# How to Solve Transformer Flux and Magnetic Circuits

Understanding how to solve transformer flux and magnetic circuits is of utmost importance for aspiring engineers preparing for the FE Electrical and PE Power exams, as per the guidelines set by the NCEES®.

The second part of our guide revolves around the mathematical problems and solutions related to series & parallel magnetic circuits, transformer flux, and equivalent circuits.

If you have not covered the basics of magnetic flux and reluctance, read the first part of this guide to learn the fundamentals of electric flux.

This study guide will help you discover valuable ways, with examples, to learn how to solve transformer flux and magnetic circuits. Let’s dive deep into the details.

## Flux and Reluctance in Series and Parallel Magnetic Circuits

Knowing the variation in flux and resistance for a series and parallel circuit is another vital aspect that helps further discover how to solve transformer flux and magnetic circuits.

### Series Magnetic Circuit

A series magnetic circuit is a type of magnetic circuit where magnetic components (such as magnetic cores or materials) are connected in series, and the same magnetic flux (ψ) passes through each part of the circuit sequentially.

In a series magnetic circuit, the individual magnetic components are arranged in a series configuration, meaning the magnetic flux continuously flows from one component to the next.

#### Flux in Series Magnetic Circuits

In a series magnetic circuit, the total magnetic flux (ψ) is the same in all the circuit parts. Since the components are connected in series, the magnetic flux passing through each part is equal. Therefore, the total magnetic flux (ψ_total) in the series magnetic circuit is the sum of the individual fluxes in each component.

Mathematically, for a series magnetic circuit with ‘n’ magnetic components, the total magnetic flux is given by:

ψ_total = ψ1 + ψ2 + ψ3 + … + ψn

Where:

- ψ_total is the total magnetic flux in the series magnetic circuit (measured in Weber, Wb).
- ψ1, ψ2, ψ3, …, ψn are the individual magnetic fluxes through each component (measured in Weber, Wb).

#### Magneto Reluctance in Series Magnetic Circuits

Reluctance (S) is the opposition a magnetic circuit offers to the flow of magnetic flux. It is analogous to resistance in electric circuits. Each magnetic component in a series magnetic circuit contributes its own reluctance to the total reluctance of the circuit.

The total reluctance (S_total) of the series magnetic circuit is the sum of the reluctances of all the components in the circuit.

Mathematically, for a series magnetic circuit with ‘n’ magnetic components, the total reluctance is given by:

S_total = S1 + S2 + S3 + … + Sn

Where:

- S_total is the total reluctance of the series magnetic circuit (measured in Ampere-turns per Weber, A-turn/Wb).
- S1, S2, S3, …, Sn are the reluctances of each component in the series magnetic circuit (measured in Ampere-turns per Weber, A-turn/Wb).

### Parallel Magnetic Circuit

A parallel magnetic circuit is a magnetic circuit where different magnetic components are arranged in parallel, and the same magnetic flux (ψ) divides among the parallel paths.

In a parallel magnetic circuit, each magnetic component has its path for magnetic flux, and the total magnetic flux is the sum of the fluxes in each parallel path.

#### Flux in Parallel Magnetic Circuits

In a parallel magnetic circuit, the total magnetic flux (ψ) is the sum of the magnetic fluxes in each parallel path. Each component has its own magnetic flux, and the total magnetic flux in the circuit is the sum of the individual fluxes in each parallel path.

Mathematically, for a parallel magnetic circuit with ‘n’ parallel paths, the total magnetic flux is given by:

ψ_total = ψ1 + ψ2 + ψ3 + … + ψn

Where:

- ψ_total is the total magnetic flux in the parallel magnetic circuit (measured in Weber, Wb).
- ψ1, ψ2, ψ3, …, ψn are the individual magnetic fluxes through each parallel path (measured in Weber, Wb).

#### Magneto Reluctance in Parallel Magnetic Circuits

Reluctance (S) is the opposition a magnetic circuit offers to the flow of magnetic flux. It is analogous to resistance in electric circuits. Each parallel path in a parallel magnetic circuit contributes its own reluctance to the total reluctance of the circuit.

The total reluctance (S_total) of the parallel magnetic circuit is determined by the total magnetomotive force (MMF) applied to the circuit and the total magnetic flux in the parallel paths.

Mathematically, for a parallel magnetic circuit with ‘n’ parallel paths, the total reluctance is given by:

S_total = ψ_total / (MMF_total)

Where:

- S_total is the total reluctance of the parallel magnetic circuit (measured in Ampere-turns per Weber, A-turn/Wb).
- ψ_total is the total magnetic flux in the parallel magnetic circuit (measured in Weber, Wb).
- MMF_total is the magnetomotive force applied to the parallel magnetic circuit (measured in Ampere-turns, A-turn).

## Transformer Flux and Magnetic Circuits

Transformers are static devices that transfer electrical energy between two sets of coils (windings) through a varying magnetic flux, provided both sets are on a common magnetic circuit (core).

The energy transfer process is based on electromagnetic induction, where a changing magnetic field induces a voltage in nearby coils.

Transformers are crucial in electrical power transmission and distribution systems, enabling voltage step-up or step-down as needed.

### Magnetic Circuits in Transformers

In transformers, the magnetic circuit refers to the path through which the magnetic flux flows. The transformer core is constructed using rectangular stampings of magnetic sheet steel clamped together, forming a closed loop for the magnetic flux. Copper or aluminum windings are positioned on the core, constituting the transformer’s primary, secondary, and tertiary windings.

### How to Solve Transformer Flux and Magnetic Circuits?

The magnetic circuit in a transformer is characterized by the concept of magnetic flux (ɸ), which represents the total magnetic field passing through a specific cross-sectional area. The flux is proportional to the product of the magnetic field intensity (H) and the cross-sectional area (A):

ɸ = H * A

The **magnetic field intensity (H)** is related to the magnetomotive force (MMF) required to establish the magnetic flux in the core. It is expressed as:

H = N * I

Where:

- N = Number of turns in the winding
- I = Current flowing through the winding

The **magnetic flux (ɸ)** is also related to the magnetic reluctance (R) of the magnetic circuit, which opposes the establishment of magnetic flux similar to electrical resistance. The magnetic reluctance is given by:

R = l / (μ * A)

Where:

- l = Length of the magnetic path
- μ = Permeability of the core material
- A = Cross-sectional area of the magnetic path

The permeability (μ) represents the ability of the core material to support the establishment of magnetic flux. For ferromagnetic materials, μ is significantly higher than air or nonmagnetic materials.

The sum of the fluxes in the individual windings determines the total magnetic flux (ɸ) in the transformer core. In an ideal transformer, the total flux is conserved, and the voltage and current on each side are inversely proportional to the turns ratio (N):

N = N1 / N2 = E1 / E2 = I2 / I1

Where:

- N1, N2 = Number of turns in the primary and secondary windings, respectively
- E1, E2 = Voltages in the primary and secondary windings, respectively
- I1, I2 = Currents in the primary and secondary windings, respectively

### Hysteresis Loop and Normal Magnetization Curve

Ferromagnetic materials used in transformer cores exhibit hysteresis, a lagging effect of magnetic flux density (B) behind the magnetic field intensity (H) during a sinusoidal magnetic field variation. The hysteresis loop is a graphical representation of this relationship.

During each magnetic cycle, energy is dissipated in heat due to the area enclosed by the hysteresis loop. This energy loss is known as hysteresis loss.

Let’s denote the B-H relationship as:

B = f(H)

**In general, the B-H relationship is nonlinear and can be represented as a curve. During a sinusoidal variation of magnetic field intensity, the magnetic flux density also varies sinusoidally but lags behind H.*

Mathematically,

The B-H relationship is typically expressed as a mathematical function, which can be represented by a curve or experimental data. For example, it may be given as:

B = f(H)

The normal magnetization curve represents the relationship between B and H for successive cycles of increasing magnetic field intensity. It shows that B reaches saturation at high values of H, indicating that further increases in H result in only marginal gains in B.

### Eddy-Current Loss and Exciting Current

Eddy-current loss occurs in transformer cores when the alternating magnetic field induces circulating currents (eddy currents) in the core material. These eddy currents experience resistance, leading to power dissipation in the form of heat.

The exciting current (Ie) is the small current that flows when an AC generator energizes the primary winding of an unloaded transformer. It produces an alternating mutual flux (ɸm) in the core. The exciting current comprises two components:

**Core-Loss Current (Ih+e) –**Represents the real-power component of the exciting current, arising from hysteresis and eddy-current losses.**Magnetizing Current (Im) –**Provides the MMF required to overcome the magnetic reluctance of the core and establish the magnetic flux.

Mathematically,

The eddy-current loss (Pe) in the transformer core is given by:

Pe = K * f^2 * B^2 * t^2 * V

Where:

- K = Eddy-current loss constant
- f = Frequency of the alternating magnetic field
- B = Magnetic flux density
- t = Thickness of the core laminations
- V = Volume of the core

The exciting current (Ie) can be calculated as the phasor sum of the core-loss current (Ih+e) and the magnetizing current (Im). The phasor diagram shows that Im is in phase with ɸm, while Ih+e is in phase with the voltage and leads the magnetizing current by 90°.

### Leakage Fluxes and Equivalent Circuits

Leakage fluxes in transformers refer to the magnetic flux that does not link with all the windings on the core. These leakage fluxes primarily flow through the air, as the core material has a much higher permeability than air. Leakage fluxes lead to pulsations observed around an energized transformer.

Equivalent circuits are used to represent a transformer’s behavior and performance. They consist of inductances symbolizing the leakage and magnetizing fluxes and resistors representing winding resistance and core loss. These equivalent circuits aid in understanding the transformer’s characteristics under different operating conditions.

Mathematically,

In the equivalent circuit, the transformer can be represented by various components such as inductances (L1, L2, Lm), resistances (R1, R2), and magnetizing reactance (Xm).

The leakage inductances (L1 and L2) account for the leakage fluxes, while the magnetizing inductance (Lm) represents the magnetizing flux. The winding resistances (R1 and R2) account for the ohmic losses in the transformer windings.

Under no-load conditions, the voltage V1 and the induced voltage E1 differ due to the leakage impedance R1 + jX1. When the transformer delivers a load current I2, the secondary voltage V2 is given by V2 = E2 – I2 * (R2 + jX2), where E2 is the secondary induced voltage.

### Magnetic Flux Problem 1 – Transformer Flux Calculation

A single-phase transformer has a primary winding of 500 turns and a secondary winding of 200 turns. The maximum magnetic flux density in the core is 1.2 Tesla. Calculate the maximum flux in the core when the transformer is connected to a 220V, 50Hz AC supply.

We have,

Number of turns in primary winding (N1) = 500 turns,

Number of turns in the secondary winding (N2) = 200 turns,

Maximum magnetic flux density in the core (Bmax) = 1.2 Tesla.

Supply voltage (V1) = 220V

Frequency (f) = 50Hz

**The maximum flux in the core can be calculated using the formula:**

Flux (Φ) = Bmax * A,

Where A is the cross-sectional area of the core. Assuming a rectangular core with dimensions a and b, the cross-sectional area A = a * b.

To calculate the cross-sectional area A, we can use the voltage and frequency to find the magnetic field intensity (H) in the core and then use the B-H curve for the given core material to determine the magnetic field intensity corresponding to the given magnetic flux density (Bmax).

Let’s assume the magnetic field intensity corresponding to Bmax is Hmax.

Next, we can use the following formula to find the cross-sectional area A:

A = (V1 * N2) / (4 * f * Hmax)

**Substitute the given values:**

A = (220 * 200) / (4 * 50 * Hmax)

Now, using the calculated value of A, we can find the maximum flux Φ:

Φ = Bmax * A

The maximum flux in the core is Φ = Bmax * A, where Bmax is 1.2 Tesla and A is the cross-sectional area of the core calculated using the formula A = (220 * 200) / (4 * 50 * Hmax).

### Magnetic Flux Problem 2 – Reluctance Calculation in Transformers

Consider a single-phase transformer with the following parameters:

Primary winding: N1 = 800 turns, μr1 = 1000, length l1 = 0.2 meters

Secondary winding: N2 = 200 turns, μr2 = 1200, length l2 = 0.1 meters

Let’s calculate the reluctance of each winding and the total reluctance of the transformer.

**The formula gives the reluctance (R) of a given magnetic circuit example:**

R = (l * μ0 * μr) / (A)

Where:

- l = length of the magnetic path
- μ0 = permeability of free space (4π × 10^-7 H/m)
- μr = relative permeability of the material
- A = cross-sectional area of the magnetic path

**Calculate the reluctance of the primary winding (R1):**

R1 = (l1 * μ0 * μr1) / (A1)

**Calculate the reluctance of the secondary winding (R2):**

R2 = (l2 * μ0 * μr2) / (A2)

**Calculate the total reluctance of the transformer (RTotal):**

RTotal = R1 + R2

To calculate the cross-sectional areas A1 and A2, we can assume a rectangular cross-section for each winding.

Let’s assume the width and height of the cross-section for the primary winding are a1 and b1, respectively, and for the secondary winding, they are a2 and b2, respectively.

A1 = a1 * b1

A2 = a2 * b2

The reluctance of the primary winding (R1) is calculated using the formula R1 = (l1 * μ0 * μr1) / (A1). Similarly, the magneto reluctance of the secondary winding (R2) is calculated using R2 = (l2 * μ0 * μr2) / (A2).

The total magneto reluctance of the transformer (RTotal) is obtained by adding the reluctances of the primary and secondary windings, i.e., RTotal = R1 + R2.

### Conclusion

Now you have a rich idea about how to solve transformer flux and magnetic circuits. Understanding transformer flux calculations and magnetic equivalent circuits are the critical techniques that equip power and electrical engineers to analyze transformers’ behavior, predict their performance, and ensure optimal energy transfer between circuits.

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