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

Download Inductance and Self-Inductance: Understanding the Role of Inductors in Electric Circuits and more Summaries Circuit Theory in PDF only on Docsity! 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. 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. The direction of the induced emf is such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field. The direction of the induced emf is opposite the direction of the emf of the battery. This results in a gradual increase in the current to its final equilibrium value. This effect is called self-inductance.  Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The emf εL is called a self-induced emf. 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 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 nI A μ I A    2 2oB o μ N AN 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. Effect of an Inductor in a Circuit •The inductance results in a back emf. •Therefore, the inductor in a circuit opposes changes in current in that circuit. • The inductor attempts to keep the current the same way it was before the change occurred. • The inductor can cause the circuit to be “sluggish” as it reacts to changes in the voltage. •The inductor affects the current exponentially. •The current does not instantly increase to its final equilibrium value. •If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected. 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.  1 t τε 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. 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. •Looking at this energy (in terms of rate) • Ie is the rate at which energy is being supplied by the battery. • I2R is the rate at which the energy is being delivered to the resistor. • Therefore, LI (dI/dt) must be the rate at which the energy is being stored in the magnetic field. 2 d I I ε I R LI dt   •Let U denote the energy stored in the inductor at any time. •The rate at which the energy is stored is •To find the total energy, integrate and dU d I LI dt dt  2 0 1 2 I U L I d I LI  Example: The Coaxial Cable Calculate L of a length ℓ for the cable •The total flux is •Therefore, L is 2 ln 2 b o B a o μ I B dA dr πr μ I b π a             ln 2 oB μ b L I π a          Mutual Inductance •The magnetic flux through the area enclosed by a circuit often varies with time because of time-varying currents in nearby circuits. •This process is known as mutual induction because it depends on the interaction of two circuits. •The current in coil 1 sets up a magnetic field. •Some of the magnetic field lines pass through coil 2. •Coil 1 has a current I1 and N1 turns. •Coil 2 has N2 turns. Induced emf in Mutual Inductance, cont. •In mutual induction, the emf induced in one coil is always proportional to the rate at which the current in the other coil is changing. •The mutual inductance in one coil is equal to the mutual inductance in the other coil. • M12 = M21 = M •The induced emf’s can be expressed as 2 1 1 2and d I d I ε M ε M dt dt     LC Circuits •A capacitor is connected to an inductor in an LC circuit. •Assume the capacitor is initially charged and then the switch is closed. •Assume no resistance and no energy losses to radiation. Oscillations in an LC Circuit •Under the previous conditions, the current in the circuit and the charge on the capacitor oscillate between maximum positive and negative values. •With zero resistance, no energy is transformed into internal energy. •Ideally, the oscillations in the circuit persist indefinitely. • The idealizations are no resistance and no radiation. •The capacitor is fully charged. • The energy U in the circuit is stored in the electric field of the capacitor. • The energy is equal to Q2 max / 2C. • The current in the circuit is zero. • No energy is stored in the inductor. •The switch is closed. LC Circuit Analogy to Spring-Mass System, 1 •The potential energy ½kx2 stored in the spring is analogous to the electric potential energy (Qmax) 2/(2C) stored in the capacitor. •All the energy is stored in the capacitor at t = 0. •This is analogous to the spring stretched to its amplitude. LC Circuit Analogy to Spring-Mass System, 2 •The kinetic energy (½ mv2) of the spring is analogous to the magnetic energy (½ L I2) stored in the inductor. •At t = ¼ T, all the energy is stored as magnetic energy in the inductor. •The maximum current occurs in the circuit. •This is analogous to the mass at equilibrium. LC Circuit Analogy to Spring-Mass System, 3 •At t = ½ T, the energy in the circuit is completely stored in the capacitor. •The polarity of the capacitor is reversed. •This is analogous to the spring stretched to -A. Time Functions of an LC Circuit •In an LC circuit, charge can be expressed as a function of time. • Q = Qmax cos (ωt + φ) • This is for an ideal LC circuit •The angular frequency, ω, of the circuit depends on the inductance and the capacitance. • It is the natural frequency of oscillation of the circuit. 1ω LC  Time Functions of an LC Circuit, cont. •The current can be expressed as a function of time: •The total energy can be expressed as a function of time: max dQ I ωQ sin(ωt φ) dt     2 2 2 21 2 2 max C L max Q U U U cos ωt LI sin ωt c     Charge and Current in an LC Circuit •The charge on the capacitor oscillates between Qmax and -Qmax. •The current in the inductor oscillates between Imax and -Imax. •Q and I are 90o out of phase with each other • So when Q is a maximum, I is zero, etc. The RLC Circuit •A circuit containing a resistor, an inductor and a capacitor is called an RLC Circuit. •Assume the resistor represents the total resistance of the circuit. RLC Circuit, Analysis •The total energy is not constant, since there is a transformation to internal energy in the resistor at the rate of dU/dt = -I2R. • Radiation losses are still ignored •The circuit’s operation can be expressed as 2 2 0 d Q dQ Q L R dt dt C    RLC Circuit Compared to Damped Oscillators •The RLC circuit is analogous to a damped harmonic oscillator. •When R = 0 • The circuit reduces to an LC circuit and is equivalent to no damping in a mechanical oscillator. •When R is small: • The RLC circuit is analogous to light damping in a mechanical oscillator. • Q = Qmax e-Rt/2L cos ωdt • ωd is the angular frequency of oscillation for the circuit and 1 2 2 1 2 d R ω LC L           