5. Circuit problem

In electronics, an LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R) placed in series with an alternating-current power source. The circuit is sensitive to the source frequency \(\omega\), with resonant response at a frequency which depends on the material properties of the circuit components.

Application: Imaging by electromagnetic induction with resonant circuits

The inductance can be modified by bringing a conductive sample into the magnetic field of the circuit. By finding the resonant frequency it may be possible to identify the material sample without making physical contact, with possible application in fields including medicine, security, industry, archaeology, geophysics.

The signal amplitude \(Q\) is found to obey the following relationship where \(L,C,R\) are positive constants representing the inductance, capacitance and resistance of the circuit components, and \(V,\omega\) are positive constants representing the amplitude and frequency of the power source:

(5.1)\[\begin{equation} Q = \frac{V}{\sqrt{R^2\omega^2+(\frac{1}{C}-L\omega^2)^2}}. \end{equation}\]

Exercise 5.1

Produce a sketch showing the shape of the graph \(Q(\omega)\). It may help you to set all of the constants \(R,L,C,V\) to equal one.

Hint

This function is of the following form where \(K\) is a quadratic in \(\omega^2\) :

\[\begin{equation*} Q=\frac{1}{\sqrt{K}} \end{equation*}\]

It may help you to introduce a substitution \(w=\omega^2\) and plot \(Q(w)\) instead.

Exercise 5.2

For what value of \(\omega\) is the amplitude maximised? Give your answer in terms of \(R,L,C\).

Hint

You do not need to do any calculus here! Think about the shape of the function in relation to the quadratic on the denominator.

Exercise 5.3

What effect does changing the resistance parameter \(R\) or capacitance \(C\) have on the peak amplitude response curve?