Inductance and Self-Inductance: Understanding the Role of Inductors in Electric Circuits, Summaries of Circuit Theory

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Inductor and Inductance

Prepared by: Engr.A.C.Patricio Reference: University Physics

Inductance

• Self-inductance
  • A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. - Basis of the electrical circuit element called an inductor
  • Energy is stored in the magnetic field of an inductor.
  • There is an energy density associated with the magnetic field.
• Mutual induction
  • An emf is induced in a coil as a result of a changing magnetic flux produced by a second coil.
• Circuits may contain inductors as well as resistors and capacitors.

Some Terminology

• Use emf and current when they are caused by batteries or other sources.
• Use induced emf and induced current when they are caused by changing magnetic fields.
• When dealing with problems in electromagnetism, it is important to distinguish between the two situations.

Self-Inductance

When the switch is closed, the current does not immediately reach its maximum value. Faraday’s law of electromagnetic induction can be used to describe the effect. As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time. This increasing flux creates an induced emf in the circuit.

Self-Inductance, Equations

• An induced emf is always proportional to the time rate of change of the current.
  • The emf is proportional to the flux, which is proportional to the field and the field is proportional to the current.
• L is a constant of proportionality called the inductance of the coil.
  • It depends on the geometry of the coil and other physical characteristics. L d I ε L dt  

Inductance of a Coil

• A closely spaced coil of N turns carrying current I has an inductance of
• The inductance is a measure of the opposition to a change in current. N (^) B εL L I d I dt    

Inductance of a Solenoid

Assume a uniformly wound solenoid having N turns and length ℓ.  Assume ℓ is much greater than the radius of the solenoid. The flux through each turn of area A is The inductance is This shows that L depends on the geometry of the object. B o o N   BA  μ n I A  μ I A 2 B o^2 o N^ μ N A L μ n V I    

RL Circuit, Introduction

• A circuit element that has a large self-inductance is called an inductor.
• The circuit symbol is
• We assume the self-inductance of the rest of the circuit is negligible compared to the inductor.
  • However, even without a coil, a circuit will have some self-inductance.

RL Circuit, Analysis

• An RL circuit contains an inductor and a resistor.
• Assume S 2 is connected to a
• When switch S 1 is closed (at time t = 0), the current begins to increase.
• At the same time, a back emf is induced in the inductor that opposes the original increasing current.

• Applying Kirchhoff’s loop rule to the previous circuit in the clockwise direction gives
• Looking at the current, we find 0 d I ε I R L dt     1  ε Rt L I e R   

RL Circuit, Time Constant

• The expression for the current can also be expressed in terms of the time constant, t, of the circuit.
  • where t = L / R
• Physically, t is the time required for the current to reach 63.2% of its maximum value.

ε t τ

I e

R



RL Circuit, Current-Time Graph, Charging

• The equilibrium value of

the current is e / R and is

reached as t approaches infinity.

• The current initially increases very rapidly.
• The current then gradually approaches the equilibrium value.

RL Circuit Without A Battery

• Now set S 2 to position b
• The circuit now contains just the right hand loop.
• The battery has been eliminated.
• The expression for the current becomes t t τ τ i

I e I e

R

 

Energy in a Magnetic Field

• In a circuit with an inductor, the battery must supply more energy than in a circuit without an inductor.
• Part of the energy supplied by the battery appears as internal energy in the resistor.
• The remaining energy is stored in the magnetic field of the inductor.