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Theory of Two Magnetically Combined RLC Circuits

CHAPTER 3

In many physical situations coupling can be created between several oscillatory systems. For instance, two pendulum clocks which can be mounted on a single wall structure will be combined by the flexing of the wall membrane as each swing. Similarly, gadgets frequently contain several tuned circuits that may be deliberately coupled by another circuit aspect, or even accidentally by stray areas. In all these circumstances, energy can be transferred when the consistency of one or both of the oscillators will be shifted. (Anon, 2011)

3. 1 Theory of two magnetically coupled RLC circuits

Two inductively combined RLC circuits are shown below (Number 1). Two resonant frequencies are obtained owing to the fact that there are two circuits. The parting of both frequencies is determined by the value of the shared inductance M, thought as the proportion of the

voltage in the secondary to the pace of change of primary current with time. It has a reactance at the operating frequency. (Arkadi, 2004)

 

Applying Kirchhoff's voltage laws equations for both primary and supplementary loops yield

 

 

(it is assumed here that )

These equations can be written in the matrix form as follows:

 

where, and

Following Cramer's guideline,

 

where

Thus, the perfect solution is that the occurrence response can be acquired is

 

Resonance occurs at both frequencies distributed by the next equations:

 

The habit of the circuit can qualitatively be recognized based on the mirrored impedance (or coupled impedance). An inductively combined circuit is said to "reflect" impedance in the extra into the main circuit. For a further explanation, the coupled circuits shown in Fig 2, is considered.

The positive course of the currents is chosen in to the polarity tag on the generator representing the induced voltages, so that Kirchhoff's equations are

 

is the shared impedance , includes the source impedance and the supplementary insert. These equations may be fixed for the same primary impedance

 

The mirrored impedance is then

A resistance is mirrored as a amount of resistance, whereas a capacitance is mirrored as an inductance , and an inductance mirrored as a capacitance .

At resonance condition, the mirrored impedance is resistive, and for that reason acts to lower the Q-factor of the principal, and thereby minimizing the output. That is however counteracted by a rise in coupling, which escalates the output. The lower Q-factor provides wider bandwidth. At lower frequencies than exact resonance, the shown impedance is said to be inductive, which plays a part in the inductance of the primary and therefore resonates at a lower frequency, producing a optimum in the productivity. At higher frequencies than exact resonance, the reflected impedance is said to be capacitive, which cancels part of the inductance and eventually causes the circuit to resonate at a higher frequency, producing the next peak. (Arkadi, 2004)

3. 2 Theory of couplings between two resonators

The procedure of resonators is nearly the same as that of the lumped-element resonators (series and RLC resonant circuits). Generally, two eigen frequencies can be obtained in colaboration with the coupling between two combined resonators, despite whether ther are synchronously or asynchronously tuned. The coupling coefficient , can therefore be extracted from both of these frequencies, which is often obtained using eqn () and eqn (). However, both of these frequencies can be easily and straight identified in tests without doing any calculations.

According to (Hong, 2004), the formulation for the computation of the coupling coefficient for synchronously tuned resonators will not yield the correct results when used to compute the coupling coefficient of asynchronously tuned resonators. Therefore it is of important importance to provide thorough treatment and derive a proper formulation to draw out the coupling coefficient for asynchronously tuned resonators.

In general, for different framework resonator (Body ), the coupling coefficient may have different self-resonant frequencies. It might be defined on the basis of a proportion of coupled energy to stored energy, that is,

 

 

Electric coupling magnetic coupling

where all fields are decided at resonance. The volume integrals are over whole parts with permittivity of † and permeability of Ој. However the direct analysis of from eqn. would require a complete knowledge of the field distributions and would need to perform space integral. This would certainly not be a fairly easy good article unless analytical alternatives of the domains exist.

However, Hong et al. (2004) found that there exists a relation between your coupling coefficient and resonant frequencies of the resonators which eases our activity in computing the coupling coefficient. The coupling is because of both electric and magnetic results. It is therefore essential to formulate expressions for each type of coupling independently.

3. 3 Formulation for coupling coefficients

3. 3. 1 Electric coupling

For electric coupling only, an comparable lumped-element circuit (Body ) is designed to represent the coupled resonators. Both resonators resonate at frequencies and . They may be coupled to the other person through mutual capacitance . For natural resonance that occurs, the condition is (as mentioned recently in 2. 2. 3). The resonant condition brings about an eigen equation

 

After some manipulations eqn () reduces to

 

This formula has four eigenvalues or alternatives. However, out of the four, only the two positive real alternatives are appealing to us. This is because they represent the resonant frequencies which are identifiable, namely

 

A new parameter is identified,

 

where the assumption is that .

Substituting and in eqn (),

 

Defining the electric coupling coefficient,

 

 

according to the proportion of the coupled electric energy to the average stored energy.

3. 3. 2 Magnetic coupling

A lumped-element circuit model like Figure is used showing the magnetic coupling through mutual inductance, of asynchronously tuned resonators. and are the two resonant frequencies of the uncoupled resonators. For natural resonance that occurs, the problem is, .

This leads to

 

After broadening,

 

Like in 3. 3. 1, this equation has four alternatives, of which only the two positive real ones are appealing to us,

 

We define a parameter,

 

Assuming , and recalling and , replacement in eqn ()

 

Defining the magnetic coupling coefficient as the proportion of the coupled magnetic energy to the common stored energy,

 

 

3. 3. 3 Combined coupling

There is an assortment of both electric and magnetic coupling regarding the experiments which will be performed in this task. Therefore to derive the coupling coefficient of the two resonators, we might have a circuit model as shown in Fig.

 

Fig.

The electric coupling is represented by an admittance inverter with while the magnetic coupling is represented by an impedance inverter with .

Based on the circuit model of Fig. , and supposing all interior currents circulation outward each node, a particular nodal admittance matrix can be explain with a guide at node '0'

 

with

 

 

 

 

 

 

For natural resonance, it suggests that

 

This requires that the determinant of admittance matrix to be zero, that is,

 

After some manipulations, we can occur at

 

This biquadratic formula is the eigen-equation for an asynchronously tuned coupled resonator circuit with the merged coupling. Allowing either or in eqn. reduces the equation to either coupling, which is exactly what can be expected. A couple of four alternatives of eqn. However, only the two positive ones are appealing, and they may be indicated as

 

with

 

 

 

Define

For narrow-band applications we can believe that and

the latter actually represents a ration of arithmetic mean to a geometric mean of two resonant frequencies. Thus we have in which

 

 

Now, it is clear that is nothing else however the blended coupling coefficient described as

 

 

The derived formula for extracting the coupling coefficients of any two asynchronously resonators can thus be produced as

 

This formula may also be used in computing the coupling coefficient of two synchronously tuned resonators, and in that case it reduces to

 

We will display the application of the produced formulation in this job through the building of two identical coupled spiral coil resonators and identify their respected resonant frequencies as well as determining the mixed coupling between them by using capacitors added to them.

 

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