# Serial Resonance Circuit

Resonance in a series of circuits occurs when the feeding frequency causes the voltages between L and C to be equal and opposite in the phase.

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So far, we have analyzed the behavior of a series of RLC circuits, the source voltage of which is a stable state sinusoidal source with a fixed frequency. In our training on serial RLC circuits, we also found that two or more sinusoidal signals can be combined using phasers, provided they have the same frequency source.

However, what happens to the properties of the circuit if a supply voltage with fixed amplitude but different frequencies is applied to the circuit? Also, what would be the "frequency response" behavior of the circuits on the two reactive components due to this changing frequency?

In a series of RLC circuits, there is a frequency point when the inductive coloract of the inductor is equal to the capacitor's capacitive recess as a value.In other words, X_{L} = X_{C} .The point at which this occurs is called the **Resonance Frequency** point ( ε _{r} ) of the circuit, and this resonance frequency produces a Series Resonance while we analyze a series of RLC **circuits.**

*Serial Resonance* circuits are one of the most important circuits used in electrical and electronic circuits.They can be found in various formats such as AC mains filters,noise filters, as well as radio and television tuning circuits, which produce a very selective adjustment circuit for the acquisition of different frequency channels.

### Serial RLC Circuit

First, let's define what we know about serial RLC circuits.

From the inductive colorance equation above, if **frequency** or **inductation is** increased, the total inductive colorance value of the inductor will also increase.As the frequency approaches forever, the coloring of the inductors will increase forever with the circuit element acting as an open circuit.

However, as the frequency approaches zero or DC, the coloracitation of the inductors drops to zero, causing the reverse effect to act as a short circuit.This means that inductive reactance is "**Proportional**" with frequency and is high at small and high frequencies at low frequencies, and this is shown in the following curve:

### Inductive Reactance Against Frequency

The graph of inductive rectal against frequency is a straight linear curve.The inductive reassurance value of an inductor increases linearly as the frequency on it increases.Therefore, inductive reactance is positive and directly proportional to the frequency ( X _{L} ∝ ε )

The same applies to the capacitive reactance formula above, but vice versa. If frequency or Capacitance is increased, the overall capacitive reassurance will decrease. As the frequency approaches infinity, the coloract of the capacitors practically drops to zero, causing the circuit element to behave like an excellent conductor of 0Ω.

However, as the frequency approaches zero or DC level, the coloracitation of capacitors will increase rapidly forever, causing it to act as a very large resistance, becoming more like an open circuit state.This means that capacitive reassurance is "**Inversely proportional** " with frequency for any given capacitance value, and this is shown below:

### Capacitive Reactance Against Frequency

The graph of capacitive reassurance against frequency is a hyperbolic curve.The Reactax value of a capacitor has a very high value at low frequencies, but decreases rapidly as the frequency on it increases.Therefore, capacitive reassurance is negative and inversely proportional to frequency ( X _{C} ∝ ε ^{-1} )

We can see that the values of these resistors depend on the frequency of the source.X _{L} is high at a higher frequency and X _{C} is high at a lower frequency.Then there should be a frequency point if the X _{L} value and the X _{C} value are the same. Now if we place the inductive reacquest curve above the capacitive reactance curve, the intersection point will give us the serial resonance frequency point ( ε _{r} or ω r ), as shown below, so that both curves are on _{the} same axis.

### Serial Resonance Frequency

Here: ε _{r} Hertz , L Henry and C Farad.

Electrical resonance in an AC circuit occurs when the effects of two recesses, opposite and equal, cancel each other out in X _{L} = X _{C.}In the chart above, the point at which this occurs is that the two reassurance curves cut each other.In the serial resonance circuit, the resonance frequency, point ε _{r} can be calculated as follows.

Then in resonance, mathematically we can see that two reactions cancel each other out as X_{L} – X_{C} = 0. This allows the series LC combination to act as a short circuit with R, the only opposition resistance to the current flow in the serial resonance circuit.

In complex form, the resonance frequency is the frequency at which the total impedance of a series RLC circuit becomes completely *"real",* that is, there is no imaginary impedance.This is because they are canceled in resonance.Thus, the total impedance of the serial circuit is only the value of resistance, and therefore: Z = R .

Then, in resonance, the impedance of the serial circuit is minimal and equals only to the R resistance of the circuit. Circuit impedance in resonance is called the "dynamic impedance" of the circuit, and depending on the frequency, X_{C} (typically at high frequencies) or X_{L} (typically at low frequencies) will dominate both sides of the resonance, as shown below.

### Impedance in Serial Resonance Circuit

Note that when capacitive reactance dominates the circuit, the impedance curve itself has a hyperbolic shape, but when the inductive reactant dominates the circuit, the curve is not symmetrical due to the linear response of X _{L.}

You can also note that if the impedance of circuits is minimal in resonance, as a result, **the admittans**of the circuits must be at the maximum, and one of the characteristics of a series of resonance circuits is that the admittans are very high.But this can be a bad thing because a very low resistance value in resonance means that the current passing through the circuit can be dangerously high.

From previous training on serial RLC circuits, we remember that the voltage in a series combination is the phaser sum of V_{R,}V_{L} and V_{C.} Then, if the two reactions in resonance are equal and canceled, the two voltages representing V_{L} and V_{C}must also be opposite and equal, so that they cancel each other out because phaser voltages with pure components are drawn at +90^{o} and-90^{o}respectively.

Then the reactive voltages that occur in the serial resonance circuit as V_{L} = -V_{C} are zero and the entire supply voltage decreases along the resistance. Therefore, V_{R} = V_{supply} and therefore serial resonance circuits are known as voltage resonance circuits (unlike parallel resonance circuits, which are current resonance circuits).

### Serial RLC Circuit in Resonance

Since the current flowing through a series of resonance circuits is the product of the voltage divided into impedance, the impedance in resonance is the minimum value of Z (=R). Therefore, the circuit current at this frequency will be at the maximum V/R value, as shown below.

### Serial Circuit Current in Resonance

The frequency response curve of the serial resonance circuit indicates that the magnitude of the current is a function of the frequency, and drawing it on a graph indicates to us that the answer begins close to zero, reaches the maximum value at the resonance frequency when I_{MAX} = I R is I_{R,} and falls back to zero as ε becomes infinite. The result of this is that the voltages on the inductor L and capacitor C can be many times larger than the feed voltage, even in resonance, but they are equal and cancel each other out.

Since a series of resonance circuits work only on the resonance frequency, this type of circuit is also known as a**Receiver**Circuit, since in resonance, the impedance of the circuit is minimal, easily accepting the current whose frequency is equal to the resonance frequency.

Since the maximum current passing through the resonance circuit is limited only to the value of resistance (a pure and real value), you may also notice that the source voltage and circuit current should be phased with each other at this frequency. Then, the phase angle between the voltage and current of a series of resonance circuits is also a function of the frequency for a constant supply voltage, and at the resonance frequency point it is zero: V, I and VR are in phase with each other, as shown below. As a result, if the phase angle is zero, the power factor should therefore be one.

### Phase Angle of Serial Resonance Circuit

Also note that the phase angle is positive for frequencies _{above} ε r and negative for frequencies below ε _{r,} which can be proven as follows:

## Bandwidth of the Serial Resonance Circuit

If the serial RLC circuit is driven at a variable frequency at a constant voltage, the size of the current is proportional to the I impedance, so the power absorbed by the circuit in resonance should be at its maximum value. P = I^{2}Z.

If we reduce or increase the frequency until the average power absorbed by the resistance in the serial resonance circuit is half its maximum value in resonance, we will take the maximum current reference by 0dB, producing two frequency points -3dB down from the maximum called semi-power points.

These -3dB points give us a current value of 70.7% of the maximum resonance value, defined as: 0.5( I ^{2} R ) = (0.707 x I) ^{2} R.Then the point corresponding to the lower frequency at half the power is called "lower cutting frequency", it is labeled with ε _{L,} and the point corresponding to the upper frequency at half power is called "upper cutting frequency" and labeled ε_{H.}The distance between these two points, i.e. ( ε _{H} – ε _{L} ) **Bandwidth**, (BW ) is the frequency range in which at least half of the maximum power and current is provided, as shown.

### Bandwidth of the Serial Resonance Circuit

The frequency response of the circuits of the above current size relates to the "sharpness" of the resonance in a series of resonance circuits. The sharpness of the peak is quantitatively measured and the Quality factor of the circuit is called Q. The quality factor associates the maximum or maximum energy (reactens) stored in the circuit with the energy (resistance) emitted during each oscillation cycle; this means that the resonance frequency has a ratio to bandwidth, and the higher Q, the smaller the bandwidth, Q = εr /BW.

Since bandwidth is taken between two -3dB points, the **selectivity**of the circuit is a measure of the ability to reject any frequency on both sides of these points.A more selective circuit will have a narrower bandwidth, while a less selective circuit will have a wider bandwidth.Since the selectivity of a series resonance circuit is Q = (X _{L} or X _{C} )/R, it can be controlled only by adjusting the value of the resistance, keeping all other components the same.

### Bandwidth of serial RLC Resonance Circuit

The relationship between resonance, bandwidth, selectivity and quality factor for a series of resonance circuits is then defined as follows:

1).Resonance Frequency, (ε _{r} )

2).Current, (I)

3).Lower cut frequency, (ε _{L} )

4).Upper cutting frequency, (ε _{H} )

5).Bandwidth, (BW)

6).Quality Factor, (Q)

### Serial Resonance Circuit Question Example 1

A series of resonance networks consisting of a resistance of 30Ω, a capacitor of 2uF and an inductor of 20mH are connected to a sinusoidal supply voltage with a constant output of 9 volts at all frequencies.Calculate the resonance frequency, current in resonance, voltage on the inductor and capacitor in resonance, quality factor and bandwidth of the circuit.Also draw the corresponding current waveform for all frequencies.

1. Resonance Frequency, ε _{r}

2. Circuit Current in Resonance, I _{m}

3. Inductive Reactance in Resonance, X _{L}

4. Voltages on inductor and capacitor, V _{L,} V _{C}

Note: The supply voltage can only be 9 volts, but in resonance, capacitor, V _{C} and inductor, the reactive voltages on V _{L} are 30 volt peaks!

5. Quality factor, Q

6. Bandwidth, BW

BC Upper and lower -3dB frequency points, ε _{H} and ε _{L}

8. Current WaveForm

### Serial Resonance Circuit Question Example 2

A series circuit consists of a 4Ω resistance, a 500mH inductans and a variable capacitance connected to a 100V, 50Hz feed.Calculate the capacitance required to create a series of resonance conditions and the voltages generated throughout both the inductor and the capacitor at the resonance point.

Resonance Frequency, ε _{r}

Voltages on inductor and capacitor, V _{L,} V _{C}

## Series Resonance Circuit Summary

During the analysis of serial resonance circuits in this tutorial, you may have noticed that we are looking at bandwidth, upper and lower frequencies, -3dB points and quality or Q factor.All these are terms used in the design and construction of Tape Transition Filters (BPF), and in fact resonance circuits are used in 3-element main filter designs to pass all frequencies in the "transition band" range while rejecting others.

However, the main purpose of this tutorial is to analyze and understand the concept of how **Serial Resonance** occurs in passive RLC series circuits.

- It must have at least one inductor and one capacitor for resonance to occur in any circuit.
- Resonance is the result of oscillations in a circuit as the stored energy passes from the inductor to the capacitor.
- Resonance occurs when X
_{L}= X_{C }and the virtual part of the transfer function is zero. - In resonance, the impedance of the circuit is equal to the resistance value in Z = R.
- At low frequencies, the serial circuit is capacitive as follows: X
_{C}> X_{L}, which gives the circuit a power factor. - At high frequencies, the serial circuit is inductive as follows: X
_{L}> X_{C}, which gives the circuit a delayed power factor. - The high current value in resonance produces very high voltage values along the inductor and capacitor.
- Serial resonance circuits are useful for creating high frequency selective filters.However, high current and very high component voltage values can damage the circuit.
- The most obvious feature of the frequency response of a resonance circuit is a sharp resonance peak in amplitude properties.
- Since impedance is minimal and current is maximum, serial resonance circuits are also called
**Receiving Circuits.**

In the next lesson about Parallel Resonance, we will look at how the frequency affects the properties of a parallel connected RLC circuit, and this time how a parallel resonance circuit determines the current magnification of the Q factor.