Question

Calculate the impedance of a 20 mH inductor at a frequency of 100 radians/s. Calculate the impedance of a 500 µF inductor at a frequency of 100 radians/s. What is the equivalent impedance of the inductor and capacitor connected in series? What is the equivalent impedance of the inductor and capacitor connected in parallel?

Answer 1

Given Information:  

Capacitance = C = 500 μF

Inductance = L = 20 mH

Frequency = ω = 100 rad/sec

Required Information:  

Capacitive reactance = Xc = ?

Inductive reactance = XL = ?

Equivalent series impedance = Zseries = ?

Equivalent parallel impedance = Zparallel = ?

Answer:

Capacitive reactance = Xc = -j20 Ω  

Inductive reactance = XL = j2 Ω

Equivalent series impedance = Zseries = -j18 Ω

Equivalent parallel impedance = Zparallel = j2.22 Ω

Explanation:

The capacitive reactance is calculated using

Xc = 1/jω*C

Where ω is the frequency in rad/sec and C is the capacitance of the capacitor

Xc = 1/j100*500x10⁻⁶

Xc = -j20 Ω

The inductive reactance is calculated using

XL = jω*L

Where ω is the frequency in rad/sec and L is the inductance of the inductor

XL = j100*0.02

XL = j2 Ω

The equivalent series impedance an can be calculated as

Zseries = XL + Xc

Zseries = j2 - j20 Ω

Zseries = -j18 Ω

The equivalent parallel impedance an can be calculated as

1/Zparallel = 1/XL + 1/Xc

Simplifying the above equation yields,

Zparallel = (XL*Xc)  / (XL+ Xc)

Zparallel = (j2*-j20)/(j2-j20)     ( j*j = -1)

Zparallel = 40/-j18     (1/-j = j)

Zparallel = j2.22 Ω

Answer 2

Answer:

a) Zinductor = j2 Ohm

b) Zcapacitor = -j20 Ohm

c) Zseries = -j18 Ohm

d) Zparallel = -j2.22 Ohm

Explanation:

The inductor impedance is directly proportional to the frequency, therefore:

a) frequency = 100 rad/s inductor = 20mH

Zinductor = j*frequency*inductor = j*100*20*10^(-3) = j2 Ohm

b) The capacitor inductance is inversely proportional to the frequency and it's also negative, therefore:

frequency = 100 rad/s capacitor = 500 uF

Zcapacitor = j*frequency*capacitor = -j/(100*500*10^(-6)) = -j/(0.05) = -j20 Ohm

c) The equivalent impedance of these two components in series is the sum of each part, wich goes as follow:

Zseries = Zinductor + Zcapacitor = j2 -j20 = -j18 Ohm

d) The equivalent impedance of these two components in parallel is given by the following equation:

Zparallel = (Zinductor*Zcapacitor)/(Zinductor+Zcapacitor)

Zparallel = [j2*(-j20)]/(j2-j20)

Zparallel = [-40]/(-j18) = 2.22/j = -j2.22 Ohm