## What is a magnetically coupled circuit?

When two loops of a circuit with or without any physical contact affect each other with the help of magnetic fields, the circuits are said to be magnetically coupled. Magnetically coupled circuits are important parameters in electrical engineering. The most common example of a magnetically coupled circuit is a transformer. The transformer is a device used to transfer electric energy from one circuit to another. The primary winding and secondary winding are the main elements of a transformer. The primary winding creates varying magnetic flux and the secondary winding induces the electromotive force due to the effect of magnetic flux created by the primary winding.

## Coupling of circuits

The coupling of circuits is an important parameter in the field of electrical engineering. The circuits are said to be coupled when the mutual inductance exists between the coils of the circuits. In coupling, the transfer of electrical energy takes place from one circuit to another. The various types of coupling in electrical works are classified as - coupling due to electrical conduction and coupling due to electromagnetic induction. The coupling due to electrical conduction is further classified as - Direct coupling and Resistive coupling. The coupling due to electromagnetic induction is further classified as - Capacitive coupling and Magnetic coupling (Inductive coupling). In a direct coupling, the electrical energy is transferred from one circuit to another by providing physical contact between the circuits with the help of a conductive medium or conductor connected to it.

### Capacitive coupling

In capacitive coupling, the electrical energy is transferred from one circuit to another with the help of displacement current. The displacement current is induced by the electric field present in the circuit. For analog circuits, a coupling capacitor is used between two circuits. Using this method, only Alternating Current (AC) signals are transferred through the circuit and the Direct Current (DC) signals are blocked. The capacitive coupling is used in analog circuits where AC signals need to be obtained in the next circuits without the interference of the DC signals, such as in Class A amplifier. The capacitive coupling is also used in digital circuits where DC-balanced signals are needed.

### Magnetic coupling (Inductive coupling)

When the change in current in one wire creates a change in voltage across the ends of another wire with the help of electromagnetic induction, the circuits or loops are said to be magnetically coupled or inductively coupled. The magnetic coupling is used in electric motors, electric generators, inductive charging products, induction cookers, metal detectors, and so on.

## Magnetically coupled circuits

### Inductance

The tendency of an electric conductor to oppose the change in current flowing through a circuit is known as inductance. The production of an Electromotive Force (EMF) due to a change in the magnetic field is known as electromagnetic induction. The magnetic field in the circuit is produced around the conductor due to the flowing electric current. If multiple circuits are placed near one another, then the magnetic field lines from one circuit can pass to another. The inductances are of two types - self-inductances and mutual inductances. The inductance is proportional to the energy stored in the magnetic field of the given current.

### Self Inductance

In self-inductance, the voltage induced in the coil is related to the current flowing in the same coil. Consider a single inductor having $N$ number of turns, current $i$ flows through it and a magnetic flux $\varphi $ is produced around the inductor. According to Faraday's law, the following equation is given,

$v=N.\frac{d\varphi}{dt}$

where v is the voltage induced in the coil and $t$ is the time.

But, as per the definition, the change in flux is caused by the change in current. Hence,

$\frac{d\varphi}{dt}=\frac{d\varphi}{di}.\frac{di}{dt}$

Substituting this value in the above equation,

$v=N.\frac{d\varphi}{di}.\frac{di}{dt}$

where $N.\frac{d\varphi}{di}$ is known as self-inductance.

### Mutual inductance

In mutual inductance, the magnetic flux produced in one coil due to flowing current produces an induced voltage in the coil, the voltage is induced in one inductor due to another inductor. In mutual inductance, two cases are studied in general, when the current travels in the same direction in both the coils and when the current travels in the opposite direction in both the coils. Mathematically, the mutual inductance is defined as the ratio of EMF induced in one coil to the rate of change of current in another coil.

### Coupling coefficients

The coupling coefficient is used to measure the coupling between two coils. The coupling between two coils is mathematically given using the following equation,

$k=\frac{M}{\sqrt{{L}_{1}{L}_{2}}}$

where $k$ is the coupling coefficient, $M$ is the mutual inductance, ${L}_{1}$ is the self-inductance of the first coil, and ${L}_{2}$ is the self-inductance of the second coil. The values of k vary between 0 to 1. The values of the coupling coefficient and their representations are as follows -

- $k=0$ means that the coils are not coupled.
- $k=1$ means that the coils are perfectly coupled.
- $k0.5$ means that the coils are loosely coupled.
- $k0.5$ means that the coils are tightly coupled.

## Dot convention

Dot convention is used to determine the polarity of the mutually induced voltage using dotted terminals. For the determination of the direction of the current, the terminals of the two magnetically coupled coils are dotted. When the current enters from any of the terminals, the direction of the magnetic flux is noted. The polarity of the self-induced voltage of the circuit is identified from the direction of the current, whereas the polarity of the mutually induced voltage is noted from the dotted terminals based on the dot convention. For applying the dot convention, the currents in both circuits have to be meshed. Then the Kirchhoff's Voltage Law is applied along with the mutually induced voltage with proper polarity. At last, the equivalent circuits are generated for both circuits. Following is the procedure for creating the dotted markings-

- A terminal, say A, is selected from one coil and marked.
- Based on the right-hand rule, the direction of magnetic flux is determined.
- In the same way, another terminal, say B, is selected from another coil and the current is applied to the terminal. The direction of the magnetic flux is again determined and noted.
- If the direction of fluxes of both the terminals is the same, then the dot is placed at B and if the direction of fluxes is opposite, then the dot is placed at A.

If the current enters the dotted terminals of one coil, the polarity of the voltage in the second coil is considered as positive, whereas if the current leaves a dotted terminal of one coil, the voltage in the second coil is considered as negative.

## Context and Applications

The magnetically coupled circuits are useful for the students undergoing the following courses-

- Bachelors of Engineering or Technology (Electrical)
- Masters of Technology (Power System and Power Electronics)

## Practice Problems

**Q1.** Which of the following is a device used to transfer electric energy from one circuit to another?

a) Transformer

b) Inductor

c) Energy dissipator

d) Resistor

**Answer**- a

**Explanation**- Transformer is a device used to transfer electric energy from one circuit to another.

**Q2. **In which of the following couplings, the electrical energy is transferred from one circuit to another by providing physical contact between the circuits?

a) energy coupling

b) direct coupling

c) inductor coupling

d) capacitive coupling

**Answer**- b

**Explanation- **In direct coupling, the electrical energy is transferred from one circuit to another by providing physical contact between the circuits.

**Q3. **Which of the following is not an application of magnetic coupling?

a) Electric motor

b) Electric generator

c) Metal detectors

d) Electronic filters

**Answer**- d

**Explanation**- Electronic filters are not an application of magnetic coupling.

**Q4**. Which of the following is the tendency of an electric conductor to oppose the change in current flowing through a circuit?

a) Inductance

b) Energy dissipation

c) Magnetic field generation

d) Impedance

**Answer**- a

**Explanation**- Inductance is the tendency of an electric conductor to oppose the change in current flowing through a circuit.

**Q5**. Which of the following is used to determine the polarity of the mutually induced voltage?

a) Impedance

b) Compatibility equations

c) Parallel flux

d) Dot convention

**Answer**- d

**Explanation**- Dot convention is used to determine the polarity of the mutually induced voltage.

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