Damped LC tank with external excitation
We consider a resistor, an inductance and a cap in series excited by an external AC signal.
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This circuit is mathematically described by
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Solving the homogeneous part of the differential equation
We solve the homogeneous part of the differential equation Eq. 1 first
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Substitution into the differential equation gives
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This equation is true for all

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is true. We solve this as follows:
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We define
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The general solution for the homogeneous differential equation can therefore be written as
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or after replacing the factors
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Since the differential equation is linear we can rewrite this as follows
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or even shorter like so
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with the two freely choosable constants


![]() | (7) |
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Finding a particular solution for the inhomogeneous differential equation
We find a particular solution for the inhomogeneous differential equation by trying a suitable approach.
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A suitable approach for the above equation is
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Substituted into the differential equation we get
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The value

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The angle

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Combining the general and the particular solution
We combine the general solution for the homogeneous differential equation
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and the particular solution of the inhomogeneous differential equation
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into a total solution by summation:
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Since the e-Function approaches 0 for large values of

![]() | (9) |
We devide by

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We determine the peak voltage in the cap and the peak current with respect to

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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![]() | |
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This gets us the following expression for the input power.
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The circulating power in the circuit is defined as
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with
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We have scaled


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