Resonance and Coupled Circuits

Contents

**Describe Resonance in Series-RLC circuits and its properties.** 1

**Describe the condition of Resonance in Series-RLC circuits (Series-Resonance)** 2

**Describe variation of R, XL, and XC with frequency** 2

**Describe the expression of Half Power frequencies in Series-RLC Resonating circuits** 3

**Describe Quality Factor ‘Q’ and Bandwidth of Series-RLC Resonating circuit** 4

**Describe the relationship among Quality Factor ‘Q’, Bandwidth and Resonating frequency** 5

**Explain Selectivity of Series-RLC Resonating circuit** 6

**Describe the condition of Resonance in Parallel-RLC Circuits (Parallel-Resonance)** 7

**Explain properties of Resonance of Parallel-RLC circuits** 8

**Describe Variation of R, Z, Capacitive, and Inductive Susceptance with frequency** 9

**Quality Factor ‘Q’ of Parallel Resonating circuit** 9

**Describe Bandwidth of Parallel-RLC Circuits and its relation with Quality Factor ‘Q’** 10

**Selectivity of Parallel-RLC Circuits** 11

**Describe Self-Inductance and Mutual-Inductance** 12

**Define Coefficient of Coupling** 12

**Describe Modelling of Coupled Circuits and Series Connection of Coupled Coils** 13

**Describe Transformer and Electrical Equivalent of Coupled circuits** 14

**Describe Single-Tuned Coupled Circuits** 15

**Describe Double-Tuned Coupled Circuits** 16

**Explain Conductively Coupled Equivalent Circuits** 16

**Describe Resonance in Series-RLC circuits and its properties.**

Resonance in a series RLC circuit occurs when the impedance of the circuit is purely resistive, which means that the reactance of the circuit is zero. This occurs when the frequency of the applied AC voltage is equal to the resonant frequency of the circuit.

In a series RLC circuit, the impedance is given by the equation:

Z = R + j(X_{l} – X_{c})

where R is the resistance of the circuit, Xl is the inductive reactance of the circuit, and Xc is the capacitive reactance of the circuit.

At resonance, the reactances cancel out, so the impedance of the circuit is purely resistive, and is given by:

Z = R

This means that the current in the circuit is maximised, and the power delivered to the circuit is also maximised. The resonance frequency is given by:

f = 1/(2π√(LC))

where L is the inductance of the circuit and C is the capacitance of the circuit.

**Describe the condition of Resonance in Series-RLC circuits (Series-Resonance)**

Some key properties of resonance in a series RLC circuit include:

- Maximum current: The current in the circuit is maximum at resonance, and is given by:I = V/R

where V is the applied voltage.

- Maximum power transfer: The power delivered to the circuit is maximum at resonance, and is given by: P = I^2R
- Bandwidth: The bandwidth of the circuit is defined as the range of frequencies for which the circuit has a given level of response. In a series RLC circuit, the bandwidth is defined as the difference between the two frequencies at which the current has dropped to 70.7% of its maximum value. The bandwidth is given by:

BW = Δf = f2 – f1

where, f1 and f2 are the frequencies at which the current has dropped to 70.7% of its maximum value.

In summary, resonance in a series RLC circuit occurs when the impedance of the circuit is purely resistive, which means that the reactance of the circuit is zero. This occurs at the resonant frequency, and has properties such as maximum current, maximum power transfer, and a defined bandwidth.

**Describe variation of R, XL, and XC with frequency**

In a series RLC circuit, the resistance R remains constant with frequency, as it is independent of the frequency of the applied voltage.

The inductive reactance X_{l}, on the other hand, increases with frequency, and is given by:

X_{l} = 2πfL

where f is the frequency of the applied voltage and L is the inductance of the circuit. As the frequency increases, the inductive reactance increases, which means that the impedance of the circuit also increases.

The capacitive reactance Xc, on the other hand, decreases with frequency, and is given by:

X_{c} = 1/(2πfC)

where f is the frequency of the applied voltage and C is the capacitance of the circuit. As the frequency increases, the capacitive reactance decreases, which means that the impedance of the circuit also decreases.

At low frequencies, the inductive reactance dominates, and the impedance of the circuit is mainly inductive. As the frequency increases, the capacitive reactance becomes more significant, and the impedance of the circuit decreases. At the resonant frequency, the inductive and capacitive reactances cancel out, and the impedance of the circuit is purely resistive.

In summary, in a series RLC circuit, the resistance R remains constant with frequency, while the inductive reactance X_{l} increases with frequency and the capacitive reactance Xc decreases with frequency. This variation of R, X_{l}, and X_{c} with frequency is key to understanding the properties of resonance in series RLC circuits.

**Describe the expression of Half Power frequencies in Series-RLC Resonating circuits**

In a series RLC resonant circuit, the half power frequencies are the frequencies at which the power transferred to the circuit is half of the maximum power transfer at resonance. These frequencies are also known as the bandwidth frequencies, as they define the range of frequencies over which the circuit will respond with at least half of the maximum power.

The half power frequencies can be determined by finding the frequencies at which the current in the circuit is 0.707 times the maximum current at resonance. This current is known as the half power current.

The half power frequencies can be calculated using the following equation:

f_{1,2} = f_{r} / (2πQ)

where, f1 and f2 are the lower and upper half power frequencies, respectively, f_{r} is the resonant frequency of the circuit, and Q is the quality factor of the circuit.

In summary, the half power frequencies in a series RLC resonant circuit are the frequencies at which the power transferred to the circuit is half of the maximum power transfer at resonance. They can be calculated using the resonant frequency and quality factor of the circuit, and are inversely proportional to the quality factor.

**Describe Quality Factor ‘Q’ and Bandwidth of Series-RLC Resonating circuit**

The quality factor, or Q factor, is a dimensionless parameter that describes the efficiency of a resonant circuit. In a series RLC resonant circuit, the Q factor is defined as the ratio of the energy stored in the circuit to the energy dissipated per cycle:

Q = (energy stored per cycle) / (energy dissipated per cycle)

The energy stored in the circuit is proportional to the square of the amplitude of the current, while the energy dissipated in the circuit is proportional to the resistance.

The Q factor of a series RLC circuit can also be expressed in terms of the resonant frequency and the bandwidth of the circuit:

Q = f_{r} / Δf

where f_{r} is the resonant frequency of the circuit, and Δf is the bandwidth of the circuit.

The bandwidth of a series RLC circuit is the range of frequencies over which the circuit will respond with a power transfer greater than or equal to half of the maximum power transfer at resonance. The bandwidth is measured in hertz (Hz), and is defined as:

Δf = f2 – f1

where f1 and f2 are the lower and upper half-power frequencies, respectively.

The Q factor and the bandwidth are related to the properties of the circuit components. A higher Q factor indicates a more efficient circuit, with a narrower bandwidth and a higher resonant frequency. The bandwidth is affected by the resistance of the circuit and the losses in the components, and a higher resistance or higher losses lead to a wider bandwidth. The Q factor and bandwidth are important parameters in the design of resonant circuits, as they determine the selectivity and the sensitivity of the circuit to different frequencies.

**Describe the relationship among Quality Factor ‘Q’, Bandwidth and Resonating frequency**

In a series RLC resonant circuit, the quality factor (Q), bandwidth (Δf), and resonant frequency (fr) are interrelated. The relationship can be expressed mathematically as:

Q = f_{r} / Δf

This means that the quality factor of a circuit is proportional to the resonant frequency and inversely proportional to the bandwidth. A higher Q factor indicates a more efficient circuit, with a narrower bandwidth and a higher resonant frequency. Conversely, a lower Q factor indicates a less efficient circuit, with a wider bandwidth and a lower resonant frequency.

The Q factor, bandwidth, and resonant frequency of a series RLC circuit are determined by the values of the circuit components. The inductance (L), capacitance (C), and resistance (R) of the circuit affect the resonant frequency, while the resistance and the losses in the components affect the bandwidth.

In general, a series RLC circuit with a high Q factor has a sharper resonance curve and is more selective in passing signals at the resonant frequency while rejecting signals at other frequencies. On the other hand, a series RLC circuit with a low Q factor has a flatter resonance curve and is less selective.

The relationship among Q, bandwidth, and resonant frequency is important in the design of filters, oscillators, and other circuits that rely on resonance. By choosing appropriate values of L, C, and R, the Q factor, bandwidth, and resonant frequency of a circuit can be tailored to meet specific design requirements.

**Explain Selectivity of Series-RLC Resonating circuit**

The selectivity of a series RLC resonant circuit refers to its ability to selectively pass or reject signals at certain frequencies. Selectivity is determined by the Q factor and the bandwidth of the circuit, which are related to the resistance, inductance, and capacitance of the circuit components.

At resonance, the circuit has maximum impedance, which means that it offers high resistance to the flow of current at the resonant frequency. This causes the circuit to selectively pass signals at the resonant frequency while rejecting signals at other frequencies. The sharper the resonance curve, the more selective the circuit is in passing signals at the resonant frequency and rejecting signals at other frequencies.

The selectivity of a series RLC resonant circuit can be increased by increasing the Q factor or decreasing the bandwidth. A higher Q factor results in a sharper resonance curve and a narrower bandwidth, which leads to higher selectivity. This is because the circuit can more effectively discriminate between signals at the resonant frequency and signals at other frequencies.

However, a higher Q factor also means that the circuit has higher losses and is more sensitive to changes in the values of the components, which can make it more difficult to design and tune. On the other hand, a lower Q factor results in a flatter resonance curve and a wider bandwidth, which leads to lower selectivity.

The selectivity of a series RLC resonant circuit is an important factor in the design of filters, oscillators, and other circuits that require frequency discrimination. By choosing appropriate values of the components, the selectivity of the circuit can be optimised to meet specific design requirements.

**Describe the condition of Resonance in Parallel-RLC Circuits (Parallel-Resonance)**

In a parallel RLC circuit, resonance occurs when the circuit impedance is at its minimum value. At resonance, the circuit offers very low impedance to the flow of current, which causes the circuit to draw a large amount of current from the source.

The condition for resonance in a parallel RLC circuit can be expressed in terms of the circuit’s admittance (Y), which is the reciprocal of the impedance (Z). The admittance of a parallel RLC circuit is given by:

Y = G + j(B – wC)

where G is the conductance of the circuit, B is the susceptance of the circuit, w is the angular frequency of the input signal, and C is the capacitance of the circuit.

At resonance, the imaginary part of the admittance is equal to zero, which means that the subscept is equal to wC. This occurs when the reactive components of the circuit cancel each other out, resulting in a purely resistive circuit.

The condition for resonance in a parallel RLC circuit can also be expressed in terms of the circuit’s natural frequency (ω_{0}), which is the frequency at which the circuit resonates without any external signal. The natural frequency of a parallel RLC circuit is given by:

ω_{0} = 1 / sqrt(LC)

At resonance, the frequency of the input signal is equal to the natural frequency of the circuit. This causes the circuit to resonate and draw a large amount of current from the source.

Overall, the condition for resonance in a parallel RLC circuit is when the circuit offers very low impedance to the flow of current, which occurs when the reactive components cancel each other out and the circuit is purely resistive.

**Explain properties of Resonance of Parallel-RLC circuits**

The resonance of a parallel RLC circuit has several important properties, including:

- High Impedance: At resonance, the impedance of the parallel RLC circuit is at its maximum value. This means that the circuit is effectively blocked to the flow of current, and very little current flows through the circuit.
- Resonant Frequency: The resonant frequency of a parallel RLC circuit is the frequency at which the circuit’s impedance is at its maximum value. This frequency is determined by the values of the circuit’s resistance, inductance, and capacitance.
- Maximum Current: At resonance, the parallel RLC circuit draws the maximum amount of current from the input source. This is because the circuit impedance is at its maximum value, which means that the current flowing through the circuit is at its minimum value.
- Bandwidth: The bandwidth of a parallel RLC circuit is the range of frequencies over which the circuit’s impedance is less than its maximum value. The bandwidth is determined by the circuit’s quality factor (Q), which is a measure of how “sharp” the circuit’s resonant peak is.
- Selectivity: The selectivity of a parallel RLC circuit is a measure of how well the circuit can discriminate between signals of different frequencies. A high-Q parallel RLC circuit has a narrow bandwidth and high selectivity, while a low-Q circuit has a wider bandwidth and lower selectivity.
- Phase Angle: At resonance, the phase angle between the voltage and current in a parallel RLC circuit is zero. This means that the voltage and current are in phase with each other, and the circuit behaves as a purely resistive circuit.

These properties of resonance in parallel RLC circuits are important in many applications, such as filters, tuning circuits, and impedance matching circuits.

**Describe Variation of R, Z, Capacitive, and Inductive Susceptance with frequency**

In a parallel RLC circuit, the impedance (Z), capacitive susceptance (Bc), and inductive susceptance (Bl) vary with frequency. The variation of these parameters is as follows:

- Resistance (R): The resistance in a parallel RLC circuit is constant and does not vary with frequency.
- Impedance (Z): The impedance of a parallel RLC circuit is a complex quantity that depends on the values of resistance, capacitance, and inductance. At resonance, the impedance of the circuit is at its maximum value, which is equal to the resistance of the circuit.
- Capacitive Susceptance (Bc): The capacitive susceptance of a parallel RLC circuit varies inversely with frequency. At low frequencies, the capacitive susceptance is large and dominates the behaviour of the circuit. As the frequency increases, the capacitive susceptance decreases, and the inductive susceptance begins to dominate the behaviour of the circuit.
- Inductive Susceptance (Bl): The inductive susceptance of a parallel RLC circuit varies directly with frequency. At low frequencies, the inductive susceptance is small and has little effect on the circuit. As the frequency increases, the inductive susceptance increases, and the capacitive susceptance begins to dominate the behaviour of the circuit.

The variation of R, Z, and susceptance with frequency is important in understanding the behaviour of a parallel RLC circuit. At resonance, the circuit is effectively blocked to the flow of current, and very little current flows through the circuit. At frequencies above and below resonance, the circuit behaves as a combination of a capacitor and an inductor, and the phase angle between the voltage and current changes. The frequency-dependent behaviour of a parallel RLC circuit is used in many applications, such as filters, tuning circuits, and impedance matching circuits.

**Quality Factor ‘Q’ of Parallel Resonating circuit**

In a parallel resonating circuit, the Quality Factor, denoted by ‘Q’, is a measure of the sharpness of the resonance. The Quality Factor is defined as the ratio of the energy stored in the circuit to the energy lost per cycle, and it is given by the following formula:

Q = (1 / R) x sqrt(L / C)

where R is the resistance of the circuit, L is the inductance, and C is the capacitance.

The Quality Factor can also be expressed in terms of the bandwidth of the circuit. The bandwidth is the range of frequencies over which the circuit will resonate. The relationship between the Quality Factor and the bandwidth is given by:

Q = f_{0} / BW

where f_{0} is the resonant frequency of the circuit and BW is the bandwidth.

A higher Quality Factor indicates a more selective or sharp resonance, meaning that the circuit has a narrower bandwidth and higher energy storage capacity. Conversely, a lower Quality Factor indicates a broader bandwidth and lower energy storage capacity.

The Quality Factor is an important parameter in the design and analysis of parallel resonating circuits, as it affects the selectivity, stability, and efficiency of the circuit. It can be adjusted by varying the resistance, inductance, or capacitance of the circuit.

**Describe Bandwidth of Parallel-RLC Circuits and its relation with Quality Factor ‘Q’**

In a parallel RLC (resistor-inductor-capacitor) circuit, the bandwidth is the range of frequencies over which the circuit will resonate. It is defined as the difference between the upper and lower frequencies at which the circuit’s power transfer is half of its maximum value.

The bandwidth of a parallel RLC circuit is related to the Quality Factor, Q, of the circuit. The relationship between Q and bandwidth can be expressed as:

BW = f_{0} / Q

where f_{0} is the resonant frequency of the circuit.

As the Quality Factor increases, the bandwidth of the circuit decreases. This means that the circuit becomes more selective, meaning that it can filter out signals outside its resonance frequency more effectively. Conversely, a lower Quality Factor results in a wider bandwidth, which means that the circuit is less selective and can allow a greater range of frequencies to pass through.

In practical applications, the bandwidth of a parallel RLC circuit is an important consideration, as it determines the range of frequencies that the circuit can pass or reject. The bandwidth can be adjusted by varying the values of the resistance, inductance, and capacitance in the circuit.

**Selectivity of Parallel-RLC Circuits**

The selectivity of a parallel RLC (Resistor-Inductor-Capacitor) circuit refers to its ability to pass or reject signals at different frequencies. It is a measure of how effectively the circuit can discriminate between signals at the resonant frequency and those at other frequencies.

The selectivity of a parallel RLC circuit is determined by its bandwidth and Quality Factor (Q). A higher Q value indicates a more selective circuit, as it means that the circuit has a narrower bandwidth and can filter out signals at frequencies that are further away from the resonant frequency.

The selectivity of a parallel RLC circuit can be improved by increasing its Q value, which can be achieved by reducing the resistance in the circuit and/or increasing the inductance and capacitance values. By doing so, the circuit can be tuned to a specific frequency, which results in a sharper resonance and higher selectivity.

On the other hand, a circuit with a lower Q value will have a wider bandwidth, which means that it is less selective and can allow a greater range of frequencies to pass through. In some applications, such as audio signal processing, a wider bandwidth may be desirable to allow for a broader range of frequencies to be passed through.

Overall, the selectivity of a parallel RLC circuit is an important consideration in many applications, such as radio and communication systems, where the circuit must be able to pass or reject signals at specific frequencies.

**Describe Self-Inductance and Mutual-Inductance**

Self-inductance and mutual inductance are two types of inductance that occur in electrical circuits.

Self-inductance is the property of a circuit element that causes a change in current flowing through it to induce an electromotive force (EMF) in the same circuit element. It occurs in a single coil or solenoid due to the magnetic field generated by the current in the coil itself. Self-inductance is measured in henries.

Mutual inductance, on the other hand, is the property of two coils that causes a change in current flowing through one coil to induce an EMF in the other coil. It occurs when the two coils are in close proximity and share a common magnetic field. Mutual inductance is also measured in henries.

Both self-inductance and mutual inductance are important in the design and operation of many electrical devices, including transformers, motors, and generators.

**Define Coefficient of Coupling**

The coefficient of coupling, denoted as k, is a measure of the extent to which two coils in a transformer are coupled magnetically. It is a dimensionless quantity that ranges between 0 and 1, where 0 indicates no magnetic coupling between the coils and 1 indicates perfect magnetic coupling.

The coefficient of coupling is defined as the ratio of the magnetic flux linking the secondary coil to the total magnetic flux produced by the primary coil. In other words, it is the fraction of the magnetic field produced by the primary coil that is linked with the secondary coil.

The value of the coefficient of coupling depends on the physical distance between the coils, the shape of the coils, and the orientation of the coils with respect to each other. A higher coefficient of coupling indicates better magnetic coupling between the coils, which results in a more efficient transfer of energy from the primary to the secondary coil in a transformer.

**Describe Modelling of Coupled Circuits and Series Connection of Coupled Coils**

In coupled circuits, the behaviour of one circuit is affected by the magnetic coupling with another circuit. The magnetic coupling can be modelled using ideal transformers, where each coil is represented by an inductor and the coupling between the coils is represented by the turns ratio and the coefficient of coupling.

In a series connection of coupled coils, the two coils are connected in series so that the same current flows through both coils. The voltage across each coil depends on the inductance of the coil and the current flowing through it. However, because of the magnetic coupling between the coils, the voltage across each coil also depends on the voltage induced in the other coil.

The behaviour of the series connection of coupled coils can be analysed using Kirchhoff’s laws and the voltage-current relationship of the ideal transformer. The voltage across the primary coil is equal to the sum of the voltage drop across the inductor and the voltage induced in the secondary coil, multiplied by the turns ratio and the coefficient of coupling. Similarly, the voltage across the secondary coil is equal to the sum of the voltage drop across the inductor and the voltage induced in the primary coil, multiplied by the turns ratio and the coefficient of coupling.

The coupled circuits can also be analysed using the concept of impedance transformation. The impedance of the primary and secondary coils can be transformed to the primary side or the secondary side using the turns ratio, and the effect of the coupling can be included in the transformed impedance using the coefficient of coupling.

The modelling of coupled circuits and the analysis of the series connection of coupled coils are important in the design of transformers, power supplies, and communication circuits.

**Describe Dot Convention**

Dot convention is a convention used in circuit analysis to indicate the polarity of the voltage across a mutual inductance. The convention involves placing a dot on one end of each coil in the transformer, and connecting the dots with a line. The polarity of the voltage across the two coils depends on the relative direction of the current in each coil with respect to the dots.

According to the dot convention, if the current in one coil enters the dot, then the voltage induced in the other coil has a positive polarity when the current in that coil also enters the dot. If the current in the other coil leaves the dot, then the voltage has a negative polarity.

The dot convention helps to ensure that the signs of the voltages and currents in the coupled circuits are correctly accounted for when writing the equations for circuit analysis.

**Describe Transformer and Electrical Equivalent of Coupled circuits**

A transformer is an electrical device that uses mutual inductance to transfer electrical energy between two or more circuits. A transformer consists of two or more coils of wire, called windings, which are wound around a common magnetic core. The windings are electrically insulated from each other, but are magnetically coupled together through the core.

The primary winding of a transformer is the winding that is connected to the input voltage source, while the secondary winding is the winding that is connected to the load. The voltage and current in the secondary winding are proportional to the voltage and current in the primary winding, according to the turns ratio of the transformer.

The electrical equivalent of coupled circuits consists of two or more circuits that are coupled together through mutual inductance. The circuits can be modelled using ideal transformers and a system of equations based on Kirchhoff’s laws and the dot convention.

In the electrical equivalent circuit, the mutual inductance between the circuits is represented by an ideal transformer, with the primary and secondary windings connected to the two circuits. The transformer turns ratio represents the ratio of the inductances of the two circuits. The voltages and currents in the two circuits can be related to each other using the transformer equations.

The electrical equivalent of coupled circuits is commonly used in the analysis of circuits with mutual inductance, such as transformers, coupled resonant circuits, and coupled transmission lines.

**Describe Single-Tuned Coupled Circuits**

Single-tuned coupled circuits are a type of resonant circuit that includes two inductors coupled through a shared magnetic field. The coupling occurs when the magnetic field from one inductor induces a voltage in the other inductor.

In a single-tuned circuit, one of the inductors is used to tune the circuit to the desired resonant frequency. The two inductors are then connected in parallel, with a capacitor placed across the parallel combination. This creates a resonant circuit that is tuned to the frequency at which the inductors are resonant.

The resonant frequency of the circuit depends on the values of the inductors, the capacitance, and the coupling coefficient between the inductors. The coupling coefficient is a measure of how closely the two inductors are coupled, and it affects the frequency response of the circuit.

Single-tuned circuits are commonly used in radio frequency (RF) applications, such as in radio and television receivers, where they are used to select a specific channel or frequency. They can also be used in power electronics for applications such as power factor correction or voltage regulation.

**Describe Double-Tuned Coupled Circuits**

Double-tuned coupled circuits are a type of resonant circuit that consists of two tuned circuits connected to each other through a coupling transformer. The resonant frequencies of both tuned circuits are different, and they are designed in such a way that they are close to each other.

In double-tuned coupled circuits, the coupling between the two circuits is provided by a transformer with a mutual inductance M. The first tuned circuit consists of an inductor L1 and a capacitor C1, while the second tuned circuit consists of an inductor L2 and a capacitor C2. The mutual inductance M represents the magnetic coupling between the two inductors L1 and L2.

The double-tuned circuit has two resonant frequencies, one at the frequency of the first tuned circuit and the other at the frequency of the second tuned circuit. When the circuit is excited at either of these resonant frequencies, a large voltage is developed across the load resistance RL.

Double-tuned coupled circuits are used in radio and television receivers as intermediate frequency (IF) amplifiers. The bandwidth of double-tuned coupled circuits is narrow compared to single-tuned circuits, which helps in rejecting unwanted signals.

**Explain Conductively Coupled Equivalent Circuits**

Conductively coupled equivalent circuits are used to analyse coupled circuits, in which the coupling between the circuits is achieved through a common conductor, such as a ground connection. In conductively coupled circuits, the coupling is achieved through a direct connection between the circuits, which allows for the transfer of energy and signals.

In conductively coupled equivalent circuits, the two coupled circuits are represented as individual circuits, with an added mutual conductance term. The mutual conductance term represents the coupling between the circuits and is denoted by Gm. The mutual conductance term is calculated based on the mutual conductance of the two circuits and the coupling coefficient.

The equivalent circuit of a conductively coupled circuit includes the resistance and reactance of each circuit, as well as the mutual conductance term Gm. The values of the circuit elements are determined through circuit analysis techniques, such as Kirchhoff’s laws and nodal analysis.

Conductively coupled equivalent circuits are commonly used in electronic circuits, such as amplifier and oscillator circuits. They allow for the analysis of the coupling between circuits and the transfer of energy between them, which is important in the design and optimization of electronic circuits.