier? Principle: A junction diode offers a low resistance to current in one direction(when forward biased) and a high resistance in the other direction(when reverse biased). Thus, the diode acts as a rectifier.   6. Why zener diode is not used in Rectifier? No, we don't prefer to use a Zener Diode in a rectifier circuit because for a rectifier circuit a high maximum peak inverse voltage is required. Unlike the normal p-n junction diode, a Zener diode has a low peak inverse voltage. This is an undesirable property for the rectifier circuit.   7. What are the signs of a bad Rectifier? You'll note signs right away like poor starts, fluctuating meter readings, and dimmed headlights. around 13 volts, the bike will start to drain the battery. When this happens, it's only a matter of time before the engine stops completely.   8. What causes a rectifier to fail? Ground connections are important for good voltage, and if there is faulty voltage, the regulator rectifier can run hot. Bad grounding, corroded battery connection and poor or loose battery connections will cause faulty voltage.   9. Will a bad rectifier cause no spark? A bad regulator/ rectifier will result in a dead battery, and once the battery is competely dead you will not get a spark.   10. What is the difference between diode and rectifier? A diode is a switching device, while a rectifier is generally used for the conversion of AC voltage to DC voltage. ... A diode allows the flow of current only when it is forward biased. The diode blocks the reverse flow of current. A rectifier, on the other hand, consists of a transformer, a diode, and a filter circuit.   You May Also Like: Transformers Basics: Construction, Types, Materials and Design Characteristics and Functions of Diodes Switched Mode Power Supply Tutorial: Principles & Functions of SMPS Circuits