Magnetic Fields, Forces, and Transformers Study Guide

Magnetic Fields and Concepts

• Magnetic Field Definition: A force field that exists around a magnet. It is also present around current-carrying conductors and current-carrying coils.

• Magnetic Poles: A bar magnet consists of two poles designated as North (N) and South (S).

• Field Concentration: Magnetic fields are at their strongest concentration at the poles.

• Field Directions: Magnetic field lines travel from the North pole and enter the South pole.

• Visualization: Iron filings can be used to show the magnetic field line patterns around a bar magnet or current-carrying conductor.

  • Around a bar magnet, iron filings reveal the lines of force connecting the poles.

  • Around a current-carrying conductor, iron filings form concentric rings.

Conductors and Magnetic Interaction

• Flux Direction in Conductors:

  • Flux lines move in a clockwise direction when current is flowing away from the viewer.

  • Flux lines move in a counter-clockwise direction when current is flowing towards the viewer.

• Forces Between Conductors:

  • Repulsion: Exists between conductors carrying current in opposite directions.

  • Attraction: Exists between conductors carrying current in the same direction.

• Solenoids and Coils: A solenoid (a coil of wire) establishes a magnetic flux similar to a bar magnet.

• Right-Hand Grip Rule: This rule is utilized to determine the magnetic poles of a solenoid or coil.

Characteristics of Magnets and Magnetic Materials

• Pole Existence: Every magnet must have a North-seeking pole and a South-seeking pole.

• Interaction Rules: Like poles repel one another; unlike poles attract one another.

• Temporary Magnetization: A piece of soft iron placed within a magnetic field becomes temporarily magnetized. For example, if a magnet's South pole is near a paper clip, it induces a South pole at the far end of that paper clip.

• Magnetic Permeability: Iron and steel are considered good magnetic materials because magnetic flux lines travel through them with ease.

Magnetic Flux and Flux Density

• Magnetic Flux ($\phi$): This describes or quantifies the number of lines of force in a magnetic field.

  • Unit: Weber [Wb].

• Flux Density ($B$): This measures the strength of the magnet. It is defined as the flux passing uniformly and normally (at right angles) through a surface area ($A$).

  • Unit: Tesla [T].

  • Formula: $B = \frac{\phi}{A}$

  • Relationship: $\phi = B \times A$ (units in Webers).

• Angular Flux: When flux and area are not at a right angle, the formula is:

  • $\phi = B \times A \cos(\theta)$

  • $\theta$ is the angle between the vector of $B$ and the vector of $A$.

• Example 1: A rectangular coil measuring $200\text{ mm}$ by $100\text{ mm}$ is at right angles to a uniform flux density of $0.05\text{ Tesla}$.

  • $\phi = BA = 0.05 \times 200 \times 100 \times 10^{-6} = 1\,mWb$

Magnetic Field Strength and Magnetomotive Force

• Magnetomotive Force (MMF): The force that produces the magnetic field around a current-carrying conductor.

  • Formula: $MMF = N \times I$

  • N: Number of turns of wire.

  • I: Current in amperes.

  • Units: Defined as the Ampere [A]. It is frequently referred to as the Ampere-turn [At] to prevent confusion with standard current.

• Magnetic Field Strength ($H$): The force experienced by an imaginary unit magnetic pole placed in a magnetic field.

  • Unit: Ampere per meter [A/m].

  • Formula: $H = \frac{NI}{l}$

  • l: The length of the material in meters ($m$).

• Example 2: A magnetic path length $l = 1\,m$ has a coil with $N = 20$ turns and a current $I = 3\,mA$.

  • $H = \frac{NI}{l} = \frac{20 \times 3 \times 10^{-3}}{1} = 0.06\,A/m$

Permeability

• Absolute Permeability ($\mu$): A measure of how easily magnetic flux lines can be established within a specific material.

  • Formula: $\mu = \frac{B}{H}$

  • Unit: Henry per meter [H/m].

• Permeability of Free Space ($\mu_0$): This applies to air, vacuums, or non-magnetic materials.

  • Value: $\mu_0 = 4\pi \times 10^{-7}\,H/m$

• Relative Permeability ($\mu_r$): The ratio of absolute permeability to the permeability of free space.

  • Formula: $\mu_r = \frac{\mu}{\mu_0}$

  • Note: $\mu_r$ has no units. $\mu = \mu_r \times \mu_0$.

• Enhancing Magnetic Fields: To increase the power of a magnetic field:

  • Increase the number of turns ($B \propto N$).

  • Increase current through the wire ($B \propto I$).

  • Shorten the length of the solenoid over which turns are concentrated ($B \propto \frac{1}{l}$).

  • Introduce a material with high permeability inside the solenoid.

  • Combined formula: $B = \mu \frac{NI}{l}$

Electromagnetic Induction and Faraday's Law

• Faraday's Discovery (1831): Michael Faraday found that moving a magnet through a coil induces a voltage across that coil. This induced voltage causes current to flow if there is a complete path.

• Core Requirement: An electromotive force (emf) is generated only when flux is in motion relative to the conductor or coil. Stationary settings generate no emf.

• Faraday’s Law Statement: The emf induced in an electric circuit is proportional to the rate of change of flux linking that circuit.

  • Formula: $e_L = -N \frac{d\phi}{dt}$ (units in Volts [V]).

  • N: Number of coil turns.

  • $\frac{d\phi}{dt}$: Rate of change of flux in [Wb/s].

• Example 3: A coil with $100$ turns is linked by a flux changing at a rate of $10\,mWb/s$.

  • $e_L = -1\,V$

Self-Inductance

• Inductance ($L$): The ability of a coil to oppose any change in current within itself.

  • Unit: Henry [H].

• Physical Inductance Formula: Inductance varies based on physical dimensions.

  • $L = \mu \frac{N^2 A}{l}$

  • A: Cross-sectional area of the coil [$m^2$].

  • l: Mean length of the core [$m$].

• Material properties:

  • Nonmagnetic materials: $\mu_r = 1$.

  • Ferromagnetic materials: $\mu_r \geq 100$.

  • Air core: $L = \mu_0 \frac{N^2 A}{l}$.

  • Ferromagnetic core: $L = \mu_r L_{\text{air}}$.

• Example 4: An air-cored solenoid with $900$ turns, $l = 15\,cm$, and diameter $d = 1.5\,cm$.

  • Inductance ($\mu_r = 1$) is $1.2\,mH$.

  • If iron core ($\mu_r = 1200$) is inserted, inductance becomes $1.44\,H$.

Lenz’s Law

• Definition: The emf induced in a circuit by a flux linkage will have a polarity that tends to set up a current that opposes the change of flux linkage.

Alternating Current (AC) and Direct Current (DC)

• Alternating Current Characteristics:

  • The voltage across two points changes polarity repetitively.

  • The direction of current flow changes direction repetitively.

• Direct Current Characteristics:

  • The value of the voltage remains constant.

  • The direction of current flow remains constant.

• Sinusoidal Waveforms: The most common AC waveform used in power utilities and electronic testing.

  • Sources: AC generators (alternators) for power supply; oscillators for laboratory signals.

  • Parameters:

    • Period ($T$): The time for one full cycle.

    • Peak Value ($V_P$): The maximum voltage level.

    • Peak-to-peak Value ($V_{PP}$): The total voltage between the positive and negative peaks.

Transformer Fundamentals

• Definition: A device that transfers electrical energy between AC circuits through mutual induction, allowing for voltage to be stepped up (increased) or stepped down (reduced).

• Mutual Induction: A changing current in one coil creates a changing magnetic field that induces a voltage in a nearby second coil.

• Components:

  • Primary Winding: Connected to the signal source or power supply.

  • Secondary Winding: Connected to the load.

  • Laminated Iron Core: Provides a common magnetic circuit with high mutual inductance.

• Characteristics:

  • Electrical Isolation: No physical electrical connection between the primary and secondary coils.

  • Magnetic Coupling: Flux from the first coil is linked to the second coil.

• Turns Ratio ($n$): The relationship between the number of turns in the primary ($N_P$) and secondary ($N_S$).

  • $n = \frac{N_S}{N_P}$

• Step-Up Transformer:

  • Secondary voltage ($E_S$) is greater than primary voltage ($E_P$).

  • Turns ratio ($n$) for step-up is less than one ($n < 1$) when defined as $E_P/E_S$.

  • Example 5: $200$-turn primary, $600$-turn secondary, $12\,V$ input results in $36\,V$ secondary.

• Step-Down Transformer:

  • Secondary voltage ($E_S$) is less than primary voltage ($E_P$).

  • Turns ratio ($n$) for step-down is greater than one ($n > 1$) when defined as $E_P/E_S$.

  • Example 6: Stepping down $230\,V$ to $24\,V$ requires a turns ratio of $9.56$.

Ideal Transformer Relationships

• Power Equality: An ideal transformer does not absorb energy; therefore, power in the primary equals power in the secondary ($Power_P = Power_S$).

  • $E_P I_P = E_S I_S$

• Relationship Summary:

  • $n = \frac{N_P}{N_S} = \frac{E_P}{E_S} = \frac{I_S}{I_P}$

• Example 7: A circuit with $V_g = 600\,V$, $E_S = 120\,V$, and load $R_L = 12\,k\Omega$.

  • Turns ratio: $0.2$ ($120/600$).

  • Secondary current: $50\,mA$.

  • Primary current: $0.25\,A$.

Summary of Essential Formulae

• Magnetic Flux Density: $B = \mu \frac{NI}{l}$ [T]

• Magnetomotive Force: $MMF = NI$ [A]

• Magnetic Field Strength: $H = \frac{NI}{l}$ [A/m]

• Permeability: $\mu = \frac{B}{H} = \mu_0 \mu_r$ [H/m]

• Magnetic Flux: $\phi = B \times A$ [Wb]

• Induced EMF (Faraday’s Law): $e_L = N \frac{d\phi}{dt}$ or $e_L = N \frac{\Delta \phi}{\Delta t}$ [V]

• (Self-) Inductance: $L = \mu \frac{N^2 A}{l}$ [H]