Thomson's equation of oscillation

With the Thomson's equation of vibration , the leaves resonant frequency of a resonant circuit (series resonant circuit and an ideal parallel resonant circuit) with the capacitance C and the inductance L calculated. It was first formulated in 1853 by the British physicist William Thomson and reads: ${\displaystyle f_{0}}$  

${\displaystyle f_{0}={\frac {1}{2\pi {\sqrt {LC}}}}}$

Or transformed for the period duration (oscillation time):

${\displaystyle T={\frac {1}{f_{0}}}=2\pi {\sqrt {LC}}}$

Derivation

In the case of resonance, the resonance resistance is as large as the series resistance. The capacitive resistance of the capacitor and the inductive resistance of the coil within the resonant circuit compensate each other to zero:

${\displaystyle X_{L}+X_{C}=0\qquad \Leftrightarrow \qquad \omega _{0}L-{\frac {1}{\omega _{0}C}}=0}$

${\displaystyle \omega _{0}L={\frac {1}{\omega _{0}C}}}$

${\displaystyle 2\pi f_{0}L={\frac {1}{2\pi f_{0}C}}}$, there applies ${\displaystyle \omega =2\pi f}$

${\displaystyle {f_{0}}^{2}={\frac {1}{4\pi ^{2}LC}}}$

${\displaystyle f_{0}={\frac {1}{2\pi {\sqrt {LC}}}}}$, The form is also common: ${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}}$

Derivation according to the law of conservation of energy

If we consider the electrical oscillating circuit as a closed system, the sum of all forms of energy in this system is constant at all times t .

${\displaystyle \!\,E_{\mathrm {mag} }(t)+E_{\rm {el}}(t)=E_{\rm {Gesamt}}}$

${\displaystyle E_{\mathrm {mag} }}$: magnetic field energy of the coil

${\displaystyle E_{\mathrm {el} }}$: electrical field energy of the capacitor

${\displaystyle E_{\mathrm {Gesamt} }}$: Total energy of the system (constant)

If you insert the corresponding formulas, you get the following differential equation:

${\displaystyle {\frac {1}{2}}LI^{2}(t)+{\frac {1}{2C}}Q^{2}(t)=E_{\mathrm {Gesamt} }}$

Out

${\displaystyle I(t)={\frac {dQ(t)}{dt}}={\dot {Q}}(t)}$

follows:

${\displaystyle {\frac {1}{2}}L{\dot {Q}}^{2}(t)+{\frac {1}{2C}}Q^{2}(t)=E_{\mathrm {Gesamt} }}$

Now we derive this equation according to the time and get:

${\displaystyle L{\dot {Q}}{\ddot {Q}}(t)+{\frac {1}{C}}Q{\dot {Q}}(t)=0}$

${\displaystyle I(t)\left(L{\ddot {Q}}+{\frac {1}{C}}Q(t)\right)=0}$

${\displaystyle L{\ddot {Q}}+{\frac {1}{C}}Q(t)=0}$Because in the resonant circuit applies: . ${\displaystyle I(t)\neq 0}$

To solve this equation, we need to establish a relationship between and . To do this, we use a sine function as a solution, as it is well suited for describing an oscillation due to its periodicity. ${\displaystyle Q(t)}$ ${\displaystyle {\ddot {Q}}(t)}$

${\displaystyle Q(t)={\hat {Q}}\cdot \sin(\omega t+\varphi )}$

${\displaystyle {\dot {Q}}(t)=\omega {\hat {Q}}\cdot \cos(\omega t+\varphi )}$

${\displaystyle {\ddot {Q}}(t)=-\omega ^{2}{\hat {Q}}\cdot \sin(\omega t+\varphi )=-\omega ^{2}\cdot Q(t)}$

${\displaystyle {\hat {Q}}}$: maximum charge (amplitude)

${\displaystyle \omega }$: Angular frequency

${\displaystyle \varphi }$: Phase shift

Insertion results in:

${\displaystyle {\frac {1}{C}}Q(t)-\omega ^{2}LQ(t)=0}$

${\displaystyle Q(t)\left({\frac {1}{C}}-\omega ^{2}L\right)=0}$

${\displaystyle {\frac {1}{C}}-\omega ^{2}L=0}$, since the following applies in the oscillating circuit: ${\displaystyle Q(t)\neq 0}$

It follows with : ${\displaystyle \omega =2\pi f}$

${\displaystyle {\frac {1}{C}}-4\pi ^{2}f_{0}^{2}L=0}$

${\displaystyle {f_{0}}^{2}={\frac {1}{4\pi ^{2}LC}}}$

${\displaystyle f_{0}={\frac {1}{2\pi {\sqrt {LC}}}}}$

The Thomson oscillation equation only applies to series oscillating circuits and ideal parallel oscillating circuits. In the case of more complex topologies , the frequency must be derived from. ${\displaystyle X_{L}=X_{C}}$

Furthermore, when using Thomson's oscillation equation, it must be ensured that the respective system is in the case of oscillation - the damping due to the ohmic resistance is not too great. If the damping is not too great, the changed resonance frequency in the parallel resonant circuit can be calculated with the loss resistance R _L of L :

${\displaystyle \omega _{D}=\omega _{0}{\sqrt {1-R_{L}^{2}{\frac {C}{L}}}}}$

literature

• Lothar Papula: Mathematics for Engineers and Natural Scientists . 12th edition. tape 1 . Vieweg + Teubner, 2009, ISBN 978-3-8348-0545-4 .

Web links

• Other derivation of the formula from the energy law ( LEIFI )