Theory of Two Magnetically Coupled RLC Circuits

Theory of Two Magnetically Coupled RLC CircuitsCHAPTER 3In many physical situations wedlock laughingstock be created surrounded by deuce or more oscillatory systems. For instance, dickens pendulum clocks which argon mounted on the same wall will be joined by the flexing of the wall as each swing. Similarly, electronic devices frequently agree several tuned tours that may be deliberately couple by another(prenominal) perimeter element, or even accidentally by stray fields. In all these situations, cipher quarter be transferred when the frequency of one or two of the oscillators will be shifted. (Anon, 2011)3.1 Theory of two magnetisedally pair RLC perimetersTwo inductively conjugated RLC circuits ar shown below (Figure 1). Two aromatic frequencies atomic number 18 obtained owing to the fact that in that respect are two circuits. The sepa symmetryn of the two frequencies depends on the value of the mutual elicitation M, fixd as the ratio of thevoltage in the minute-string coil to the rate of change of primary genuine with time. It has a reactance at the operating frequency. (Arkadi, 2004)Applying Kirchhoffs voltage law compares for both the primary and secondary loops yield(it is assumed here that )These equations can be written in the hyaloplasm form as followswhere, and Following Cramers rule,where Thus, the solution from which the frequency response can be obtained isResonance occurs at the two frequencies given by the following equationsThe behavior of the circuit can qualitatively be understood on the basis of the reflected electrical resistance (or twin resistor). An inductively join circuit is said to reflect impedance in the secondary into the primary circuit. For a further explanation, the coupled circuits shown in Fig 2, is considered.The validating direction of the currents is chosen into the polarity mark on the generator representing the generate voltages, so that Kirchhoffs equations are is the mutual impedance , includes the source impedance and the secondary load. These equations may be solved for the equivalent primary impedanceThe reflected impedance is then A resistance is reflected as a resistance, whereas a capacity is reflected as an inductance , and an inductance reflected as a capacitance .At resonance condition, the reflected impedance is resistive, and therefore acts to reject the Q-factor of the primary, and thereby reducing the output. This is just counteracted by an increase in brotherhood, which increases the output. The lower Q-factor gives a wider bandwidth. At lower frequencies than exact resonance, the reflected impedance is said to be inductive, which contributes to the inductance of the primary and hence resonates at a lower frequency, producing a peak in the output. At higher frequencies than exact resonance, the reflected impedance is said to be capacitive, which cancels part of the inductance and eventually ca single-valued functions the circuit to resonate at a higher frequency, producing the second peak. (Arkadi, 2004)3.2 Theory of couples amongst two resonatorsThe operation out of resonators is very similar to that of the lumped-element resonators (series and RLC resonating circuits). Generally, two eigen frequencies can be obtained in association with the coupling between two coupled resonators, despite whether ther are synchronously or asynchronously tuned. The coupling coefficient , can therefore be extracted from these two frequencies, which can be obtained victimization eqn () and eqn (). However, these two frequencies can also be easily and directly identify in experiments without doing any calculations.According to (Hong, 2004), the formula for the computation of the coupling coefficient for synchronously tuned resonators does not yield the appropriate results when used to compute the coupling coefficient of asynchronously tuned resonators. thus it is of fundamental importance to present comprehensive treatment and derive a proper face to extract the coupling coefficient for asynchronously tuned resonators.In general, for different coordinate resonator (Figure ), the coupling coefficient may have different self- reverberating frequencies. It may be defined on the basis of a ratio of coupled energy to stored energy, that is,Electric coupling charismatic couplingwhere all fields are determined at resonance. The volume integrals are over entire regions with permittivity of and permeability of . However the direct evaluation of from eqn. would require a complete companionship of the field distributions and would need to perform space integral. This would certainly not be an easy piece of work unless analytical solutions of the fields exist.However, Hong et al. (2004) found that there exists a relation between the coupling coefficient and resonant frequencies of the resonators which eases our task in computing the coupling coefficient.The coupling is due to both electric and magnetic effects. It is th erefore essential to formulate expressions for each type of coupling separately.3.3 verbalism for coupling coefficients3.3.1 Electric couplingFor electric coupling alone, an equivalent lumped-element circuit (Figure ) is designed to represent the coupled resonators. The two resonators resonate at frequencies and . They are coupled to each other through mutual capacitance . For native resonance to occur, the condition is (as mentioned previously in 2.2.3). The resonant condition leads to an eigen equation afterwards some manipulations eqn () reduces toThis equation has four-spot eigenvalues or solutions. However, out of the four, moreover the two positive real solutions are of interest to us. This is because they represent the resonant frequencies which are identifiable, namelyA new parameter is defined,where it is assumed that . alter and in eqn (),Defining the electric coupling coefficient,according to the ratio of the coupled electric energy to the average stored energy.3. 3.2 Magnetic couplingA lumped-element circuit model like Figure is used to show the magnetic coupling through mutual inductance, of asynchronously tuned resonators. and are the two resonant frequencies of the uncoupled resonators. For inbred resonance to occur, the condition is, .This leads toAfter expanding,Like in 3.3.1, this equation has four solutions, of which only(prenominal) the two positive real ones are of interest to us,We define a parameter,Assuming , and recalling and , substitute in eqn ()Defining the magnetic coupling coefficient as the ratio of the coupled magnetic energy to the average stored energy,3.3.3 blend couplingThere is a mixture of both electric and magnetic coupling in the case of the experiments that will be performed in this project. because to derive the coupling coefficient of the two resonators, we may have a circuit model as shown in Fig.Fig.The electric coupling is represented by an admittance inverter with while the magnetic coupling is re presented by an impedance inverter with .Based on the circuit model of Fig. , and assuming all midland currents flow outward each lymph node, a definite nodal admittance matrix can be define with a reference at node 0withFor natural resonance, it implies thatThis requires that the determinant of admittance matrix to be zero, that is,After some manipulations, we can arrive atThis biquadratic equation is the eigen-equation for an asynchronously tuned coupled resonator circuit with the motley coupling. Letting either or in eqn. reduces the equation to either coupling, which is what should be expected. There are four solutions of eqn. However, only the two positive ones are of interest, and they may be expressed aswithDefine For narrow-band applications we can assume that and the latter actually represents a ration of an arithmetic mean to a geometric mean of two resonant frequencies. Thus we have in whichNow, it is clear that is nothing else but the mixed coupling coefficient defined asThe derived formula for extracting the coupling coefficients of any two asynchronously resonators can thus be formulated asThis formula can also be used in computing the coupling coefficient of two synchronously tuned resonators, and in that case it reduces toWe will demonstrate the application of the derived formulation in this project through the construction of two identical coupled spiral coil resonators and identify their respective resonant frequencies as headspring as determining the mixed coupling between them through the use of capacitors added to them.