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

$I_{L}={\frac {1}{L}}\int Vdt$

Reactance

Reactance is defined as the ratio of Voltage over current

X_{L}={\frac {L{\frac {dI}{dt}}}{I}}

$X_{L}=\omega L\angle 90^{\circ }$
$X_{L}=j\omega L$
$X_{L}=sL$

Impedance

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

$Z_{L}=R_{L}+X_{L}$
$Z_{L}={\sqrt {R_{L}^{2}+(\omega L)^{2}}}\angle \arctan(\omega {\frac {L}{R}})$
$Z_{L}=R_{L}+j\omega L$
$Z_{L}=R_{L}+sL$

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 = [(R_L]⅓</math> ,

V_{L}={\frac {V}{2}}

I_{L}={\frac {V}{2R_{L}}}

With the value of I at three frequency points ω = 0, $\infty$ , 1 / CR_C 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]	$L={\frac {\mu _{0}KN^{2}A}{l}}$	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)
Straight wire conductor	$L=5.08\cdot l\left(\ln {\frac {4l}{d}}-1\right)$	L = inductance (nH) l = length of conductor (in) d = diameter of conductor (in)
Short air-core cylindrical coil	$L={\frac {r^{2}N^{2}}{9r+10l}}$	L = inductance (µH) r = outer radius of coil (in) l = length of coil (in) N = number of turns
Multilayer air-core coil	$L={\frac {0.8r^{2}N^{2}}{6r+9l+10d}}$	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	$L={\frac {r^{2}N^{2}}{(2r+2.8d)\times 10^{5}}}$	L = inductance (H) r = mean radius of coil (m) N = number of turns d = depth of coil (outer radius minus inner radius) (m)
Flat spiral air-core coil	$L={\frac {r^{2}N^{2}}{8r+11d}}$	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)	$L=\mu _{0}\mu _{r}{\frac {N^{2}r^{2}}{D}}$	L = inductance (H) μ₀ = permeability of free space = 4 $\pi$ × 10^-7 H/m μ_r = 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 2 Nagaoka, Hantaro. The Inductance Coefficients of Solenoids. 27. Journal of the College of Science, Imperial University, Tokyo, Japan. p. 18.

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