Difference between revisions of "Resonance"

From Microlab Classes
Jump to navigation Jump to search
Line 24: Line 24:
 
Where <math>Q_S</math> is the series circuit quality factor:
 
Where <math>Q_S</math> is the series circuit quality factor:
  
{{NumBlk|::|<math>Q_S = \frac{\omega_0 L}{R_L} = \frac{1}{\omega_0 R_L C</math>|{{EquationRef|7}}}}
+
{{NumBlk|::|<math>Q_S = \frac{\omega_0 L}{R_L} = \frac{1}{\omega_0 R_L C}</math>|{{EquationRef|7}}}}
  
 
Note that:
 
Note that:
 
# For small values of <math>R_L</math> results in large values of <math>Q</math>, leading to large voltages across the inductor and capacitor.  
 
# For small values of <math>R_L</math> results in large values of <math>Q</math>, leading to large voltages across the inductor and capacitor.  
# The voltage across the inductor is exactly <math>180\textdegree</math> out of phase with the capacitor voltage, thus cancelling each other out.
+
# The voltage across the inductor is exactly <math>180^\circ</math> out of phase with the capacitor voltage, thus cancelling each other out.
  
 
== Parallel Resonant Circuits ==
 
== Parallel Resonant Circuits ==

Revision as of 18:24, 10 September 2020

Series Resonant Circuits

Consider the series resonant circuit shown in Fig. 1. We can calculate the total impedance seen by the source , as:

 

 

 

 

(1)

Combining the imaginary terms of the impedance, we get:

 

 

 

 

(2)

We can see that the imaginary component of the impedance becomes zero at the resonant frequency, , equal to:

 

 

 

 

(3)

Note that the cancellation is narrowband, since perfect cancellation occurs only at a single frequency. Thus, from the point of view of the voltage source , the impedance is purely real. We can then calculate the current, as:

 

 

 

 

(4)

We then use this current to calculate the voltage across the inductor and capacitor:

 

 

 

 

(5)

 

 

 

 

(6)

Where is the series circuit quality factor:

 

 

 

 

(7)

Note that:

  1. For small values of results in large values of , leading to large voltages across the inductor and capacitor.
  2. The voltage across the inductor is exactly out of phase with the capacitor voltage, thus cancelling each other out.

Parallel Resonant Circuits