3.2 Mathematical model

Course subject(s) 3. Extending the model

For a base clock for a computer, we want to build an LC-oscillator that generates $100$ MHz oscillations.

The components that are given are:

a 5V battery: $V_{B} = 5$ V,
an inductor: $L = 0.004 μ$ H,
the resistance: $R = 0.1 Ω$ .

Our problem is

Find the capacitance $C$ of the capacitor ( $[C] =$ henry) such that the current in the electrical circuit oscillates with a frequency of $100$ MHz.

The following physical laws relate the current $I (t)$ running in the circuit (measured in ampere), the charge on (one side of) the capacitor $Q (t)$ (in coulomb) and the voltage drops over the different elements (in volts).

The current, $I (t)$ , is related to the charge on the capacitor, $Q (t)$ , according to $I (t) = \frac{d Q}{d t} .$

The voltage drop over the capacitor, $V (t)$ , is related to the charge on the capacitor by: $V (t) = \frac{Q (t)}{C} .$

The voltage drop over the inductor, $V_{L} (t)$ , is related to the change in the current: $V_{L} (t) = L \frac{d I}{d t} .$

And an expression for voltage drop over the resistor, $V_{R} (t)$ is given by: $V_{R} (t) = R I (t) .$

Kirchhoff’s voltage law states that the voltage drop around a loop in a circuit equals zero: $V + V_{L} + V_{R} - V_{B} = 0.$

There are different options for a set of dependent variables to calculate in. We choose to calculate in current $I (t)$ and the voltage drop over the capacitor $V (t)$ .

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