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 dimension permability, length of inductor, and number of turns and inversely proportional to cross-sectional area.

${\displaystyle L={\frac {B}{I}}}$ = μN^2${\displaystyle {\frac {A}{l}}}$

Characteristics

Inductance

Inductance is the ability to generate a magnetic field for a given current.

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

Magnetic Field

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

B = I L

Voltage

${\displaystyle V_{L}=L{\frac {dI}{dt}}={\frac {dB}{dt}}}$

Current

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

Reactance

Reactance is defined as the ratio of Voltage over current

${\displaystyle X_{L}={\frac {L{\frac {dI}{dt}}}{I}}}$
${\displaystyle X_{L}=\omega L\angle 90^{\circ }}$
${\displaystyle X_{L}=j\omega L}$
${\displaystyle X_{L}=sL}$

Impedance

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

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

Frequency Respond

Inductor is a device depends on frequency ${\displaystyle \omega }$

• ${\displaystyle \omega =0,X_{L}=0}$, Inductor Closed circuit, I ≠ 0
• ${\displaystyle \omega =\infty ,X_{L}=\infty }$, Inductor Opened circuit, I = 0
• ${\displaystyle \omega ={\frac {R_{L}}{L}}}$
${\displaystyle X_{L}=R_{L}}$ ,
Z_L = [(RL]⅓[/itex] ,
${\displaystyle V_{L}={\frac {V}{2}}}$
${\displaystyle I_{L}={\frac {V}{2R_{L}}}}$

With the value of I at three frequency points ω = 0, ${\displaystyle \infty }$ , 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 = ${\displaystyle \omega {L}{R_{L}}}$ = 2π f${\displaystyle {\frac {L}{R_{L}}}}$

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

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

Induced Voltage

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

-ξ = ${\displaystyle N{\frac {dB}{dt}}={\frac {d\phi }{dt}}}$ 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

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

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

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

Độ Dẩn Từ của vật liệu

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

Construction Formula Dimensions
Cylyndrical Coil [1] ${\displaystyle L={\frac {\mu _{0}KN^{2}A}{l}}}$
• L = inductance in henries (H)
• μ0 = permeability of free space = 4${\displaystyle \pi }$ × 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 ${\displaystyle L=l\left(\ln {\frac {4l}{d}}-1\right)\cdot 200\times 10^{-9}}$
• L = inductance (H)
• l = length of conductor (m)
• d = diameter of conductor (m)
${\displaystyle L=5.08\cdot l\left(\ln {\frac {4l}{d}}-1\right)}$
• L = inductance (nH)
• l = length of conductor (in)
• d = diameter of conductor (in)
Short air-core cylindrical coil ${\displaystyle L={\frac {r^{2}N^{2}}{9r+10l}}}$
• L = inductance (µH)
• r = outer radius of coil (in)
• l = length of coil (in)
• N = number of turns
Multilayer air-core coil ${\displaystyle L={\frac {0.8r^{2}N^{2}}{6r+9l+10d}}}$
• 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 ${\displaystyle L={\frac {r^{2}N^{2}}{(2r+2.8d)\times 10^{5}}}}$
• L = inductance (H)
• r = mean radius of coil (m)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (m)
${\displaystyle L={\frac {r^{2}N^{2}}{8r+11d}}}$
• 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) ${\displaystyle L=\mu _{0}\mu _{r}{\frac {N^{2}r^{2}}{D}}}$
• L = inductance (H)
• μ0 = permeability of free space = 4${\displaystyle \pi }$ × 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

${\displaystyle {\frac {1}{L_{\mathrm {eq} }}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}}$

Series Connection

${\displaystyle L_{\mathrm {eq} }=L_{1}+L_{2}+\cdots +L_{n}\,\!}$

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.