Practical Electronics/Inductors

Inductor

This device generates a magnetic field as current passes through it similar to the magnetic field of a magnet. An inductor stores electrical energy in the form of a magnetic field.

Inductor's Symbol

The symbol for inductance is L and is measured in Henry which has the symbol H.

Inductor's Construction

An inductor is a device made from a wire conductor with several turns that has the dimensiion permability, length of inductor, and number of turns and inversely proportional to cross-sectional area.

$L = \frac{B}{I}$ = μN^2$\frac{A}{l}$

Characteristics

Inductance

Inductance is the ability to generartes Magnetic Field B for a given Current

$L = \frac{B}{I}$

Magnetic Field

When a voltage is applied across the inductor, current generates Electric Field . Change of Electric Field in the turns generates Magnetic Field perpendicular to Electric Field

B = I L

Voltage

$V_L = L \frac{dI}{dt} = \frac{dB}{dt}$

Current

$I_L = \frac{1}{L} \int V dt$

Reactance

Reactance is defined as the ratio of Voltage over current

$X_L = \frac {L \frac{dI}{dt}}{I}$
$X_L = \omega L \ang 90^\circ$
$X_L = j \omega L$
$X_L = s L$

Impedance

Impedance is defined as the sum of Reactance and Resistance of Inductor . Since all conductor has Resistance

$Z_L = R_L + X_L$
$Z_L = \sqrt{R_L^2 + (\omega L)^2} \ang \arctan ( \omega \frac{L}{R} )$
$Z_L = R_L + j \omega L$
$Z_L = R_L + s L$

Frequency Respond

Inductor is a device depends on frequency $\omega$

• $\omega = 0 , X_L = 0$, Inductor Closed circuit, I ≠ 0
• $\omega = \infty , X_L = \infty$, Inductor Opened circuit, I = 0
• $\omega = \frac{R_L}{L}$
$X_L = R_L$ ,
Z_L = [(RL]⅓[/itex] ,
$V_L = \frac {V}{2}$
$I_L = \frac {V}{2R_L}$

With the value of I at three frequency points ω = 0, $\infty$ , 1 / CRC I - f curve can be drawn to give a picture of current in the inductor over time

Phase Angle

When a Voltage is applied across inductor , current generates magnetic field . Change in curent generate change in magnetic field which generate voltage across inductor . Therefore, current will lead voltage

For ideal losses inductor which has no internal resistance, Current will lead Voltage an angle 90 . For Non - Ideal inductor which has an internal resistance, Current will lead Voltage an angle θ

Tanθ = $\omega {L}{R_L}$ = 2π f$\frac{L}{R_L}$

Phase angle relates to time frequecy or time and the value of R and L . When there is a change in phase angle Time and frequency also change

$f = \frac{Tan\theta}{2\pi}\frac{R_L}{L}$
$t =\frac{2\pi}{Tan\theta}\frac{L}{R_L}$

Induced Voltage

Induced Voltage is defined as the voltage of the turns which oppose the current flow

-ξ = $N \frac{dB}{dt} = \frac{d\phi}{dt}$ where Φ = NB

Từ Dung

Từ Dung là tính chất Vật lý của Cuộn Từ đại diện cho Từ Lượng sinh ra bởi một Dòng Điện trên Cuộn Từ . Từ Dung đo bằng đơn vị Henry H và có ký hiệu mạch điện L

$L = \frac{B}{I}$

Cuộn Từ tạo từ một cộng dây dẩn điện có kích thứớc Chiều dài , l , Điện tích , A , với vài vòng quấn N . Khi mắc với điện

$L = \mu N^2 \frac{l}{A}$

Độ Dẩn Từ của vật liệu

$\mu = \frac{B}{I} \frac{A}{l} \frac{1}{N^2}$

Construction Formula Dimensions
Cylyndrical Coil [1] $L=\frac{\mu_0KN^2A}{l}$
• L = inductance in henries (H)
• μ0 = permeability of free space = 4$\pi$ × 10-7 H/m
• K = Nagaoka coefficient[1]
• N = number of turns
• A = area of cross-section of the coil in square metres (m2)
• l = length of coil in metres (m)
Straight wire conductor $L = l\left(\ln\frac{4l}{d}-1\right) \cdot 200 \times 10^{-9}$
• L = inductance (H)
• l = length of conductor (m)
• d = diameter of conductor (m)
$L = 5.08 \cdot l\left(\ln\frac{4l}{d}-1\right)$
• L = inductance (nH)
• l = length of conductor (in)
• d = diameter of conductor (in)
Short air-core cylindrical coil $L=\frac{r^2N^2}{9r+10l}$
• L = inductance (µH)
• r = outer radius of coil (in)
• l = length of coil (in)
• N = number of turns
Multilayer air-core coil $L = \frac{0.8r^2N^2}{6r+9l+10d}$
• L = inductance (µH)
• r = mean radius of coil (in)
• l = physical length of coil winding (in)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (in)
Flat spiral air-core coil $L=\frac{r^2N^2}{(2r+2.8d) \times 10^5}$
• L = inductance (H)
• r = mean radius of coil (m)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (m)
$L=\frac{r^2N^2}{8r+11d}$
• L = inductance (µH)
• r = mean radius of coil (in)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (in)
Toroidal core (circular cross-section) $L=\mu_0\mu_r\frac{N^2r^2}{D}$
• L = inductance (H)
• μ0 = permeability of free space = 4$\pi$ × 10-7 H/m
• μr = relative permeability of core material
• N = number of turns
• r = radius of coil winding (m)
• D = overall diameter of toroid (m)

Network

Inductors can be connected in series to increase inductance or in parallel to decrease inductance

Parallel Connection

$\frac{1}{L_\mathrm{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_n}$

Series Connection

$L_\mathrm{eq} = L_1 + L_2 + \cdots + L_n \,\!$

References

1. a b Nagaoka, Hantaro. The Inductance Coefficients of Solenoids[1]. 27. Journal of the College of Science, Imperial University, Tokyo, Japan. p. 18.