Why does the impedance of a parallel RLC circuit approach infinity at resonance?

Question

Pretty new to this stuff so apologies if I get some words wrong.

In the book I'm reading they described a parallel circuit with a resistor capacitor and an inductor powered by a current source. They say the capacitor has impedance \$\frac{1}{j\omega C}\$ and the inductor has impedance \$j\omega L\$ and the resistor has resistance \$R_{p}\$. The resonant frequency is when the magnitudes of the two impedances of the capacitor and the inductor are equal i.e. \$\omega_0^2 = \frac{1}{LC}\$. But when I use the parallel resistance formula I get the impedance of the circuit should be

$$\left(\frac{1}{R}_p + j\omega C + \frac{1}{j\omega L}\right)^{-1}$$

But at resonant frequency where \$\omega C = \frac{1}{\omega L}\$ this comes out to \$R_{p}\$, not infinity. So I'm struggling to understand what they're trying to say here.

To clarify some things here's the page from the book I'm reading. (Electronics with Digital and Analog Integrated Circuits by Higgins)

Answer 1

What book? Do they actually say the impedance of the RLC circuit approaches infinity or just that the impedance of the LC section approaches infinity. Your calculation is correct but you may be misinterpreting the book.

Answer 2

R acts a damping factor to parallel LC and determines the bandwidth, BW, and risetime and R/Zc(fo)=Q=fo/BW

Thus if R were ideally infinite ! Q is infinite and any fo oscillation remains constant forever with Zc(fo)=-ZL(fo) in the complex (reactive) domain, and \$f_o 2\pi=\omega_o\$

Answer 3

You are correct, and the book appears to be wrong. In a parallel RLC circuit, when the driving frequency is the resonant frequency, the impedance is the resistance of R, not \$\infty\$. In a pure LC circuit, with ideal components, the impedance approaches \$\infty\$.

The impedance at any frequency is $$Z= R || X_C || X_L$$

(where \$X_C\$ and \$X_L\$ have imaginary values.)

At resonance (and assuming ideal components)

$$X_C = -X_L$$

so

$$Z_0 = R || X_C ||( -X_C) = \frac{\displaystyle 1}{\displaystyle \frac{\displaystyle 1}{\displaystyle R}+\frac{\displaystyle 1}{\displaystyle X_C} + \frac{\displaystyle 1}{\displaystyle -X_C}} = R$$

If instead, X_L and X_C are given a real, positive values, the formulas become

$$X_C = X_L$$

and

$$Z_0 = R || (jX_C) ||( -jX_C) = \frac{\displaystyle 1}{\displaystyle \frac{\displaystyle 1}{\displaystyle R}+\frac{\displaystyle 1}{\displaystyle jX_C} + \frac{\displaystyle 1}{\displaystyle -jX_C}} = R$$

Both ways of solving for \$Z_0\$ obviously give the same answer,

$$Z_0=R$$.

For clarity, what I assume to be a parallel RLC circuit looks like this:

^{simulate this circuit – Schematic created using CircuitLab}