series resonance in a series resonant circuit

8/10/2019 Series Resonance in a Series Resonant Circuit

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Capacitors Circuits

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From the above equation for inductive reactance, if either the Frequencyor the Inductanceis increased the overall inductive

reactance value of the inductor would also increase. As the frequency approaches infinity the inductors reactance would also

increase towards infinity with the circuit element acting like an open circuit. However, as the frequency approaches zero or

DC, the inductors reactance would decrease to zero, causing the opposite effect acting like a s hort circuit. This means then

that inductive reactance is "Proportional" to frequency and is small at low frequencies and high at higher frequencies and

this dem onstrated in the following curve:

Inductive Reactance against Frequency

The graph of inductive reactance against frequency is a

straight line linear curve. The inductive reactance value of

an inductor increases linearly as the frequency across it

increases. Therefore, inductive reactance is positive and is

directly proportional to frequency (X )

The same is also true for the capacitive reactance formula above but in reverse. If either the Frequencyor the Capacitanceis

increased the overall capacitive reactance would decrease. As the frequency approaches infinity the capacitors reactance

would reduce to zero causing the circuit element to act like a perfect conductor of 0's. However, as the frequency

approaches zero or DC level, the capacitors reactance would rapidly increase up to infinity causing it to act like a very large

resistance acting like an open circuit condition. This means then that capacitive reactance is " Inversely proportional" to

frequency for any given value of capacitance and this shown below:

Capacitive Reactance against Frequency

The graph of capacitive reactance against frequency is a

hyperbolic curve. The Reactance value of a capacitor has a

very high value at low frequencies but quickly decreases as

the frequency across it increases. Therefore, capacitive

reactance is negative and is inversely proportional to

frequency ( X )

We can see that the values of these resistances depends upon the frequency of the supply. At a higher frequency X is high

and at a low frequency X is high. Then there must be a frequency point were the value of X is the same as the value of X

and there is. If we now place the curve for inductive reactance on top of the curve for capacitive reactance so that both curves

are on the sam e axes, the point of intersection will give us the series resonance frequency point, ( or ) as s hown below.

Series Resonance Frequency

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where: is in Hertz, Lis in Henries and Cis in Farads.

Electrical resonance occurs in an AC circuit when the two reactances which are oppos ite and equal cancel each other out as

X = X and the point on the graph at which this happens is were the two reactance curves cross each other. In a series

resonant circuit, the resonant frequency, point can be calculated as follows.

We can see then that at resonance, the two reactances cancel each other out thereby making a series LC combination act as

a short circuit with the only oppos ition to current flow in a series resonance circuit being the resistance, R. In complex form,

the resonant frequency is the frequency at which the total impedance of a series RLC circuit becomes purely "real", that is no

imaginary impedances exist. This is because at resonance they are cancelled out. So the total impedance of the series

circuit becomes just the value of the resistance and therefore: Z = R.

Then at resonance the impedance of the series circuit is at its minimum value and equal only to the resistance, Rof the

circuit. The circuit impedance at resonance is called the "dynamic impedance" of the circuit and depending upon the

frequency, X (typically at high frequencies) or X (typically at low frequencies) will dominate either side of resonance as

shown below.

Impedance in a Series Resonance Circuit

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Note that when the capacitive reactance dominates the circuit the impedance curve has a hyperbolic shape to itself, but when

the inductive reactance dominates the circuit the curve is non-symmetrical due to the linear response of X . You may also

note that if the circuits im pedance is at its minim um at resonance then consequently, the circuits admittancemust be at its

maximum and one of the characteristics of a series resonance circuit is that admittance is very high. But this can be a bad

thing because a very low value of resis tance at resonance means that the circuits current may be dangerously high.

We recall from the previous tutorial about series RLC circuits that the voltage across a s eries combination is the phasor sum

of V , V and V . Then if at resonance the two reactances are equal and cancelling, the two voltages representing V and

V must also be opposite and equal in value thereby cancelling each other out because with pure components the phasor

voltages are drawn at +90 and -90 respectively. Then in a series resonancecircuit V = -V therefore, V = V .

Series RLC Circuit at Resonance

Since the current flowing through a s eries resonance circuit is the product of voltage divided by impedance, at resonance the

impedance, Zis at its minimum value, ( =R). Therefore, the circuit current at this frequency will be at its maximum value of

V/Ras s hown below.

Series Circuit Current at Resonance

The frequency response curve of a series resonance circuit shows that the magnitude of the current is a function of frequency

and plotting this onto a graph shows us that the response s tarts a t near to zero, reaches m aximum value at the resonance

frequency when I = I and then drops again to nearly zero as becomes infinite. The result of this is that the magnitudes

of the voltages across the inductor, Land the capacitor, Ccan becom e many times larger than the supply voltage, even at

resonance but as they are equal and at oppos ition they cancel each other out.

As a series resonance circuit only functions on resonant frequency, this type of circuit is also known as an Acceptor Circuit

because at resonance, the impedance of the circuit is at its minim um s o easily accepts the current whose frequency is equalto its resonant frequency. The effect of resonance in a series circuit is als o called "voltage resonance".

You may also notice that as the maximum current through the circuit at resonance is limited only by the value of the

resis tance (a pure and real value), the source voltage and circuit current must therefore be in phase with each other at this

frequency. Then the phase angle between the voltage and current of a series resonance circuit is also a function of frequency

for a fixed supply voltage and which is zero at the resonant frequency point when: V, Iand V are all in phase with each other

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. , .

Phase Angle of a Series Resonance Circuit

Notice also, that the phase angle is positive for frequencies above and negative for frequencies below and this can be

proven by,

Bandwidth of a Series Resonance Circuit

If the series RLC circuit is driven by a variable frequency at a constant voltage, then the magnitude of the current, I is

proportional to the impedance, Z, therefore at resonance the power abs orbed by the circuit mus t be at its maximum value as

P = I Z. If we now reduce or increase the frequency until the average power absorbed by the resis tor in the series resonance

circuit is hal f that of its maximum value at resonance, we produce two frequency points called the half-power pointswhich

are -3dB down from maximum , taking 0dB as the maximum current reference.

These -3dB points give us a current value that is 70.7% of its m aximum resonant value as: 0.5( I R ) = (0.707 x I) R. Then

the point corresponding to the lower frequency at half the power is called the "lower cut-off frequency", labelled with the

point corresponding to the upper frequency at half power being called the "upper cut-off frequency", labelled . The distance

between these two points, i.e. ( - ) is called the Bandwidth, (BW) and is the range of frequencies over which at least

half of the maximum power and current is provided as shown.


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The frequency response of the circuits current magnitude above, relates to the "sharpness" of the resonance in a series

resonance circuit. The sharpness of the peak is m easured quantitatively and is called the Quality factor, Qof the circuit. Thequality factor relates the maximum or peak energy stored in the circuit (the reactance) to the energy dissipated (the

resis tance) during each cycle of oscillation meaning that it is a ratio of resonant frequency to bandwidth and the higher the

circuit Q, the smaller the bandwidth, Q = /BW.

As the bandwidth is taken between the two -3dB points , the selectivityof the circuit is a meas ure of its ability to reject any

frequencies either side of these points. A more selective circuit will have a narrower bandwidth whereas a less selective

circuit will have a wider bandwidth. The selectivity of a s eries resonance circuit can be controlled by adjusting the value of the

resis tance only, keeping all the other components the same, since Q = (X or X )/R.


Then the relationship between resonance, bandwidth, selectivity and quality factor for a series resonance circuit being

defined as:

1). Resonant Frequency, ( )

2). Current, (I)

3). Lower cut-off frequency, ( )

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4). Upper cut-off frequency, ( )

5). Bandwidth, (BW)

6). Quality Factor, (Q)

Example No1

A series resonance network consisting of a resistor of 30, a capacitor of 2uF and an inductor of 20mH is connected across

a s inusoidal s upply voltage which has a constant output of 9 volts at all frequencies. Calculate, the resonant frequency, the

current at resonance, the voltage across the inductor and capacitor at resonance, the quality factor and the bandwidth of the

circuit. Also sketch the corresponding current waveform for all frequencies.

Resonant Frequency,

Circuit Current at Resonance, I

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Inductive Reactance at Resonance, X

Voltages across the inductor and the capacitor, V , V

( Note: the supply voltage is only 9 volts, but at resonance the reactive voltages are 30 volts peak! )

Quality factor, Q

Bandwidth, BW

The upper and lower -3dB frequency points, and

Current Waveform

Example No2

A series circui t consis ts of a resis tance of 4, an inductance of 500m H and a variable capacitance connected across a 100V,

50Hz supply. Calculate the capacitance require to give series resonance and the voltages generated across both the

inductor and the capacitor.

Resonant Frequency,

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Resistor DC Voltage Electrical Resonance
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Basic Electronics Tutorials by Wayne Storr. Last updated: September 2013 ,

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series resonance in a series resonant circuit

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