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Starting in 1996, Alexa Internet has been donating their crawl data to the Internet Archive. Flowing in every day, these data are added to the Wayback Machine after an embargo period.

Collection: Alexa Crawls

Starting in 1996, Alexa Internet has been donating their crawl data to the Internet Archive. Flowing in every day, these data are added to the Wayback Machine after an embargo period.

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The Wayback Machine - https://web.archive.org/web/20120110062257/http://fourier.eng.hmc.edu:80/e84/lectures/ch3/node8.html

Series and Parallel Resonance

Series Resonance

Consider an RCL series circuit consists of a resistor $R$, an inductor $L$, and a capacitor $C$ connected in series to a voltage source. The overall impedance of the three elements is

$$Z=R+j\omega L+\frac{1}{j\omega C}=R+j\left(\omega L-\frac{1}{\omega C}\right) =\vert Z\vert e^{j\angle Z}$$

where

$$\vert Z\vert=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\righ... ...=0 90^\circ & \omega \rightarrow \infty \end{array} \right.$$

In particular the resonant frequency is defined as

$$\omega_0\stackrel{\triangle}{=}\frac{1}{\sqrt{LC}}$$

When $\omega=\omega_0$, the impedances of the capacitor and the inductor have the same magnitude but opposite phase:

$$Z_L=j\omega_0L=j\sqrt{\frac{L}{C}},\;\;\;\;\;\; Z_C=\frac{1}{j\omega_0C}=-j\sqrt{\frac{L}{C}}$$

and they add up to zero $Z_L+Z_C=0$. Now the total impedance is minimized:

$$Z=Z_R+Z_C+Z_L=Z_R=R$$

and the current $\dot{I}=\dot{V}/Z=\dot{V}/R$ is maximized. The current $\dot{I}$ and voltage $\dot{V}$ are in phase. At the resonant frequency $\omega=\omega_0=1/\sqrt{LC}$, the ratio of the magnitude of the inductor/capacitor impedance and the resistance is defined as the quality factor:

$$Q\stackrel{\triangle}{=}\frac{\vert Z_L\vert}{R}=\frac{\vert ... ...ga_0L}{R}=\frac{1}{\omega_0CR} =\frac{1}{R}\sqrt{\frac{L}{C}}$$

When $\omega=\omega_0$, the voltages across each of the three components are:

$$\dot{V}_R=\dot{I} Z_R =\dot{I} R=\dot{V},\;\;\;\;\mbox{i.e.}\;\;\;\; \dot{I}=\frac{\dot{V}}{R}$$

$$\dot{V}_L=\dot{I} Z_L=\frac{\dot{V}}{R}j\omega_0 L=jQ\dot{V}$$

$$\dot{V}_C=\dot{I} Z_C=\frac{\dot{V}}{R}\frac{1}{j\omega_0 C} =-jQ\dot{V}$$

The magnitude of $V_L$ and $V_C$ are $Q$ times larger than that of $\dot{V}_R$, which is equal to the source voltage $\dot{V}$. But as $V_L$ and $V_C$ are in opposite polarity ($180^\circ$ out of phase), they cancel each other.

The RCL series circuit is a band-pass filter with the passing band centered around the resonant frequency $\omega_0=1/\sqrt{CL}$. The bandwidth is determined by the quality factor $Q$. The larger $Q$, the narrower the bandwidth. The impedance $Z=Z_R+R_L+Z_C$ as a function of $\omega$ is shown below:

and the admittances $Y=1/Z$ for different $Q$ ($R$) and $C$ are shown below:

The bandpass effect can be intuitively explained. When $\omega$ is high, the inductor's impedance $\omega L$ is high, and when $\omega$ is low, the capacitor's impedance $1/\omega C$ is high. When $\omega=\omega_0$ the overall impedance is the smallest. If the input is a voltage source $v(t)$, the current through the circuit will reach a maximum value when $\omega=\omega_0$.

Example: In a series RLC circuit, $R=5\Omega$, $L=4\;mH$ and $C=0.1\;\mu F$. The resonant frequency $\omega_0$ can be found to be $\omega_0=1/\sqrt{LC}=1/\sqrt{4\times 10^{-3}\times 10^{-7}}=5\times 10^4$. The quality factor is

$$Q=\frac{\omega_0L}{R}=\frac{(5\times 10^4)\times (4\times 10^{-3})}{5} =40$$

or

$$Q=\frac{1}{\omega_0CR}=\frac{1}{(5\times 10^4)\times 10^{-7}\times 5} =40$$

If the input voltage is $V_{rms}=10V$ at the resonant frequency, the current is $I=V/R=10/5=2 A$, and the voltages across each of the elements are:

• $\dot{V}_R=R\dot{I}=V=10V$
• $\dot{V}_L=j\omega L \dot{I}=j5\times 10^4\times 4\times 10^{-3} \times 2=j400V$
• $\dot{V}_C=\dot{I}/j\omega_0C=-j2/(5\times 10^4\times 0.1\times 10^{-6})=-j400V$

Or, more conveniently, the amplitudes of $\dot{V}_L$ and $\dot{V}_C$ can be found by $\vert V_L\vert=\vert V_C\vert=QV=40\times 10V=400V$. Note that although input voltage is $10V$, the voltage across L and C ($Q$ times the input) could be very high (but they are in opposite phase and therefore cancel each other).

Parallel Resonance: A GCL parallel circuit consists of a resistor $R=1/G$, an inductor $L$ and a capacitor connected in parallel to input voltage.

In this case, it is much easier to consider the conductance of the admittance $Y=1/Z$ of each of the element. The overall admittance of the three elements in parallel is

$$Y=G+j\omega C+\frac{1}{j\omega L}=G+j\left(\omega C-\frac{1}{\omega L}\right) =\vert Y\vert e^{j\angle Y}$$

where

$$\vert Y\vert=\sqrt{G^2+\left(\omega C-\frac{1}{\omega L}\right)^2},\;\;\;\; \angle Y=tan^{-1} \frac{\omega C-1/\omega L}{G}$$

In particular when $\omega$ is at the resonant frequency

$$\omega_0\stackrel{\triangle}{=}\frac{1}{\sqrt{LC}}$$

we have

$$Y_C+Y_L=j\omega C+\frac{1}{j\omega L}=j\left(\omega C-\frac{1... ...right) =j\left(\sqrt{\frac{C}{L}}-\sqrt{\frac{C}{L}}\right)=0$$

the effects of $L$ and $C$ cancel each other, and the complex admittance $Y$ becomes real and its magnitude reaches the minimum

$$\vert Y\vert=\sqrt{G^2+\left(\omega_0 C-\frac{1}{\omega_0 L}\... ...)^2}=G, \;\;\;\;\;\;\;\;\;\; \angle{Y}=\tan^{-1}\frac{0}{G}=0$$

and the current $\dot{I}$ reaches a minimum value $\dot{I}=\dot{V}G$. In particular, if the resistor does not exist, i.e., $R=\infty$ and $G=0$, then the admittance $Y=0$ and $Z=\infty$.

The Quality Factor $Q_p$ of a parallel resonance circuit is defined as the ratio of the magnitude of the inductor/capacitor susceptance and the conductance:

$$Q_p\stackrel{\triangle}{=}\frac{\omega_0C}{G}=\frac{1/\omega_... ...rac{1}{G}\sqrt{\frac{C}{L}} =R\sqrt{\frac{C}{L}}=\frac{1}{Q_s}$$

Note that $Q_p$ for a parallel RCL circuit is the reciprocal of $Q_s$ for a series RCL circuit. The currents through each of the three components are:

• 
  
  $$\dot{I}_R=\dot{V} G=\dot{I}$$
  
  

• 
  
  $$\dot{I}_C=\dot{V} Y_C=\dot{V}j\omega_0 C\vert _{\omega_0=1/\sqrt{LC}} =j \frac{\dot{I}}{G}\sqrt{\frac{C}{L}}=jQ\dot{I}$$
  
  

• 
  
  $$\dot{I}_L=\dot{V} Y_L=\dot{V}\frac{1}{j\omega_0 L}\vert _{\om... ...1/\sqrt{LC}}=-j \frac{\dot{I}}{G}\sqrt{\frac{C}{L}}=-jQ\dot{I}$$
  
  

The magnitude of the current $I_L$ through $L$ and $I_C$ through $C$ are $Q$ times larger than the current $\dot{V}_R$ through $R$ which is the same as the current source $\dot{I}$). But as $I_L$ and $I_C$ are in opposite direction, they form a loop current through $L$ and $C$ with no effect to the rest of the circuit.

The parallel RCL circuit behaves like a bandstop filter which can be intuitively understood. When $\omega$ is high, the capacitor's impedance $1/\omega C$ is low, and when $\omega$ is low, the inductor's impedance $\omega L$ is low. When $\omega=\omega_0$ the overall impedance is the largest. However, if the input is a current source, the voltage across the elements $\dot{V}=\dot{I}Z=\dot{I}/Y$ will reach a maximum value when $\omega=\omega_0$.