1 Impedance Characteristics
(written test, approx. 6 % of a 120-minute written test, SS2021)
A coil has an inductive reactance of $X_0 = X(f_0) = 80~\rm \Omega$ at a frequency $f_0 = 60 ~\rm kHz$.
Calculate the frequencies $f_1$, $f_2$, $f_3$ at which the following reactances are measured:
$X_1 = 50 ~\rm \Omega$
$X_2 = 121 ~\rm \Omega$
$X_3 = 147 ~\rm \Omega$
Path
There are multiple ways to solve this question.
One way would be, to calculate the inductance $L$ first by rearranging $X(f) = 2\pi \cdot f \cdot L$.
Another way uses ratios (or “rule of three”), since $X(f) = f \cdot k$ with a constant $k$.
Therefore one can set up two formulas $X_n = f_n \cdot k$, $X_0 = f_0 \cdot k$, and divide the formulae by each other.
This leads to:
\begin{align*}
{{X_n}\over{X_0}} &= {{f_n}\over{f_0}} \\
f_n &= {{X_n}\over{X_0}}\cdot f_0
= {{f_0}\over{X_0}}\cdot X_n \\
\end{align*}
Putting in the numbers:
\begin{align*}
f_n &= {{60 ~\rm kHz}\over{80~\rm \Omega}}\cdot X_n \\
&= 0.75 {{\rm \Omega}\over{\rm kHz}}\cdot X_n \\
\end{align*}
Result
$f_1 = 37.5~\rm kHz$
$f_2 = 90.75~\rm kHz$
$f_3 = 110.25~\rm kHz$