6.4 Inductors

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6.4 Inductors

An inductor is a passive element designed to store energy in its magnetic field. Inductors find numerous applications in electronic and power systems. They are used in power supplies, transformers, radios, TVs, radars, and electric motors.

Any conductor of electric current has inductive properties and may be regarded as an inductor. But in order to enhance the inducti ve effect, a practical inductor is usually formed into a c ylindrical coil with man y turns of conducting wire, as shown in Fig. 6.21.

An inductor consists of a coil of conducting wire.

If current is allowed to pass through an inductor, it is found that the voltage across the inductor is directly proportional to the time rate of change of the current. Using the passive sign convention,

$$v = L \frac{di}{dt}$$

(6.18)

where L is the constant of proportionality called the inductance of the inductor. The unit of inductance is the henry (H), named in honor of the American inventor Joseph Henry (1797–1878). It is clear from Eq. (6.18) that 1 henry equals 1 volt-second per ampere.

Inductance is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H).

The inductance of an inductor depends on its ph ysical dimension and construction. F ormulas for calculating the inductance of inductors of different shapes are deri ved from electromagnetic theory and can be found in standard electrical engineering handbooks. For example, for the inductor, (solenoid) shown in Fig. 6.21,

$$L = \frac{N^2 \mu A}{\ell} \tag{6.19}$$

where N is the number of turns, ℓ is the length, A is the cross-sectional area, and μ is the permeability of the core. We can see from Eq. (6.19) that inductance can be increased by increasing the number of turns of coil, using material with higher permeability as the core, increasing the cross-sectional area, or reducing the length of the coil.

Like capacitors, commercially a vailable inductors come in dif ferent values and types. Typical practical inductors ha ve inductance values ranging from a few microhenrys (μH), as in communication systems, to tens of henrys (H) as in po wer systems. Inductors may be fixed or variable. The core may be made of iron, steel, plastic, or air . The terms coil and choke are also used for inductors. Common inductors are sho wn in Fig. 6.22. The circuit symbols for inductors are shown in Fig. 6.23, following the passive sign convention.

Equation (6.18) is the v oltage-current relationship for an inductor . Figure 6.24 sho ws this relationship graphically for an inductor whose

Historical

Joseph Henry (1797–1878), an American physicist, discovered inductance and constructed an electric motor.

Born in Albany, New York, Henry graduated from Albany Academy and taught philosophy at Princeton University from 1832 to 1846. He was the first secretary of the Smithsonian Institution. He conducted several experiments on electromagnetism and developed powerful electromagnets that could lift objects weighing thousands of pounds. Interestingly, Joseph Henry discovered electromagnetic induction before Faraday but failed to publish his findings. The unit of inductance, the henry, was named after him.

inductance is independent of current. Such an inductor is kno wn as a linear inductor. For a nonlinear inductor, the plot of Eq. (6.18) will not be a straight line because its inductance v aries with current. We will assume linear inductors in this textbook unless stated otherwise.

The current-voltage relationship is obtained from Eq. (6.18) as

$$di = \frac{1}{L}v \, dt$$

Integrating gives

$$i = \frac{1}{L} \int_{-\infty}^{t} v(\tau) d\tau$$

(6.20)

$$(6.20)$$

or

$$i = \frac{1}{L} \int_{t_0}^t v(\tau) d\tau + i(t_0)$$

(6.21)

where i(t0) is the total current for −∞ < t < t0 and i(−∞) = 0. The idea of making i(−∞) = 0 is practical and reasonable, because there must be a time in the past when there was no current in the inductor.

The inductor is designed to store ener gy in its magnetic field. The energy stored can be obtained from Eq. (6.18). The power delivered to the inductor is

$$p = vi = \left(L\frac{di}{dt}\right)i\tag{6.22}$$

The energy stored is

$$w = \int_{-\infty}^{t} p(\tau) d\tau = L \int_{-\infty}^{t} \frac{di}{d\tau} i d\tau$$

= $L \int_{-\infty}^{t} i di = \frac{1}{2} L i^{2}(t) - \frac{1}{2} L i^{2}(-\infty)$ (6.23)

Figure 6.23

Circuit symbols for inductors: (a) air-core, (b) iron-core, (c) variable iron-core.

Figure 6.24 Voltage-current relationship of an inductor.

Since i(−∞) = 0,

$$w = \frac{1}{2}Li^2 \tag{6.24}$$

We should note the following important properties of an inductor.

1. Note from Eq. (6.18) that the v oltage across an inductor is zero when the current is constant. Thus,

An inductor acts like a short circuit to dc.

2. An important property of the inductor is its opposition to the change in current flowing through it.

The current through an inductor cannot change instantaneously.

According to Eq. (6.18), a discontinuous change in the current through an inductor requires an infinite voltage, which is not physically possible. Thus, an inductor opposes an abrupt change in the current through it. For example, the current through an inductor may take the form sho wn in Fig. 6.25(a), whereas the inductor current cannot take the form shown in Fig. 6.25(b) in real-life situations due to the discontinuities. Ho wever, the voltage across an inductor can change abruptly.

• 3. Like the ideal capacitor , the ideal inductor does not dissipate energy. The energy stored in it can be retrie ved at a later time. The inductor tak es power from the circuit when storing ener gy and delivers po wer to the circuit when returning pre viously stored energy.
• 4. A practical, nonideal inductor has a significant resistive component, as shown in Fig. 6.26. This is due to the fact that the inductor is made of a conducting material such as copper , which has some resistance. This resistance is called the winding resistance Rw, and it appears in series with the inductance of the inductor . The presence of Rw makes it both an ener gy storage device and an energy dissipation device. Since Rw is usually v ery small, it is ignored in most cases. The nonideal inductor also has a winding capacitance Cw due to the capacitive coupling between the conducting coils. Cw is very small and can be ignored in most cases, except at high frequencies. We will assume ideal inductors in this book.

Example 6.8 The current through a 0.1-H inductor is i(t) = 10te⁻5*^t* A. Find the voltage across the inductor and the energy stored in it.

Solution:

Since v = L di∕dt and L = 0.1 H,

$$v = 0.1 \frac{d}{dt} (10te^{-5t}) = e^{-5t} + t(-5)e^{-5t} = e^{-5t}(1 - 5t)$$

V

Figure 6.25

Current through an inductor: (a) allowed, (b) not allowable; an abrupt change is not possible.

Since an inductor is often made of a highly conducting wire, it has a very small resistance.

Figure 6.26 Circuit model for a practical inductor.

The energy stored is

$$w = \frac{1}{2}Li^2 = \frac{1}{2}(0.1)100t^2e^{-10t} = 5t^2e^{-10t} \text{ J}$$

If the current through a 1-mH inductor is i(t) = 90 sin (200t) mA, find the Practice Problem 6.8 terminal voltage and the energy stored.

Answer: 18 cos (200t) mV, 4.05 sin2 (200t) μJ.

Find the current through a 5-H inductor if the voltage across it is Example 6.9

$$v(t) = \begin{cases} 30t^2, & t > 0 \\ 0, & t < 0 \end{cases}$$

Also, find the energy stored at t = 5 s. Assume i(v) > 0.

Solution:

Since i = __1 L ∫ t0 t v(τ) dτ + i(t0) and L = 5 H, i = __1 5 ∫ 0 t 30τ ² dτ + 0 = 6 × t 3 __ 3 = 2t 3 A

The power p = vi = 60t 5 , and the energy stored is then

$$w = \int p \, dt = \int_0^5 60t^5 \, dt = 60 \frac{t^6}{6} \Big|_0^5 = 156.25 \, \text{kJ}$$

Alternatively, we can obtain the energy stored using Eq. (6.24), by writing

$$w\Big|_0^5 = \frac{1}{2}Li^2(5) - \frac{1}{2}Li(0) = \frac{1}{2}(5)(2 \times 5^3)^2 - 0 = 156.25 \text{ kJ}$$

as obtained before.

The terminal voltage of a 2-H inductor is ^{v =} 10(1 ^{− t}) V. Find the Practice Problem 6.9 current flowing through it at t = 4 s and the energy stored in it at t = 4 s. Assume i(0) = 2 A.

Figure 6.27 For Example 6.10.

Example 6.10 Consider the circuit in Fig. 6.27(a). Under dc conditions, find: (a) i, vC, and iL, (b) the energy stored in the capacitor and inductor.

Solution:

(a) Under dc conditions, we replace the capacitor with an open circuit and the inductor with a short circuit, as in Fig. 6.27(b). It is evident from Fig. 6.27(b) that

$$i = i_L = \frac{12}{1+5} = 2$$

A

The voltage vC is the same as the voltage across the 5-Ω resistor. Hence,

$$v_C = 5i = 10 \text{ V}$$

(b) The energy in the capacitor is

$$w_C = \frac{1}{2}Cv_C^2 = \frac{1}{2}(1)(10^2) = 50 \text{ J}$$

and that in the inductor is

$$w_L = \frac{1}{2}Li_L^2 = \frac{1}{2}(2)(2^2) = 4 \text{ J}$$

For Practice Prob. 6.10.

Figure 6.29 (a) A series connection of N inductors, (b) equivalent circuit for the series inductors.

Practice Problem 6.10 Determine vC, iL, and the energy stored in the capacitor and inductor in the circuit of Fig. 6.28 under dc conditions.

Answer: 15 V, 7.5 A, 450 J, 168.75 J.