Series and Parallel Resonance

Series Resonance: An RCL series circuit consists of a resistor $R$, an inductor $L$ and a capacitor connected in series to a voltage source.

The overall impedance of the three elements is

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

where

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

In particular we define the resonant frequency as

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

When $\omega=\omega_0$, the circuit is at resonance with the following properties:

• The impedances of the capacitor and the inductor have the same magnitude but opposite phase:
  
  $$Z_L=j\omega_0L=j\sqrt{\frac{L}{C}},\;\;\;\;\;\; Z_C=\frac{1}{j\omega_0C}=-j\sqrt{\frac{L}{C}}$$
  
  
  The effects of $L$ and $C$ cancel each other
  
  $$Z_L+Z_C=j\omega L+\frac{1}{j\omega C}=j(\omega L-\frac{1}{\omega C})=0$$
  
  

• The total impedance is minimized:
  
  $$Z=Z_R+Z_C+Z_L=R$$
  
  
  As the phase angle $\angle Z=0$ is zero, the current $\dot{I}$ and voltage $\dot{V}$ are in phase. Given $\dot{V}$, the current $\dot{I}=\dot{V}/Z=\dot{V}/R$ reaches maximum value.

  In particular, if $R=0$, an RCL series circuit has zero impedance at the resonant frequency $\omega=\omega_0=1/\sqrt{LC}$.

At the resonant frequency, the ratio of the magnitude of the inductor/capacitor impedance and the resistance is defined as the quality factor

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

$Q$ is inversely proportional to the resistance $R$ in the circuit.

According to the phasor version of Ohm's law, 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

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

where

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

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

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

we have

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

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

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

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 $\omega=\omega_0$.