How Do You Calculate Impedance for Series and Parallel RLC Circuits?

[physicsphy]

Homework Statement

I come across the impedance for an inductor an capacitor combined is Z = XL - XC, for a resistor and capacitor in series is Z = sqrt(R^2 + XC^2) and how do you obtain a formula for a combination of a capacitor, an inductor and a resistor in series and in parallel?

Homework Equations

I come across the impedance for an inductor an capacitor combined is Z = XL - XC, for a resistor and capacitor in series is Z = sqrt(R^2 + XC^2)

The Attempt at a Solution

in series, Z = R + wL + 1/(w*L)
in parallel Z = (R*wL)(1/wL)/(R+wL)+(1/wC)

 

[gneill]

Hi physicsphy, Welcome to Physics Forums.

Homework Statement

I come across the impedance for an inductor an capacitor combined is Z = XL - XC, for a resistor and capacitor in series is Z = sqrt(R^2 + XC^2) and how do you obtain a formula for a combination of a capacitor, an inductor and a resistor in series and in parallel?

Homework Equations

I come across the impedance for an inductor an capacitor combined is Z = XL - XC, for a resistor and capacitor in series is Z = sqrt(R^2 + XC^2)

The Attempt at a Solution

in series, Z = R + wL + 1/(w*L)
in parallel Z = (R*wL)(1/wL)/(R+wL)+(1/wC)

Strictly speaking, XL and XC are called reactances, and X = XL - XC is the total reactance for the series combination of inductor L and capacitor C.

Impedance is the 'big brother' of resistance, and and is a representation of resistance and reactance together in the form of complex numbers, thus Z = R + jX. Using impedance you can apply all the formulas you normally would for resistances, only one uses complex number math. For the basic components then,

$ZR = R$
$ZL = jωL$
$ZC = 1/(jωC) = -j/(ωC)$

As you can see, the reactances of L and C components are simply the values of the imaginary components without the "j" constants.

You can treat ZR, ZL, and ZC just as you would resistances when combining them. So impedances in series simply add: R + ZL + ZC, while impedances in parallel are combined using the reciprocal of the sum of reciprocals method:

$Z = \frac{1}{1/R + 1/ZL + 1/ZC}$

The magnitude of the impedance is found by summing the real and imaginary components in quadrature (square root of the sum of the squares, just like you would for vector components). You should be able to determine that for the series connection, the magnitude of the impedance turns out to be identical to your reactance method, namely

$|Z| = \sqrt{R^2 + (XL - XC)^2}$

For doing the parallel case it is often easier to deal with the reciprocals of the impedances (Y = 1/Z) for each component. These are called admittances, and are the complex version of conductance (G = 1/R). You can simply add parallel admittances as you did for series impedances. Pay attention to the signs assigned to the "reactance" version of admittances, since taking the reciprocal of a complex number reverses the sign of the complex component.

Once you have the net admittance, Y, you can find the impedance using Z = 1/Y.