## Complex Impedance in Alternating Current

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- Written by fisikastudycenter

physics.fisikastudycenter.com,- discussing alternating current circuit (AC circuit) using complex impedance to find the currents, angles on series or series-parallel circuit.

**Question 1**

Given an AC electric circuit contents a resistor R and an inductor L as shown below:

Where

R = 2 Ω

L = 2 mH

and the voltage source of ν = 120 sin 5000 t volts

Determine the equivalent impedance of this circuit, express the answer in rectangular form of complex impedance!

**Discussion**

Rectangular form of impedance :

Z = R + j ( X_{L} − X_{C} )

with

R = resistance (Ω)

X_{C} = capasitive reactance (Ω)

X_{L} = inductive reactance (Ω)

We have no capacitance in above circuit so X_{C} = 0

and the inductive reactance

X_{L} = ω L

X_{L} = (5000)(2x10^{−3})

X_{L} = 10 Ω

Z = R + j ( X_{L} − X_{C} )

Z = (8 + j 10) Ω

Take a note that expressing Z in complex impedance will be usefull in an advance circuit containing series and parallel as a combination, as shown in question 4, for the simple circuit as question 1, its easy enough to solve by using ordinary ways without complex impedance. Question 1, 2 and 3 will introduce us to the complex impedance forms.

**Question 2**

In below circuit R = 10 Ω, C = 5 μ F and ν = 200 sin 10000t volts

Express the circuit impedance in rectangular form of complex impedance!

**Discussion**

Z = R + j ( X_{L} − X_{C} )

X_{L} = 0

X_{C} = ^{1}/_{ω C}

X_{C} = ^{1}/_{(10000)(5x10−6)}

X_{C} = 20 Ω

Z = R + j ( X_{L} − X_{C} )

Z = 10 − j 20 Ω

**Question 3**

Express the impedance of below circuit in rectangular form of complex impedance

R = 8 Ω

L = 32 mH

C = 800 μF

ν = 120 sin 125 t volts

**Discussion**

Z = R + j ( X_{L} − X_{C} )

X_{L} = ΩL

X_{L} = (125)(32x10^{−6})

X_{L} = 4 Ω

X_{C} = ^{1}/_{ω C}

X_{C} = ^{1}/_{(125)(800 x 10−6)}

X_{C} = 10 Ω

Z = R + j ( X_{L} − X_{C} )

Z = 8 + j ( 4 − 10 )

Z = 8 − j 6 Ω

**Question 4**

Given an alternating current circuit contents R, L and C as the figure below

At frequency of ω = 400 rad/s and ν = 120 /0^{o} volts,

a) find the equivalent impedance of the circuit, express in complex

b) the total current of circuit

c) the current of resistance R = 5 Ω

d) the current of L = 25 mH

**Discussion**

a) equivalent impedance of this circuit:

R1 = 8Ω

R2 = 5Ω

X_{L} = ΩL

X_{L} = (400(25x10^{−6})

X_{L} = 10 Ω → + j 10

in the polar form we get:

b) Total current

c) Curent through L → I_{L} or I_{2} in the figure above

d) Curent through R = 5 Ω → I_{5} or I_{1} in the figure above

Prepared by fisikastudycenter.com

**Literatures **

*-Theory and Problems of Electric Circuit, Joseph A Edminister, McGraw-Hill**-Soal UN Fisika SMA 2008*