Theory about Electric Ressonance in AC

This chapter consists of the items below. If you want to go directly to an item, click it.

1 - Introduction click here!

2 - Series Resonator Circuit click here!

2.1 - Bandwidth click here!

2.2 - Power in Resonance click here!

2.3 - Quality Factor (Q) click here!

2.4 - Selectivity click here!

3 - Parallel Resonant Circuit click here!

3.1 - Ideal Parallel Resonant Circuit click here!

3.2 - Real Parallel Resonator Circuit click here!

3.2.1 - Frequency for F.P. Unit clique aqui!

3.2.2 - Frequency for Maximum Impedance clique aqui!

3.3 - Merit Factor - Q click here!

3.4 - Dynamic Impedance click here!

In this chapter we will study the resonant circuits, also known as tuned. For resonance to occur in a circuit, it is fundamental that it contains capacitor(s) and inductor(s), a condition that is necessary to have energy exchange between one reactive element and the other. As a rule, we have a combination of resistors, capacitors and inductors in the circuit. The resistor dissipates the energy exchanged between the reactive elements. Depending on the values of these components we can calculate how often the resonance happens. There are two types of resonant circuits: series and parallel. Let's discuss them separately.

2. Series Resonator Circuit

There will be resonance in a series circuit formed by R , L and C if the current in the circuit is in phase with the total voltage over the circuit. At what frequency does the resonance depend on the values of L and C, as we will see later. In the Figure 58-01 we see a typical RLC circuit. Let's take it as a reference for our study. It should be noted that in the value of R, in addition to its own value, the values of resistance of the source and the resistance of the inductor winding.

We have already seen the equation that determines the equivalent impedance of a series RLC circuit. Let's, again, reproduce it here.

eq. 58-01

In this way we can calculate the current of the circuit at any instant, at a certain frequency, using the equation below.

eq. 58-02

So if the voltage and current are in phase in the circuit, it is in resonance. And, for this to occur, it is necessary that the imaginary term of the impedance is equal to zero or, that we must satisfy X_L - X_C = 0. And from there, we conclude that in the resonance the equality below must be satisfied.

eq. 58-03

From this relation, we can easily calculate the resonance frequency of the circuit, simply by substitute the reactances of the above equality by their respective equations. Like this:

X_L = X_C ⇒ 2 π f L = 1/ (2 π f C)

By developing this relation and for a RLC series circuit, let us denote the resonance frequency by f_S, so that we can differentiate it from the frequency of resonance of a parallel circuit, represented by f_P. Assim, temos:

eq. 58-04

Notice that the resonant frequency in a RLC series circuit is independent of the value of the resistor. In addition, we can define two lateral frequencies at the resonant frequency. Let's call them f₁ and f₂. The frequency f₁ is known as LOWER cutoff frequency and the frequencyf₂ it is called UPPER cutting frequency. Also is valid the relation f₁ < f₂. So, f₁ is on the left of f_S and the other frequency, f₂, is on the right of f_S.

In the Figure 58-02 we can appreciate the frequency response of a resonant circuit. Notice what happens the maximum of the electric current in the circuit at the resonant frequency, f_S. The cut frequencies f₁ and f₂ are located on the curve where the signal has 3 dB unless its maximum intensity. In general, the frequency response at resonance is not as symmetrical as it appears in the figure. We maintained symmetry for didactic matters.

2.1 Bandwidth

We define bandwidth as being the difference between side frequencies f₁ and f₂. It is represented by Δf. Then:

eq. 58-05

Bandwidth can also be expressed by eq. 58-06 (below) based on the above definition. Simply replace the values of f₁ and f₂ by equations 57-16 and 57-17. Making an algebraic arrangement, we find:

eq. 58-06

On the other hand, there is a relation between the lateral frequencies and the resonant frequency, which is expressed as the geometric mean between the lateral frequencies. We can express it as:

eq. 58-07

2.2 Power in Resonance

As we have already said, the equivalent impedance of a series RLC circuit, at resonance, is equal to the resistance value R. This implies a minimum of impedance at the resonance. Consequently, in this condition, we have the maximum of electric current flowing through the circuit and this current is given by:

eq. 58-08

Therefore, we can write the power that the resistor dissipates in the resonance as:

P_o = (1/2) R I_max²

On the other hand, in the lateral frequencies of cut, the intensity of the electric current is root of two times less than the maximum intensity. Then, by calculating the power in these frequencies, we realize that they will be half the power at the resonant frequency. Hence these frequencies are also known as half-power frequencies. Therefore, we can write:

P₁ = P₂ = (1/2) P_o

2.3 Quality Factor

The quality factor, also known as merit factor, Q, of a resonant circuit, is defined as the ratio of the reactive power to the average power dissipated in the resistor at the resonance frequency. Like this:

Q = reactive power / average power

The lower the power dissipated for the same reactive power value, the greater the factor Q, indicating a higher concentration of energy in the resonance region. Substituting the mean and reactive powers (let us consider the inductive reactance) by its values and simplifying, we find for the calculation of Q the following expression:

eq. 58-09

We use the S index for Q, in order to differentiate the Q from a parallel circuit, which we will study later. Notice that in the above equation, X_L depends on the frequency at which the circuit works. We can compute the Q of the circuit only in function of the values of R, L and C, since in equation above, let us replace the frequency value in the inductive reactance by the equation 57-03. After some algebraic simplifications, we arrive at the expression below.

eq. 58-10

On the other hand, we can calculate the voltages on the inductor and the capacitor. For the inductor, using a voltage divider and remembering that at resonance, Z_eq = R, we can write the expression V_L = V ( X_L / R ). Using the same principle for the capacitor, we can write the two equations as follows:

eq. 58-11

eq. 58-12

It should be noted that in a RLC series circuit, at the resonant frequency, there is a possibility that the capacitor and inductor will develop high voltages, well above the voltage supplied by the circuit power supply. Let us assume that our example circuit is in resonance and we have V = 20∠0° and Q_S = 100. Therefore, substituting the values and performing the calculation will have V_L = V_C = 100 20 = 2,000 volts. This suggests that we should be extremely careful when we are working in practice with RLC series circuits in resonance.

2.4 Selectivity

Selectivity is understood as how much the circuit should be selective so that the desired frequencies are within the bandwidth. The smaller the bandwidth, the greater the selectivity. This feature is closely linked to the quality factor of the circuit. Thus, the greater the Q, the greater the selectivity. Note that for a RLC series circuit, the value of the resistor is greater, the higher the Q (for constant values of L and C). Similarly, for constant resistance values, the higher the L / C ratio, the smaller the bandwidth and the greater the selectivity.

Based on what was said above, for Q_s ≥ 10, we can find the bandwidth using the equation below:

eq. 58-13

And as a consequence, we can write that:

eq. 58-14

eq. 58-15

For any value of Q_S, we can determine the cut-off frequencies f₁ and f₂, simply knowing the parameters R, L e C. Below we see the two expressions that we should use.

eq. 58-16

eq. 58-17

And so, all these equations allow to calculate the parameters of the circuit.

3. Parallel Resonant Circuit

Let's divide our study into two topics: parallel resonant circuit Ideal and circuit resonant Real. The first one obeys the same characteristics studied in the series circuit. For the second case there are some additional considerations that will be studied in the item 3.2.

3.1 Ideal Parallel Resonant Circuit

The conditions for a parallel resonant circuit to be in resonance are the same as those required for a series resonant circuit, that is, obeys the eq. 58-03 repeated below:

eq. 58-03

We should point out some differences between the behavior of the two circuits when in RESONANCE. In the series resonant circuit there was a cancellation of the capacitor and inductor reactances and therefore the resulting impedance of these two components was NULL. Thus the equivalent impedance of the entire circuit was a pure resistive value represented by the resistance R.

In the Figure 58-03 we see an ideal parallel resonant circuit. In the parallel resonant circuit, the parallel association of the inductor with the capacitor at the resonant frequency generates an INFINITA impedance.

In this way, we can say that for a circuit series or parallel is in resonance the equivalent impedance of the same must be equal to a value pure resistive.And for the calculation of the resonance frequency, it is worth eq. 58-04, which we reproduce below.

eq. 58-04

3.2 Real Parallel Resonator Circuit

In the case of a real parallel resonant circuit we must take into account that the inductor has an electric resistance. This resistance is due to the resistance that the wire used in its confection presents. In the case of the series resonant circuit we consider that this resistance was included in the value of R. Thus, a typical circuit to study the real parallel resonant circuit is presented in the Figure 58-04.

In the case of the parallel resonant circuit, the resistor R_S can not be combined in series or parallel with the source resistance or any other resistance of the circuit. Although the resistance R_S has a very small value in relation to the other resistances of the circuit, it can present important influence in the resonance of the circuit.

Let's find a parallel circuit that is equivalent to the R - L branch in series, as shown in the figure above. For this branch, we can write the following equation:

Z_R-L = R_s + jX_L

From this impedance we can calculate the admittance of this branch. Like this:

Y_R-L = 1 / (R_s + jX_L)

To solve this equation simply multiply by the conjugate complex of the denominator. Developing and renaming, we come to:

Y_R-L = 1 / R_p + 1 / j X_Lp

The relationship between these two new variables and those known in the circuit is given by:

eq. 58-18

eq. 58-19

Thus, we can find an equivalence of a series R-L circuit for a parallel R-L circuit. If we take into account that the source has an internal resistance R_i, we can associate it with R_p, getting a new value that we will define as R. Thus:

R = R_i || R_p = R_i R_p / (R_i + R_p)

In this way, we can redesign the circuit as shown in the Figure 58-05.

At this point, it should be remembered that in the series resonant circuit the resonance frequency was that in which the impedance was minimal, the current was maximum, the input impedance was purely resistive and the circuit had a unit power factor. In the case of the parallel resonant circuit, such as the R_p in the equivalent circuit depends on the frequency, the value at which the maximum value of V_C is not necessarily the same for which the power factor is unitary. So we have two situations to consider. Let's study them separately.

3.2.1 Frequency for F.P. Unit

Considering the circuit of the figure above, we can write the total admittance of the circuit as:

Y_T = 1 / R + j (1 / X_C - 1 / X_Lp)

For the circuit to have a unit power factor, the reactive component must be NULL. So:

_Lp

X_C = X_Lp

Substituting the value of X_Lp, previously found, we have:

eq. 58-20

Remembering that:

X_C = 1 / (ω_P C) e X_Lp = ω_P L

Where, ω_P is the resonant frequency of the parallel resonant circuit. Developing the eq. 58-13, we find that the resonant frequency of the parallel resonant circuit is equal to the resonant frequency of the series resonant circuit multiplied by a factor K. This factor is given by:

eq. 58-21

Thus, we can express the resonant frequency of the parallel resonant circuit as:

eq. 58-22

Where, f_S is given by eq. 58-04, again shown below:

eq. 58-04

It should be noted that the resonant frequency f_P depends on the resistance R_S. Since the factor K is less than unity, of course that f_P is smaller than f_S. We can also conclude that as soon as R_S approximates to ZERO, f_P rapidly approaching f_S.

3.2.2 Frequency for Maximum Impedance

As R_P depends on the frequency, when f = f_P the input impedance of the resonant circuit is very close to its maximum, but did not reach it. For this case, maximum impedance, let us denote the frequency as f_m. Its value is slightly higher than f_P. The equation that determines the value of f_m is shown below.

eq. 58-23

3.3 Merit Factor, Q

The Merit or Quality factor in a parallel resonant circuit is also given by the ratio of reactive power to real. Like this:

Q_P = (V² / X_Lp ) / (V² / R )

In this equation V is the voltage in the branches in parallel and R is the value of the parallel of R_S with R_P. Rearranging the previous equation, Q_P is given by:

eq. 58-24

For the particular case of resistance R_P is much smaller than the resistance presented by the source, the above equation falls in the case of the series resonant circuit, given by eq. 58-13, reproduced below with minor modification. In general, the bandwidth is related to the resonant frequency, f_r, and the factor of merit, Q_P, of the circuit, given by:

eq. 58-25

As in the series resonant circuit, the cutoff frequencies f₁ and f₂ in the parallel resonant circuit can also be determined by the values of the circuit components. The equations that allow the calculation of f₁ and f₂ are shown below.

eq. 58-26

eq. 58-27

In the same way as was done for the series resonant circuit, using the definition of bandwidth and replacing f₁ and f₂ by equations 58-26 and 58-27, after some algebraic work we come to:

eq. 58-28

In this equation we notice how the bandwidth is closely related to the value of R. The greater the R value, the smaller the bandwidth and vice versa. In the graph shown in Figure 58-06, it is clear influence. Note, however, that the value of R does not interfere with the resonant frequency.

3.4 Dynamic Impedance

Considering the equivalence between a series and parallel circuit we can consider that there is the equality below:

Q = Q_S = Q_P

Using the equations eq. 58-09 and eq. 58-24 we arrive at the following relation:

eq. 58-29

So, doing some algebraic work on these equations, we determine the value of the dynamic impedance, Z_d, that a real parallel circuit presents, that is:

eq. 58-30

From this equation we conclude that the lower the value of the electrical resistance of the wire that is used in the manufacture of the inductor, the greater the dynamic impedance of the parallel circuit. For inductors with high Q, R_S is small, causing Z_d is very large, as is desirable to obtain high selectivity, especially in circuits used in telecommunications.

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