Damped LC tank with external excitation

We consider a resistor, an inductance and a cap in series excited by an external AC signal.

This circuit is mathematically described by

(1)

Solving the homogeneous part of the differential equation

We solve the homogeneous part of the differential equation Eq. 1 first

(2)
using the apporach


Substitution into the differential equation gives


This equation is true for all only if


is true. We solve this as follows:


We define
and thus get the following two solutions:


The general solution for the homogeneous differential equation can therefore be written as


or after replacing the factors

Since the differential equation is linear we can rewrite this as follows


or even shorter like so
(3)
(4)
(5)
(6)

with the two freely choosable constants und . Please note the dependence of the resonance frequency from the resistance.

(7)

Finding a particular solution for the inhomogeneous differential equation

We find a particular solution for the inhomogeneous differential equation now by trying a suitable approach.

A suitable approach for the above equation is


Substituted into the differential equation we get



The value is the length of the complex number on the right.


The angle is determined as follows:




Combining the general and the particular solution

We combine the general solution for the homogeneous differential equation


and the particular solution of the inhomogeneous differential equation


into a total solution by summation:

(8)

Since the e-Function approaches 0 for large values of only the particular solution remains after an initiation period.

(9)

We devide by to get the voltage in the cap, take first derivatives to get the current and take the second derivative to get the voltage over the coil.


We determine the peak voltage in the cap and the peak current with respect to (resonance frequency):



If we excite the series tank with its resonance frequency we get high voltages over the components and a very high current only limited by the resistance of the wire.

Energy Consideration

The power going into the circuit is given by


This gets us the following expression for the input power.

(10)

The circulating power in the circuit is defined as

with



We have scaled with a factor of 100 in the figure above. The Input power is very small compared to the circulating power for .
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