Difference between revisions of "Resonance"
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We then use this current to calculate the voltage across the inductor and capacitor: | We then use this current to calculate the voltage across the inductor and capacitor: | ||
− | {{NumBlk|::|<math>v_L = i_S \cdot j\omega_0 L = \frac{v_S}{R_L} \cdot j\omega_0 L = v_S \cdot | + | {{NumBlk|::|<math>v_L = i_S \cdot j\omega_0 L = \frac{v_S}{R_L} \cdot j\omega_0 L = v_S \cdot jQ_S</math>|{{EquationRef|5}}}} |
− | {{NumBlk|::|<math>v_C = i_S \cdot \frac{1}{j\omega_0 C} = -\frac{v_S}{R_L} \cdot j\frac{1}{\omega_0 C} = -v_S \cdot | + | {{NumBlk|::|<math>v_C = i_S \cdot \frac{1}{j\omega_0 C} = -\frac{v_S}{R_L} \cdot j\frac{1}{\omega_0 C} = -v_S \cdot jQ_S</math>|{{EquationRef|6}}}} |
+ | |||
+ | 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}}}} | ||
+ | |||
+ | 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. | ||
+ | # The voltage across the inductor is exactly <math>180\textdegree</math> out of phase with the capacitor voltage, thus cancelling each other out. | ||
== Parallel Resonant Circuits == | == Parallel Resonant Circuits == |
Revision as of 18:22, 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:
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(1)
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Combining the imaginary terms of the impedance, we get:
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(2)
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We can see that the imaginary component of the impedance becomes zero at the resonant frequency, , equal to:
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(3)
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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:
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(4)
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We then use this current to calculate the voltage across the inductor and capacitor:
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(5)
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(6)
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Where is the series circuit quality factor:
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(7)
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Note that:
- For small values of results in large values of , leading to large voltages across the inductor and capacitor.
- The voltage across the inductor is exactly out of phase with the capacitor voltage, thus cancelling each other out.