Damped LC tank with external excitationWe consider a resistor, an inductance and a cap in series excited by an external AC signal.
This circuit is mathematically described by
Solving the homogeneous part of the differential equationWe solve the homogeneous part of the differential equation Eq. 1 first
Substitution into the differential equation gives
This equation is true for all only if
is true. We solve this as follows:
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
with the two freely choosable constants und . Please note the dependence of the resonance frequency from the resistance.
Finding a particular solution for the inhomogeneous differential equationWe find a particular solution for the inhomogeneous differential equation 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 solutionWe combine the general solution for the homogeneous differential equation
and the particular solution of the inhomogeneous differential equation
into a total solution by summation:
Since the e-Function approaches 0 for large values of only the particular solution remains after an initiation period.
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 ConsiderationThe power going into the circuit is given by
This gets us the following expression for the input power.
The circulating power in the circuit is defined as
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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