# Impedance of an LCR circuit

So I am trying to understand how to find the impedance of the parallel LC component of this filter circuit. I am given that the transfer function $$\mathbf{T}_C = \dfrac{\mathbf{V}_{out}}{\mathbf{V}_{in}} = \dfrac{R_2}{R_2+z_p}$$ where $$z_p = R_p||j\omega L||\dfrac{1}{j\omega C} = \dfrac{R_p}{1+jR_p(\omega C-\dfrac{1}{\omega L})}$$

$$R_p$$ is supposed to represent the resistance from the inductor and capacitor in parallel.

I thought that to calculate impedance over parallel components I had to take the product of the impedances of each 'side' (here meaning just the product of the inductor's and capacitor's impedances) divided by the sum of the impedance of each 'side'.

I am confused as to what $$R_p$$ really means and why we are taking the norm of the inductors impedance. Basically, I am struggling to find out how each of the equations for the total impedance for the parallel compenents are found.

• You not calculate the norm of the induction. The || means that all components are parallel to each other. May 20, 2019 at 18:57
• Why is $R_P$ parallel to the inductor and capacitor when it is supposed to represent the resistance from both of them? (It is not an additional component in the circuit) May 20, 2019 at 19:00
• I don't know your starting position, but just judging from your circuit, the resistance, the inductor and the capacitor are all parallel. Keep in mind that this electrical networks are just representations of the real world. It does not necessarily mean that you have a resistor or inductor, but it behaves like there are three components parallel to each other. May 20, 2019 at 19:04
• @user208480 There are various models of non-ideal inductors and capacitors. The ones I have seen for inductors always contain a resistor in series with the inductor. The model you show does not, but I'm not sure how it was derived. Because there also different models of capacitors. In any event, if all you need to do is find the impedance of the parallel inductor and capacitor, why do you care about $R_p$? May 20, 2019 at 19:17
• I feel like I still have a very bad understanding of resistance/impedance when it comes to inductors/capacitors. Is it true that it is possible to have both a resistance and an impedance from inductors or capacitors? I understand that the distinction between resistance and reactance is that resistance is a DC effect but does that mean we can't have both for an AC voltage? May 20, 2019 at 19:30

I thought that to calculate impedance over parallel components I had to take the product of the impedances of each 'side' (here meaning just the product of the inductor's and capacitor's impedances) divided by the sum of the impedance of each 'side'.

This works when there are only devices... $$z = \frac{z_1 z_2}{z_1 + z_2}$$

For more than two devices, better formula is $$\frac1z = \frac1z_1 + \frac1z_2$$ This formula can be extended easily to more than two devices.

So for you circuit, result of parallel network will be $$\frac1z_p = \frac1R_p + \frac1{j\omega L} + jwC$$

You can simplify this to get the result given in the book.

Analyisis of these circuit diagrams always assume idealized elements, resistors, capacitors, inductors, etc. In the real world, inductors have series resitance in their wire, and parallel capacitance between coils, etc. In many cases some or all of these parasitic reactances can be ignored, for example, if the frequency is high enough that $$R \ll \omega L$$, the resistance is usually negligible, and if frequency is low enough that $$\omega L \ll 1/\omega C$$, the capacitance can be ignored . Similiarly, capacitors have small leakage current that can be modeled by a large parallel resistance, and also some series inductance in their wires. So it is often necessary for high frequency applications that each of these components must be represented by a more complex combination of ideal elements.

In the circuit you were given to analyze, it seems that a parallel resistor may have been used to represent dissipation in both the parallel capacitor and inductor. As has been pointed out in the comments, the $$\parallel$$ symbols indicated that these elements are combined in parallel, and are not intented to represent a norm. Combining elements in parallel is done by adding their conductaces and admittances, then inverting the result to get the equivalent impedance.

Ideal resistors obey Ohm's law for both dc and ac applied voltages, so the voltage across them is proprtional to current, and not frequency. Current, of course, will be time dependent for ac sources, and may also depend on other frequency dependent reactances in series or parallel.

I am confused as to what Rp really means

If $$R_p$$ were 'infinite', there would just be the parallel connected (ideal) inductor and (ideal) capacitor. The impedance of this combination is 'infinite' at angular frequency $$\omega_0 = \frac{1}{\sqrt{LC}}$$

However, for the parallel connected physical inductor and capacitor, the impedance is never infinite. The parallel resistance $$R_p$$ niavely models this fact by providing an upper limit on the impedance of the parallel LC combination.
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