# Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits

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1 Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits

2 Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the timevarying 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. Circuits may contain inductors as well as resistors and capacitors.

3 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. ε L di L dt L is a constant of proportionality called the inductance of the coil. It depends on the geometry of the coil and other physical characteristics. A closely spaced coil of N turns carrying current I has an inductance of L N I B εl d I dt The SI unit of inductance is the henry (H) V s 1H 1 A I 1 Let U denote the energy stored in the inductor at any time. U L I d I LI 0 2 2

4 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.

5 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 circuit in the clockwise direction gives di ε I R L dt 0 Looking at the current, we find I ε R 1 e Rt L (L / R)= t I(t) = I 0 (1 e t τ) Time Constant

6 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. I(t) = I 0 (1 e t τ) Physically, t is the time required for the current to reach 63.2% of its maximum value.

7 RL Circuit, Current-Time Graph, Discharging The time rate of change of the current is a maximum at t = 0. It falls off exponentially as t approaches infinity. In general, di dt ε e L t τ I(t) = I 0 (e t τ)

8 RC Circuits 8

9 Charging a Capacitor in an RC Circuit The charge on the capacitor varies with time. q(t) = Ce (1 e -t/rc ) = Q(1 e -t/rc ) The current can be found I t = Q RC e t τ t is the time constant t = RC The energy stored in the charged capacitor is U= ½ QV = ½ CV 2 9

10 Discharging a Capacitor in an RC Circuit When a charged capacitor is placed in the circuit, it can be discharged. q(t) = Qe -t/rc The charge decreases exponentially. I t = ε R e t τ 10

11 RL and RC Circuits, Introduction

12 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. The capacitor becomes fully charged and the cycle repeats. The energy continues to oscillate between the inductor and the capacitor. The total energy stored in the LC circuit remains constant in time and equals. 2 Q 1 U U 2 2 I C UL L C 2

13 LC Circuit Analogy to Spring-Mass System

14 LC Circuit Analogy to Spring-Mass System

15 Time Functions of an LC Circuit In an LC circuit, charge can be expressed as a function of time. Q = Q max 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. ω 1 LC The current can be expressed as a function of time: dq I ωqmax sin(ωt φ) dt The total energy can be expressed as a function of time: 2 Qmax U UC UL cos ωt LImax sin ωt 2c 2

16 Notes About Real LC Circuits!! In actual circuits, there is always some resistance. Therefore, there is some energy transformed to internal energy. Radiation is also inevitable in this type of circuit. The total energy in the circuit continuously decreases as a result of these processes.

17 The RLC Circuit A circuit containing a resistor, an inductor and a capacitor is called an RLC Circuit. Depending on the combination of R,L, and C many different type RLC circuits can be established. The basic RLC circuits are series RLC and Parallel RLC circuits. The circuit analysis of these type is way above the scope of this course. Either current or voltage in these circuits time dependent and have oscillatory behavior mainly three different type: Overdamped, Underdamped, and Critically damped depending of the values of R, L, and C values in the circuit.

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