## Exploring the Resonant Frequency of an RLC Circuit

### Key Takeaways

Understand what a resonant frequency is.

Learn how a resonant frequency affects series and parallel RLC circuits.

Determine what happens at the resonant frequency of an RLC circuit.

There are moments where the logical part of yourself is heavily burdened by unfounded fears. Imagine getting stuck in traffic on a bridge that spans miles across the ocean. As your car inches to the middle of the bridge, you suddenly feel the vehicle start to sway with the bridge. For a fleeting moment, you are terrified that an earthquake struck or the bridge is on the verge of collapse. This may not be an experience everyone has had, but it does happen to me on occasion.

If you are an engineer, your logical mind might consider a theory that revolves around resonant frequencies, which states that a bridge could vibrate when it’s subjected to an oscillating force that matches its resonant frequency.

The scenario above offers a visceral insight into our topic of what happens at the resonant frequency of an RLC circuit. Let’s explore this topic further.

### What Does Resonant Frequency Mean?

A resonant frequency is defined as the natural frequency of a system where it oscillates at the greatest amplitude. A system is said to be in resonance when an external force applied shares the same frequency as its natural frequency.

In daily life, you’ll come across mechanisms that resonate at their resonant frequency, which results in greater amplitude. Besides bridges, swings, string instruments, and RLC circuits are also known to exhibit extraordinary behavior at their resonant frequencies.

For example, if a swing is pushed at its resonant frequency, it results in the swing reaching greater heights than it would otherwise. The strings of a musical instrument interact with each other in a similar way. In electronics, you’ll come across resonant frequencies, particularly in RLC circuits.

### Resonant Frequency in a Series RLC Circuit

*Series RLC circuit resonant frequency.*

The series RLC circuit depicted above is commonly used in various PCB applications. Both inductor and capacitor display dynamic properties in reactance across a different range of frequencies.

At a specific frequency, the inductive reactance and the capacitive reactance will be of equal magnitude but in opposite phase. They are represented by the equation:

X_{L} = X_{C}

As both capacitive and inductive reactance cancel each other out, the circuit’s impedance will be purely resistive. When this phenomenon occurs, the circuit is said to be oscillating at its resonant frequency. The resonant frequency of the series RLC circuit is expressed as

f_{r} = 1/2π√(LC)

At its resonant frequency, the total impedance of a series RLC circuit is at its minimum. It also means that the current will peak at the resonant frequency as both inductor and capacitor appear as a short circuit.

Frequency response of a series RLC circuit.

A series RLC circuit, which achieves maximum power transfer at resonance, is commonly used as a bandpass filter for radio, TV, or as a noise filter.

### Resonant Frequency in a Parallel RLC Circuit

*Diagram of a parallel RLC circuit.*

A parallel RLC circuit will also exhibit peak behaviors at its resonant frequency, however, there will be big differences compared to a series RLC circuit.

The resonant frequency of a parallel RLC circuit is also expressed by:

f_{r} = 1/2π√(LC)

But, that’s where the similarities end. At resonance, both capacitive and inductive reactance will be equal to each other. The inductor and capacitor will also be conducting more current at the resonant frequency.

Current flowing across both components is 180° out of phase, which results in a mutually canceling current. Therefore, the segment of inductor and capacitor in parallel will appear as an open circuit.

When the frequency response of the parallel RLC circuit is plotted on a chart, you’ll find that the current decreases to a minimum at the resonant frequency. This is the opposite of the response of a series RLC circuit.

*The frequency response of a parallel RLC circuit.*

The parallel RLC circuit is also dubbed an anti-resonance circuit. It’s used as a rejector circuit to suppress current at a specific frequency from passing through.

Whether you’re designing a series or parallel RLC circuit, you’ll need a good PCB design and analysis software. Allegro, by Cadence, has a robust selection of schematic, PCB, and simulation tools that will be instrumental in designing resonance circuits and other types of PCB designs.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.

## About the Author

Cadence PCB solutions is a complete front to back design tool to enable fast and efficient product creation. Cadence enables users accurately shorten design cycles to hand off to manufacturing through modern, IPC-2581 industry standard.

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## RLC circuit

Resistor Inductor Capacitor Circuit

An **RLC circuit** is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

The circuit forms a harmonic oscillator for current, and resonates in a similar way as an LC circuit. Introducing the resistor increases the decay of these oscillations, which is also known as damping. The resistor also reduces the peak resonant frequency. In ordinary conditions, some resistance is unavoidable even if a resistor is not specifically included as a component; an ideal, pure LC circuit exists only in the domain of superconductivity, a physical effect demonstrated to this point only at temperatures far below and/or pressures far above what are found naturally anywhere on the Earth's surface.

RLC circuits have many applications as oscillator circuits. Radio receivers and television sets use them for tuning to select a narrow frequency range from ambient radio waves. In this role, the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a band-pass filter, band-stop filter, low-pass filter or high-pass filter. The tuning application, for instance, is an example of band-pass filtering. The RLC filter is described as a *second-order* circuit, meaning that any voltage or current in the circuit can be described by a second-order differential equation in circuit analysis.

The three circuit elements, R, L and C, can be combined in a number of different topologies. All three elements in series or all three elements in parallel are the simplest in concept and the most straightforward to analyse. There are, however, other arrangements, some with practical importance in real circuits. One issue often encountered is the need to take into account inductor resistance. Inductors are typically constructed from coils of wire, the resistance of which is not usually desirable, but it often has a significant effect on the circuit.

### Basic concepts[edit]

### Resonance[edit]

An important property of this circuit is its ability to resonate at a specific frequency, the resonance frequency, *f*_{0}. Frequencies are measured in units of hertz. In this article, angular frequency, *ω*_{0}, is used because it is more mathematically convenient. This is measured in radians per second. They are related to each other by a simple proportion,

Resonance occurs because energy for this situation is stored in two different ways: in an electric field as the capacitor is charged and in a magnetic field as current flows through the inductor. Energy can be transferred from one to the other within the circuit and this can be oscillatory. A mechanical analogy is a weight suspended on a spring which will oscillate up and down when released. This is no passing metaphor; a weight on a spring is described by exactly the same second order differential equation as an RLC circuit and for all the properties of the one system there will be found an analogous property of the other. The mechanical property answering to the resistor in the circuit is friction in the spring–weight system. Friction will slowly bring any oscillation to a halt if there is no external force driving it. Likewise, the resistance in an RLC circuit will "damp" the oscillation, diminishing it with time if there is no driving AC power source in the circuit.

The resonance frequency is defined as the frequency at which the impedance of the circuit is at a minimum. Equivalently, it can be defined as the frequency at which the impedance is purely real (that is, purely resistive). This occurs because the impedances of the inductor and capacitor at resonance are equal but of opposite sign and cancel out. Circuits where L and C are in parallel rather than series actually have a maximum impedance rather than a minimum impedance. For this reason they are often described as antiresonators, it is still usual, however, to name the frequency at which this occurs as the resonance frequency.

### Natural frequency[edit]

The resonance frequency is defined in terms of the impedance presented to a driving source. It is still possible for the circuit to carry on oscillating (for a time) after the driving source has been removed or it is subjected to a step in voltage (including a step down to zero). This is similar to the way that a tuning fork will carry on ringing after it has been struck, and the effect is often called ringing. This effect is the peak natural resonance frequency of the circuit and in general is not exactly the same as the driven resonance frequency, although the two will usually be quite close to each other. Various terms are used by different authors to distinguish the two, but resonance frequency unqualified usually means the driven resonance frequency. The driven frequency may be called the undamped resonance frequency or undamped natural frequency and the peak frequency may be called the damped resonance frequency or the damped natural frequency. The reason for this terminology is that the driven resonance frequency in a series or parallel resonant circuit has the value^{[1]}

This is exactly the same as the resonance frequency of an LC circuit, that is, one with no resistor present. The resonant frequency for an RLC circuit is the same as a circuit in which there is no damping, hence undamped resonance frequency. The peak resonance frequency, on the other hand, depends on the value of the resistor and is described as the damped resonant frequency. A highly damped circuit will fail to resonate at all when not driven. A circuit with a value of resistor that causes it to be just on the edge of ringing is called critically damped. Either side of critically damped are described as underdamped (ringing happens) and overdamped (ringing is suppressed).

Circuits with topologies more complex than straightforward series or parallel (some examples described later in the article) have a driven resonance frequency that deviates from , and for those the undamped resonance frequency, damped resonance frequency and driven resonance frequency can all be different.

### Damping[edit]

Damping is caused by the resistance in the circuit. It determines whether or not the circuit will resonate naturally (that is, without a driving source). Circuits that will resonate in this way are described as underdamped and those that will not are overdamped. Damping attenuation (symbol α) is measured in nepers per second. However, the unitless damping factor (symbol ζ, zeta) is often a more useful measure, which is related to α by

The special case of *ζ* = 1 is called critical damping and represents the case of a circuit that is just on the border of oscillation. It is the minimum damping that can be applied without causing oscillation.

### Bandwidth[edit]

The resonance effect can be used for filtering, the rapid change in impedance near resonance can be used to pass or block signals close to the resonance frequency. Both band-pass and band-stop filters can be constructed and some filter circuits are shown later in the article. A key parameter in filter design is bandwidth. The bandwidth is measured between the cutoff frequencies, most frequently defined as the frequencies at which the power passed through the circuit has fallen to half the value passed at resonance. There are two of these half-power frequencies, one above, and one below the resonance frequency

where Δ*ω* is the bandwidth, *ω*_{1} is the lower half-power frequency and *ω*_{2} is the upper half-power frequency. The bandwidth is related to attenuation by

where the units are radians per second and nepers per second respectively.^{[citation needed]} Other units may require a conversion factor. A more general measure of bandwidth is the fractional bandwidth, which expresses the bandwidth as a fraction of the resonance frequency and is given by

The fractional bandwidth is also often stated as a percentage. The damping of filter circuits is adjusted to result in the required bandwidth. A narrow band filter, such as a notch filter, requires low damping. A wide band filter requires high damping.

### Q factor[edit]

The Q factor is a widespread measure used to characterise resonators. It is defined as the peak energy stored in the circuit divided by the average energy dissipated in it per radian at resonance. Low-Q circuits are therefore damped and lossy and high-Q circuits are underdamped. Q is related to bandwidth; low-Q circuits are wide-band and high-Q circuits are narrow-band. In fact, it happens that Q is the inverse of fractional bandwidth

Q factor is directly proportional to selectivity, as the Q factor depends inversely on bandwidth.

For a series resonant circuit, the Q factor can be calculated as follows:^{[2]}

### Scaled parameters[edit]

The parameters ζ, *F*_{b}, and Q are all scaled to *ω*_{0}. This means that circuits which have similar parameters share similar characteristics regardless of whether or not they are operating in the same frequency band.

The article next gives the analysis for the series RLC circuit in detail. Other configurations are not described in such detail, but the key differences from the series case are given. The general form of the differential equations given in the series circuit section are applicable to all second order circuits and can be used to describe the voltage or current in any element of each circuit.

### Series circuit[edit]

**Figure 1:**RLC series circuit

- V, the voltage source powering the circuit
- I, the current admitted through the circuit
- R, the effective resistance of the combined load, source, and components
- L, the inductance of the inductor component
- C, the capacitance of the capacitor component

In this circuit, the three components are all in series with the voltage source. The governing differential equation can be found by substituting into Kirchhoff's voltage law (KVL) the constitutive equation for each of the three elements. From the KVL,

where V_{R}, V_{L} and V_{C} are the voltages across R, L and C respectively and *V*(*t*) is the time-varying voltage from the source.

Substituting , and into the equation above yields:

For the case where the source is an unchanging voltage, taking the time derivative and dividing by L leads to the following second order differential equation:

This can usefully be expressed in a more generally applicable form:

α and *ω*_{0} are both in units of angular frequency. α is called the *neper frequency*, or *attenuation*, and is a measure of how fast the transient response of the circuit will die away after the stimulus has been removed. Neper occurs in the name because the units can also be considered to be nepers per second, neper being a unit of attenuation. *ω*_{0} is the angular resonance frequency.^{[3]}

For the case of the series RLC circuit these two parameters are given by:^{[4]}

A useful parameter is the *damping factor*, ζ, which is defined as the ratio of these two; although, sometimes α is referred to as the damping factor and ζ is not used.^{[5]}

In the case of the series RLC circuit, the damping factor is given by

The value of the damping factor determines the type of transient that the circuit will exhibit.^{[6]}

### Transient response[edit]

*L*= 1,

*C*= 1and

*ω*

_{0}= 1.

The differential equation has the characteristic equation,^{[7]}

The roots of the equation in s-domain are,^{[7]}

The general solution of the differential equation is an exponential in either root or a linear superposition of both,

The coefficients *A*_{1} and *A*_{2} are determined by the boundary conditions of the specific problem being analysed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time.^{[8]} The differential equation for the circuit solves in three different ways depending on the value of ζ. These are overdamped (*ζ* > 1), underdamped (*ζ* < 1), and critically damped (*ζ* = 1).

#### Overdamped response[edit]

The overdamped response (*ζ* > 1) is^{[9]}

The overdamped response is a decay of the transient current without oscillation.^{[10]}

#### Underdamped response[edit]

The underdamped response (*ζ* < 1) is^{[11]}

By applying standard trigonometric identities the two trigonometric functions may be expressed as a single sinusoid with phase shift,^{[12]}

The underdamped response is a decaying oscillation at frequency *ω*_{d}. The oscillation decays at a rate determined by the attenuation α. The exponential in α describes the envelope of the oscillation. *B*_{1} and *B*_{2} (or *B*_{3} and the phase shift φ in the second form) are arbitrary constants determined by boundary conditions. The frequency *ω*_{d} is given by^{[11]}

This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency, *ω*_{0}, which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped resonance frequency to distinguish it.^{[13]}

#### Critically damped response[edit]

The critically damped response (*ζ* = 1) is^{[14]}

The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting. *D*_{1} and *D*_{2} are arbitrary constants determined by boundary conditions.^{[15]}

### Laplace domain[edit]

The series RLC can be analyzed for both transient and steady AC state behavior using the Laplace transform.^{[16]} If the voltage source above produces a waveform with Laplace-transformed *V*(*s*) (where s is the complex frequency*s* = *σ* + *jω*), the KVL can be applied in the Laplace domain:

where *I*(*s*) is the Laplace-transformed current through all components. Solving for *I*(*s*):

And rearranging, we have

#### Laplace admittance[edit]

Solving for the Laplace admittance*Y*(*s*):

Simplifying using parameters α and *ω*_{0} defined in the previous section, we have

#### Poles and zeros[edit]

The zeros of *Y*(*s*) are those values of s where *Y*(*s*) = 0:

The poles of *Y*(*s*) are those values of s where *Y*(*s*) → ∞. By the quadratic formula, we find

The poles of *Y*(*s*) are identical to the roots *s*_{1} and *s*_{2} of the characteristic polynomial of the differential equation in the section above.

#### General solution[edit]

For an arbitrary *V*(*t*), the solution obtained by inverse transform of *I*(*s*) is:

- In the underdamped case,
*ω*_{0}>*α*: - In the critically damped case,
*ω*_{0}=*α*: - In the overdamped case,
*ω*_{0}<*α*:

where *ω*_{r} = √*α*^{2} − *ω*_{0}^{2}, and cosh and sinh are the usual hyperbolic functions.

#### Sinusoidal steady state[edit]

*ω*

_{0}= 1 rad/s, damping ratio

*ζ*= 0.4.

Sinusoidal steady state is represented by letting *s* = *jω*, where j is the imaginary unit. Taking the magnitude of the above equation with this substitution:

and the current as a function of ω can be found from

There is a peak value of |*I*(*jω*)|. The value of ω at this peak is, in this particular case, equal to the undamped natural resonance frequency:^{[17]}

From the frequency response of the current, the frequency response of the voltages across the various circuit elements can also be determined.

### Parallel circuit[edit]

**Figure 2.**RLC parallel circuit

V– the voltage source powering the circuit

I– the current admitted through the circuit

R– the equivalent resistance of the combined source, load, and components

L– the inductance of the inductor component

C– the capacitance of the capacitor component

The properties of the parallel RLC circuit can be obtained from the duality relationship of electrical circuits and considering that the parallel RLC is the dual impedance of a series RLC. Considering this, it becomes clear that the differential equations describing this circuit are identical to the general form of those describing a series RLC.

For the parallel circuit, the attenuation α is given by^{[18]}

and the damping factor is consequently

Likewise, the other scaled parameters, fractional bandwidth and Q are also reciprocals of each other. This means that a wide-band, low-Q circuit in one topology will become a narrow-band, high-Q circuit in the other topology when constructed from components with identical values. The fractional bandwidth and Q of the parallel circuit are given by

Notice that the formulas here are the reciprocals of the formulas for the series circuit, given above.

### Frequency domain[edit]

**Figure 3.**Sinusoidal steady-state analysis. Normalised to

*R*= 1 Ω,

*C*= 1 F,

*L*= 1 H, and

*V*= 1 V.

The complex admittance of this circuit is given by adding up the admittances of the components:

The change from a series arrangement to a parallel arrangement results in the circuit having a peak in impedance at resonance rather than a minimum, so the circuit is an anti-resonator.

The graph opposite shows that there is a minimum in the frequency response of the current at the resonance frequency when the circuit is driven by a constant voltage. On the other hand, if driven by a constant current, there would be a maximum in the voltage which would follow the same curve as the current in the series circuit.

### Other configurations[edit]

**Figure 4.**RLC parallel circuit with resistance in series with the inductor

A series resistor with the inductor in a parallel LC circuit as shown in Figure 4 is a topology commonly encountered where there is a need to take into account the resistance of the coil winding. Parallel LC circuits are frequently used for bandpass filtering and the Q is largely governed by this resistance. The resonant frequency of this circuit is^{[19]}

This is the resonant frequency of the circuit defined as the frequency at which the admittance has zero imaginary part. The frequency that appears in the generalised form of the characteristic equation (which is the same for this circuit as previously)

is not the same frequency. In this case it is the natural undamped resonant frequency:^{[20]}

The frequency *ω*_{m} at which the impedance magnitude is maximum is given by^{[21]}

where *Q _{L}* =

*ω′*

_{0}

*L*/

*R*is the quality factor of the coil. This can be well approximated by

^{[21]}

Furthermore, the exact maximum impedance magnitude is given by^{[21]}

For values of Q_{L} greater than unity, this can be well approximated by^{[21]}

**Figure 5.**RLC series circuit with resistance in parallel with the capacitor

In the same vein, a resistor in parallel with the capacitor in a series LC circuit can be used to represent a capacitor with a lossy dielectric. This configuration is shown in Figure 5. The resonant frequency (frequency at which the impedance has zero imaginary part) in this case is given by^{[22]}

while the frequency *ω*_{m} at which the impedance magnitude is minimum is given by

where *Q _{C}* =

*ω′*

_{0}

*RC*.

### History[edit]

The first evidence that a capacitor could produce electrical oscillations was discovered in 1826 by French scientist Felix Savary.^{[23]}^{[24]} He found that when a Leyden jar was discharged through a wire wound around an iron needle, sometimes the needle was left magnetized in one direction and sometimes in the opposite direction. He correctly deduced that this was caused by a damped oscillating discharge current in the wire, which reversed the magnetization of the needle back and forth until it was too small to have an effect, leaving the needle magnetized in a random direction.

American physicist Joseph Henry repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently.^{[25]}^{[26]} British scientist William Thomson (Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency.^{[23]}^{[25]}^{[26]}

British radio researcher Oliver Lodge, by discharging a large battery of Leyden jars through a long wire, created a tuned circuit with its resonant frequency in the audio range, which produced a musical tone from the spark when it was discharged.^{[25]} In 1857, German physicist Berend Wilhelm Feddersen photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations.^{[23]}^{[25]}^{[26]} In 1868, Scottish physicist James Clerk Maxwell calculated the effect of applying an alternating current to a circuit with inductance and capacitance, showing that the response is maximum at the resonant frequency.^{[23]}

The first example of an electrical resonance curve was published in 1887 by German physicist Heinrich Hertz in his pioneering paper on the discovery of radio waves, showing the length of spark obtainable from his spark-gap LC resonator detectors as a function of frequency.^{[23]}

One of the first demonstrations of resonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889^{[23]}^{[25]} He placed two resonant circuits next to each other, each consisting of a Leyden jar connected to an adjustable one-turn coil with a spark gap. When a high voltage from an induction coil was applied to one tuned circuit, creating sparks and thus oscillating currents, sparks were excited in the other tuned circuit only when the inductors were adjusted to resonance. Lodge and some English scientists preferred the term "*syntony*" for this effect, but the term "*resonance*" eventually stuck.^{[23]}

The first practical use for RLC circuits was in the 1890s in spark-gap radio transmitters to allow the receiver to be tuned to the transmitter. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Anglo Italian radio pioneer Guglielmo Marconi.^{[23]}

### Applications[edit]

### Variable tuned circuits[edit]

A very frequent use of these circuits is in the tuning circuits of analogue radios. Adjustable tuning is commonly achieved with a parallel plate variable capacitor which allows the value of C to be changed and tune to stations on different frequencies. For the IF stage in the radio where the tuning is preset in the factory, the more usual solution is an adjustable core in the inductor to adjust L. In this design, the core (made of a high permeability material that has the effect of increasing inductance) is threaded so that it can be screwed further in, or screwed further out of the inductor winding as required.

### Filters[edit]

Figure 6.RLC circuit as a low-pass filter | Figure 7.RLC circuit as a high-pass filter |

Figure 8.RLC circuit as a series band-pass filter in series with the line | Figure 9.RLC circuit as a parallel band-pass filter in shunt across the line |

Figure 10.RLC circuit as a series band-stop filter in shunt across the line | Figure 11.RLC circuit as a parallel band-stop filter in series with the line |

In the filtering application, the resistor becomes the load that the filter is working into. The value of the damping factor is chosen based on the desired bandwidth of the filter. For a wider bandwidth, a larger value of the damping factor is required (and vice versa). The three components give the designer three degrees of freedom. Two of these are required to set the bandwidth and resonant frequency. The designer is still left with one which can be used to scale R, L and C to convenient practical values. Alternatively, R may be predetermined by the external circuitry which will use the last degree of freedom.

#### Low-pass filter[edit]

An RLC circuit can be used as a low-pass filter. The circuit configuration is shown in Figure 6. The corner frequency, that is, the frequency of the 3 dB point, is given by

This is also the bandwidth of the filter. The damping factor is given by^{[27]}

#### High-pass filter[edit]

A high-pass filter is shown in Figure 7. The corner frequency is the same as the low-pass filter:

The filter has a stop-band of this width.^{[28]}

#### Band-pass filter[edit]

A band-pass filter can be formed with an RLC circuit by either placing a series LC circuit in series with the load resistor or else by placing a parallel LC circuit in parallel with the load resistor. These arrangements are shown in Figures 8 and 9 respectively. The centre frequency is given by

and the bandwidth for the series circuit is^{[29]}

The shunt version of the circuit is intended to be driven by a high impedance source, that is, a constant current source. Under those conditions the bandwidth is^{[29]}

#### Band-stop filter[edit]

Figure 10 shows a band-stop filter formed by a series LC circuit in shunt across the load. Figure 11 is a band-stop filter formed by a parallel LC circuit in series with the load. The first case requires a high impedance source so that the current is diverted into the resonator when it becomes low impedance at resonance. The second case requires a low impedance source so that the voltage is dropped across the antiresonator when it becomes high impedance at resonance.^{[30]}

### Oscillators[edit]

For applications in oscillator circuits, it is generally desirable to make the attenuation (or equivalently, the damping factor) as small as possible. In practice, this objective requires making the circuit's resistance R as small as physically possible for a series circuit, or alternatively increasing R to as much as possible for a parallel circuit. In either case, the RLC circuit becomes a good approximation to an ideal LC circuit. However, for very low-attenuation circuits (high Q-factor), issues such as dielectric losses of coils and capacitors can become important.

In an oscillator circuit

or equivalently

As a result,

### Voltage multiplier[edit]

In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit

If R is small, consisting only of the inductor winding resistance say, then this current will be large. It will drop a voltage across the inductor of

An equal magnitude voltage will also be seen across the capacitor but in antiphase to the inductor. If R can be made sufficiently small, these voltages can be several times the input voltage. The voltage ratio is, in fact, the Q of the circuit,

A similar effect is observed with currents in the parallel circuit. Even though the circuit appears as high impedance to the external source, there is a large current circulating in the internal loop of the parallel inductor and capacitor.

### Pulse discharge circuit[edit]

An overdamped series RLC circuit can be used as a pulse discharge circuit. Often it is useful to know the values of components that could be used to produce a waveform. This is described by the form

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## Resonance

Resonant circuits are used to respond selectively to signals of a given frequency while discriminating against signals of different frequencies. If the response of the circuit is more narrowly peaked around the chosen frequency, we say that the circuit has higher "selectivity". A "quality factor" Q, as described below, is a measure of that selectivity, and we speak of a circuit having a "high Q" if it is more narrowly selective.

An example of the application of resonant circuits is the selection of AM radio stations by the radio receiver. The selectivity of the tuning must be high enough to discriminate strongly against stations above and below in carrier frequency, but not so high as to discriminate against the "sidebands" created by the imposition of the signal by amplitude modulation. |

The selectivity of a circuit is dependent upon the amount of resistance in the circuit. The variations on a series resonant circuit at right follow an example in Serway & Beichner. The smaller the resistance, the higher the "Q" for given values of L and C. The parallel resonant circuit is more commonly used in electronics, but the algebra necessary to characterize the resonance is much more involved. |

Using the same circuit parameters, the illustration at left shows the power dissipated in the circuit as a function of frequency. Since this power depends upon the square of the current, these resonant curves appear steeper and narrower than the resonance peaks for current above. The quality factor Q is defined by where Δω is the width of the resonant power curve at half maximum. |

Since that width turns out to be Δω =R/L, the value of Q can also be expressed as

AC Circuits

Reference

Serway & Beichner

Ch 33

In the RLC series circuit, when the circuit current is in phase with the applied voltage, the circuit is said to be in **Series Resonance.** The resonance condition arises in the series RLC circuit when the inductive reactance is equal to the capacitive reactance.

**X _{L} = X_{C} or (X_{L }– X_{C} = 0)**

A series resonant circuit has the capability to draw heavy current and power from the mains; it is also called **acceptor circuit**. The series resonance RLC circuit is shown in the figure below:

At the resonance : X_{L }– X_{C }= 0 or X_{L} = X_{C}

The Impedance will be:

Where Zr is the resonance impedance of the circuit.

Putting the value of X_{L }– X_{C} = 0 in equation (1) we will get:

Zr = R

Current I = V/ Zr = V/R

Since at resonance the opposition to the flow of current is only resistance (R) of the circuit. At this condition, the circuit draws the maximum current.

**Also See**: What is Resonant Frequency?

### Effects of Series Resonance

The following effects of the series resonance condition are given below:

- At resonance condition, XL = XC the impedance of the circuit is minimum and is reduced to the resistance of the circuit. i.e Zr = R
- At the resonance condition, as the impedance of the circuit is minimum, the current in the circuit is maximum. i.e Ir = V/Zr = V/R
- As the value of resonant current Ir is maximum hence, the power drawn by the circuit is also maximized. i.e Pr = I
^{2}Rr - At the resonant condition, the current drawn by the circuit is very large or we can say that the maximum current is drawn. Therefore, the voltage drop across the inductance L i.e (V
_{L}= IX_{L }= I x 2πfrL) and the capacitance C i.e (V_{C}= IX_{C}= I x I/2πfrC) will also be very large.

In the power system, at the resonant condition, the excessive voltage is built up across the inductive and capacitive component of the circuit such as circuit breaker, reactors, etc., may cause damage. Therefore, the series resonant condition is avoided in the power system.

However, in some of the electronics devices such as antenna circuit of radio and TV receiver, tunning circuit, etc. The series resonant condition is used to increase the signal voltage and current at the desired frequency (fr).

## Resonance rlc series

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Circuits in which the inductive reactance equals the capacitive reactance (XL=XC) are called **resonant circuits**. They can be series or parallel circuits and either **RLC** or **LC** circuits. The impedance vector for a typical series RLC resonant circuit is shown in Figure 1 and is summarized as follows:

- X
_{L}and X_{C}are 180 degrees out of phase. - X
_{L}and X_{C}are equal in value (100 Ω), resulting in a net reactance of zero ohm. - The only opposition to current is then R (10 Ω).
- Z is equal to R and is at its
**minimum**value, allowing the greatest amount of current to flow.

**Figure 1** Impedance vector for a series *RLC* resonant circuit.

The voltage vector for the series *RLC* resonant circuit is shown in Figure **2** and is summarized as follows:

- The same amount of current flows through all components.
- The voltage drop across the inductor is

- The voltage drop across the capacitor is

- Voltages across
*X*and_{L}*X*are equal (120 V) and 180 degrees out of phase with each other so that each cancels the other._{C} - With the effects of both the inductor and capacitor canceled out, the only current-limiting component will be the resistor, and the total applied supply voltage appears across the resistor.
- Therefore, the
**phase angle**between the circuit current and the supply voltage will be**zero**and the power factor will be 1, or 100 percent. - The circuit can be considered to be
**purely resistive**in nature with inductive reactive VARs of the inductor being canceled out by the inductive capacitive VARs of the capacitor. The true power is

**Figure 2** Voltage vector for the series *RLC* resonant circuit.

**Series RLC Resonant Circuit Calculations Example 1**

For the series *RLC resonant* circuit shown in Figure **3**, determine:

- Impedance (
*Z*). - Current (
*I*). - Voltage drop across the resistor (ER), inductor (EL) and capacitor (EC).
- Apparent, true, and net reactive power.
- Power factor.

**Figure 3** Circuit for example 1.

**Solution:**

\[\text{a}\text{. Z=R=24 }\!\!\Omega\!\!\text{ }\]

\[\text{b}\text{. I=}\frac{{{\text{E}}_{\text{T}}}}{\text{R}}\text{=}\frac{\text{120V}}{\text{24 }\!\!\Omega\!\!\text{ }}\text{=5A}\]

\[\begin{align}& \text{c}\text{. }{{\text{E}}_{\text{R}}}\text{=I }\!\!\times\!\!\text{ R=5A }\!\!\times\!\!\text{ 24 }\!\!\Omega\!\!\text{ =120V} \\& {{\text{E}}_{\text{L}}}\text{=I }\!\!\times\!\!\text{ }{{\text{X}}_{\text{L}}}\text{=5A }\!\!\times\!\!\text{ 40 }\!\!\Omega\!\!\text{ =200V} \\& {{\text{E}}_{\text{C}}}\text{=I }\!\!\times\!\!\text{ }{{\text{X}}_{\text{C}}}\text{=5A }\!\!\times\!\!\text{ 40 }\!\!\Omega\!\!\text{ =200V} \\\end{align}\]

\[\begin{matrix} Apparent\text{ }Power & ={{E}_{T}}~\times ~I & =120V~\times ~5A=600VA \\ True\text{ }Power & ={{E}_{R}}~\times ~I & =120V~\times ~5A=600W \\ Net\text{ }Reactive\text{ }Power & =\left( I\times {{E}_{L}} \right)-\left( I\times {{E}_{C}} \right) & =0VARs \\\end{matrix}\]

\[\text{e}\text{. Power Factor=}\frac{\text{W}}{\text{VA}}\text{=}\frac{\text{600W}}{\text{600VA}}\text{=1}\]

Recall that inductive reactance varies ** directly** as the frequency of the AC supply voltage (XL=2πfL), while capacitive reactance varies

**as the frequency (XC=1/2πfC).**

*inversely*When an inductor and capacitor are connected in series in a circuit, there will be **one resonant frequency** at which the inductive reactance and capacitive reactance will become equal. The reason for this is that as frequency increases, inductive reactance increases and capacitive reactance decreases.

The following formula is used to determine the resonant frequency when the values of inductance and capacitance are known:

As an example, suppose that a fixed AC voltage of variable frequency is applied to a series *RLC* circuit. As the frequency of the applied voltage is increased, the inductive reactance *X*_{L}**increases** but the capacitive reactance *X*_{C}**decreases,** as illustrated in Figure **4**. You can see from this graph that at the resonant frequency XL=XC.

**Figure 4** How *X _{L}* and

*X*vary with change in frequency.

_{C}**Resonant Frequency of RLC Series Circuit Example 2**

Calculate the resonant frequency of a *RLC* series circuit containing a 750-mH inductor and a 47-μF capacitor.

**Solution:**

In certain applications a series resonant circuit is used to achieve an increase in voltage at the resonant frequency. As an example, in the series resonant circuit of Figure **5**, the voltage across *X _{L}* and

*X*is much

_{C}**This seemingly impossible condition is caused by the interaction between the capacitor and inductor. The voltage across the inductor and capacitor is**

*higher than the applied total voltage.***while the applied voltage is only 120 volts.**

*1,200 volts,*In some control applications, the voltage across either *X _{L}* or

*X*is used as a signal voltage to perform some function.

_{C}**Figure 5** High voltage across reactive elements.

A typical **frequency response curve** for a series *RLC* circuit is shown in Figure **6** and summarized as follows:

- The frequency is varied, and the values of current at the different frequencies plotted on the graph.
*The magnitude of the current is a function of the frequency.*- The response curve starts near zero, reaches maximum value at the resonance frequency, and then drops to near zero as the frequency becomes infinite.
- There is a small range of frequencies, called the resonant band or
**band pass,**on either side of resonance where the current is almost the same as it is at resonance. - The circuit can be used to isolate or
**filter**out certain frequencies.

**Figure 6** Frequency response curve for a series *RLC* resonant circuit.

In a series *RLC* circuit at resonance, the two reactances, *X _{L}* and

*X*are equal and canceling. In addition, the two voltages representing

_{C}*V*and

_{L}*V*are also opposite and equal in value, thereby canceling each other out.

_{C}Figure **7** compares the circuit condition that exists above or below resonance with that when the circuit is at resonance and is summarized as follows:

- When operating above its resonant frequency, a series
*RLC*circuit has the dominate characteristics of a series*R*circuit._{L} - When operating below its resonant frequency, a series
*RLC*circuit has the dominate characteristics of a series*R*circuit._{C} - When operating at its resonant frequency:
- – Reactance (
*X*) is zero as XL=XC. - – Impedance is minimum and current is maximum as
*Z*=*R*. - – The voltage measured across the two series reactive components
*L*and*C*is zero. - – All the supply voltage is dropped across the resistor.
- – The phase angle between the supply voltage and current is zero and the power factor is 1.

- – Reactance (

**Figure 7** Series *RLC* resonant circuit characteristics.

**Review Questions**

- In a series resonant
*RLC*circuit how does the value of*X*compare with that of_{L}*X*?_{C} - Maximum current will flow when a series
*RLC*circuit is at resonance. Why? - What are the circuit voltage conditions that always exist across the inductor, capacitor, and resistor in a resonant
*RLC*circuit? - In a series resonant
*RLC*circuit the apparent power (VA) is the same value as the true power (watts). Why? - State the phase relationship of each of the following for a series resonant
*RLC*circuit:*X*and_{L}*X*._{C}*E*and_{L}*I*.*E*and_{C}*I*.*E*and_{L}*E*._{C}*R*and*Z*.*E*and_{T}*E*._{R}

- Determine each of the following for the
*RLC*circuit shown in Figure**8**, when at resonance:- The resonant frequency.
- The circuit impedance.
- The circuit current.
- The voltage drops across the resistor, inductor and capacitor.
- Net reactive power.
- True power.
- Power factor.

**Figure 8** Circuit for review question 6.

7. Answer the following with reference to the circuit shown in Figure **9**.

- What is the resonant frequency of the circuit? Why?
- What should the reading on the voltmeter be? Why?
- Assume the frequency of the applied voltage is increased to 400 Hz. Would the circuit become inductive or capacitive? Why?
- Assume the frequency of the applied voltage is decreased to 60 Hz. Would the circuit become inductive or capacitive? Why?
- If the frequency of the applied voltage is increased above 318 Hz, what happens to the value of the impedance and current? Why?
- If the frequency of the applied voltage is decreased below 318 Hz, what happens to the value of the impedance and current? Why?

**Figure 9** Circuit for review question 7.

8. A series *RLC* circuit is connected to a variable frequency AC power supply with an ammeter connected to measure current flow and a voltmeter connected to measure the voltage drop across the resistor. Outline how you would proceed to set the circuit to resonance.

9. What should you be aware of when measuring voltages in a series resonant circuit?

10. Give two practical applications for series resonant circuits.

11. A 24-Ω resistor, an inductor with a reactance of 120 Ω, and a capacitor with a reactance of 120 Ω are in series across a 60-V source. The circuit is at resonance. Determine the voltage across the inductor.

**Review Questions – Answers**

- X
_{L }= X_{C} - At resonance X
_{L}is cancelled out by X_{C}so only the resistance limits the current flow in the circuit. - The voltage across the inductor is equal to that across the capacitor. The voltage across the resistor is the same value as the applied voltage.
- At resonance the reactive power generated by inductance component is cancelled out by the reactive power generated by the capacitance component.
- (a) 180° out of phase, (b) 90° out of phase, (c) 90° out of phase, (d) 180° out of phase, (e) in phase, (f) in phase
- (a) 159 Hz, (b) 30 Ω, (c) 4 A, (d) E
_{R}= 120 V, E_{L}= 160 V, E_{C}= 160 V, (e) zero, (f) 480 Watt, (g) 1 or 100% - (a) The applied 318 Hz is the resonant frequency, because the voltage across the inductor is equal to the voltage across the capacitor.

(b) Zero, because the two voltages E_{C} and E_{L} are of equal value and cancel each other out.

(c) Inductive because X_{L} would become greater than X_{C}.

(d) Capacitive because X_{C} would become greater than X_{L}.

(e) The impedance will increase and the current will decrease because there will be some reactance in addition to resistance limiting the current flow.

(f) The impedance will increase and the current will decrease because there will be some reactance in addition to resistance limiting the current flow.

8. Adjust the frequency of the power supply until the current and voltage across the resistor are at maximum values. This is the resonant frequency.

9. Voltages across XL and XC are equal and 180 degrees out of phase with each other resulting in the circuit to be purely resistive in nature with inductive reactive VARs of the inductor being canceled out by the inductive capacitive VARs of the capacitor.

10. The series resonant circuit can be used to select a particular frequency in a communication circuit or to generate a voltage higher than the supply voltage.

11. 300 volts

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## Network Theory - Series Resonance

**Resonance** occurs in electric circuits due to the presence of energy storing elements like inductor and capacitor. It is the fundamental concept based on which, the radio and TV receivers are designed in such a way that they should be able to select only the desired station frequency.

There are **two types** of resonances, namely series resonance and parallel resonance. These are classified based on the network elements that are connected in series or parallel. In this chapter, let us discuss about series resonance.

### Series Resonance Circuit Diagram

If the resonance occurs in series RLC circuit, then it is called as **Series Resonance**. Consider the following **series RLC circuit**, which is represented in phasor domain.

Here, the passive elements such as resistor, inductor and capacitor are connected in series. This entire combination is in **series** with the input sinusoidal voltage source.

Apply **KVL** around the loop.

$$V - V_R - V_L - V_C = 0$$

$$\Rightarrow V - IR - I(j X_L) - I(-j X_C) = 0$$

$$\Rightarrow V = IR + I(j X_L) + I(-j X_C)$$

$\Rightarrow V = I[R + j(X_L - X_C)]$**Equation 1**

The above equation is in the form of ** V = IZ**.

Therefore, the **impedance Z** of series RLC circuit will be

$$Z = R + j(X_L - X_C)$$

### Parameters & Electrical Quantities at Resonance

Now, let us derive the values of parameters and electrical quantities at resonance of series RLC circuit one by one.

### Resonant Frequency

The frequency at which resonance occurs is called as **resonant frequency f_{r}**. In series RLC circuit resonance occurs, when the imaginary term of impedance

*Z*is zero, i.e., the value of $X_L - X_C$ should be equal to zero.

$$\Rightarrow X_L = X_C$$

Substitute $X_L = 2 \pi f L$ and $X_C = \frac{1}{2 \pi f C}$ in the above equation.

$$2 \pi f L = \frac{1}{2 \pi f C}$$

$$\Rightarrow f^2 = \frac{1}{(2 \pi)^2 L C}$$

$$\Rightarrow f = \frac{1}{(2 \pi) \sqrt{LC}}$$

Therefore, the **resonant frequency f_{r}** of series RLC circuit is

$$f_r = \frac{1}{(2 \pi) \sqrt{LC}}$$

Where, ** L** is the inductance of an inductor and

**is the capacitance of a capacitor.**

*C*The **resonant frequency f_{r}** of series RLC circuit depends only on the inductance

**and capacitance**

*L***. But, it is independent of resistance**

*C***.**

*R*### Impedance

We got the **impedance Z** of series RLC circuit as

$$Z = R + j(X_L - X_C)$$

Substitute $X_L = X_C$ in the above equation.

$$Z = R + j(X_C - X_C)$$

$$\Rightarrow Z = R + j(0)$$

$$\Rightarrow Z = R$$

At resonance, the **impedance Z** of series RLC circuit is equal to the value of resistance

**, i.e.,**

*R***.**

*Z = R*### Current flowing through the Circuit

Substitute $X_L - X_C = 0$ in Equation 1.

$$V = I[R + j(0)]$$

$$\Rightarrow V = IR$$

$$\Rightarrow I = \frac{V}{R}$$

Therefore, **current** flowing through series RLC circuit at resonance is $\mathbf{\mathit{I = \frac{V}{R}}}$.

At resonance, the impedance of series RLC circuit reaches to minimum value. Hence, the **maximum current** flows through this circuit at resonance.

### Voltage across Resistor

The voltage across resistor is

$$V_R = IR$$

Substitute the value of ** I** in the above equation.

$$V_R = \lgroup \frac{V}{R} \rgroup R$$

$$\Rightarrow V_R = V$$

Therefore, the **voltage across resistor** at resonance is ** V_{R} = V**.

### Voltage across Inductor

The voltage across inductor is

$$V_L = I(jX_L)$$

Substitute the value of ** I** in the above equation.

$$V_L = \lgroup \frac{V}{R} \rgroup (jX_L)$$

$$\Rightarrow V_L = j \lgroup \frac{X_L}{R} \rgroup V$$

$$\Rightarrow V_L = j QV$$

Therefore, the **voltage across inductor** at resonance is $V_L = j QV$.

So, the **magnitude** of voltage across inductor at resonance will be

$$|V_L| = QV$$

Where ** Q** is the

**Quality factor**and its value is equal to $\frac{X_L}{R}$

### Voltage across Capacitor

The voltage across capacitor is

$$V_C = I(-j X_C)$$

Substitute the value of *I* in the above equation.

$$V_C = \lgroup \frac{V}{R} \rgroup (-j X_C)$$

$$\Rightarrow V_C = -j \lgroup \frac{X_C}{R} \rgroup V$$

$$\Rightarrow V_C = -jQV$$

Therefore, the **voltage across capacitor** at resonance is $\mathbf{\mathit{V_C = -jQV}}$.

So, the **magnitude** of voltage across capacitor at resonance will be

$$|V_C| = QV$$

Where ** Q** is the

**Quality factor**and its value is equal to $\frac{X_{C}}{R}$

**Note** − Series resonance RLC circuit is called as **voltage magnification** circuit, because the magnitude of voltage across the inductor and the capacitor is equal to *Q* times the input sinusoidal voltage *V*.

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