How Does Frequency Affect Impedance in a Parallel RLC Circuit?

[NullSpace0]

A 1000 ohm resistor, 500 picofarad capacitor, and 2 millihenry inductor are all conducted in parallel. What is the impedance if the frequency is 10 kilocycles per second? 10 megacycles per sec? At what frequency is the absolute value of impedance the greatest?

I know I start by considering admittances since those add in parallel. I will end up with a complex number. Can I take its modulus and then the reciprocal? Or do I take the reciprocal to get impedance and then take the modulus? Or do I take the reciprocal to get impedance and leave it as the complex number?

 

[Simon Bridge]

You have a complex admittance.
The complex impedance is the reciprocal of the complex admittance
"Impedence" and "complex impedance" can be used interchangeably: you'll have to use the context of your classes to decide if the modulus is expected or not (that's how I'd interpret it.)

note; if $y=a+jb$
then, using $|z|=|1/y|$ (i.e. reciprocal then modulus):$$z=1/y = 1/(a+jb) = (a-jb)/(a^2-b^2)\Rightarrow |z| = \sqrt{a^2+b^2}/(a^2-b^2)$$... something like that - what happens if you do it in the other order (i.e. modulus then reciprocal). Is $|1/y|=1/|y|$)?

See:
http://en.wikipedia.org/wiki/Electrical_impedance

 

[NullSpace0]

Ahah! I was certainly wrong to think I could do the modulus and then take the reciprocal, that would give something like sqrt(a^2+b^2), and it would ignore the denominator that you got in the modulus of z itself.

So if I were to take the modulus, it should ALWAYS be after I've gotten all the way to the end of the problem in complex numbers?

 

[Simon Bridge]

It usually works out simpler that way anyhow.

 

[Nickie]

The impedance of a parallel circuit can be calculated by using the formula Z = 1/√(1/R^2 + (1/XL - 1/XC)^2), where R is the resistance, XL is the inductive reactance, and XC is the capacitive reactance.

At a frequency of 10 kilocycles per second, the inductive reactance can be calculated as XL = 2πfL = 2π(10,000)(0.002) = 40 ohms. Similarly, the capacitive reactance can be calculated as XC = 1/(2πfC) = 1/(2π(10,000)(500x10^-12)) = 318.31 ohms. Plugging these values into the formula, we get Z = 1/√(1/1000^2 + (1/40 - 1/318.31)^2) = 984.71 ohms.

At a frequency of 10 megacycles per second, the inductive reactance can be calculated as XL = 2πfL = 2π(10,000,000)(0.002) = 12,566.37 ohms. Similarly, the capacitive reactance can be calculated as XC = 1/(2πfC) = 1/(2π(10,000,000)(500x10^-12)) = 3.18 ohms. Plugging these values into the formula, we get Z = 1/√(1/1000^2 + (1/12,566.37 - 1/3.18)^2) = 1,000.00 ohms.

The absolute value of impedance will be greatest when the inductive reactance and capacitive reactance are equal, i.e. when XL = XC. This occurs at a frequency of 1/(2π√(LC)) = 1/(2π√(0.002x500x10^-12)) = 1,000,000 cycles per second. At this frequency, the impedance will be equal to the resistance, which in this case is 1000 ohms.

To answer your question about taking the modulus and reciprocal, the impedance is a complex number and its modulus represents the magnitude of the impedance, while its reciprocal represents the admittance of the circuit. Both can be