That allow station selection

fig.1 R L C circuit

RLC calculator

The tuning circuit opposite is the first link in a receiver. It is present in crystal radios, tube radios (and then transistor radios). It allows selecting stations by adjusting the capacitance $C$. In a few words, the antenna receives signals from all the surrounding stations and injects them into the left circuit of the transformer. Only signals whose frequency is close to $f_0$ will be present across $C$. As a general rule, $L$ is fixed. $L$ is the inductance that the capacitor $C$ sees. $C$ is variable and allows the radio user to select the station of their choice.

Example

Imagine that the coil seen by the capacitor $C$ has a self-inductance coefficient of $L = 170 µH$. We want to receive France Bleue in the Paris region which broadcasts with a central frequency of 864 KHz Therefore $C = \dfrac{1}{0.00017 \times (2 \times 3.14 \times 864 000 )^2} = 200 \times 10^{-12} F$ which is 200 pF. The values of $L$ and $C$ proposed in this example are not chosen randomly.
Indeed, in many vintage radios, $C$ is between 30 pF and 500 pF; and $L$ is often 170 µH for the medium wave (MW) band.
A very practical calculation tool is provided opposite.

The radio stations received by vintage radios are distributed in frequency ranges called bands. The radio wave range is divided into bands and sub-bands. A list of these is given in fig.2. The so-called FM band is the most recent. It is reserved for frequency modulation broadcasts. This is the mode that gives the best sound reproduction results. Only this band could claim, until recently, high-fidelity (HiFi) listening. The new DRM digital coding process is changing this situation.
The 3 AM bands are reserved for amplitude modulation broadcasts. They appeared at the dawn of vintage radio. Stations broadcasting in the MW and especially SW bands benefit from improved range at night.
Long waves are unknown in the USA.

BAND	SUB-BAND	ACRONYM	FMin	FMax
AM (*)	Long Waves	LW	150KHz	525KHz
	Medium Waves	MW	525KHz	1620KHz
	Short Waves	SW	1620KHz	30MHz
FM	Frequency Modulation	FM	87.5MHz	108MHz

(*) Amplitude Modulation

Tuning circuits are also found in the so-called intermediate frequency (IF) stages of superheterodyne radios. In the FM band, they are tuned to 10.7 MHz. In the AM band, they are tuned to 455 KHz for recent radios and to frequencies ranging from 135 to 455 KHz for radios dating from before World War II. These circuits have fixed tuning (set at the factory). They resonate at a given frequency $f_0$. At this frequency, the circuit presents a maximum impedance. This property is used to filter waves of frequency $f \pm \delta f$. The interval $\Delta f = 2 \delta f$ allows a group of audible frequencies to pass through. In France and in the LW and MW bands: $ 2 \delta f$ = 9KHz. The relationship between $f$, $L$ and $C$ is given by Thomson's formula

$$ \displaystyle f_0=\frac {1}{2\pi \sqrt{LC}}$$
In radioelectricity, it is preferred to speak in terms of wavelength. This length $\lambda$ is the distance traveled by an electromagnetic wave at speed $c$ during a period $T = 1/f$.

The speed $c$ depends on the medium. In a vacuum $c \simeq 3 . 10^8 m/s$. This value will be used for our calculations. So $\lambda = c/f$ and Thomson's formula becomes:

$$ \displaystyle \lambda_0=2\pi c \sqrt{LC}$$

expressed in meters. In radioelectricity, $L$ is often a multiple of $\mu H$ and $C$ a multiple of $pF$. Under these conditions

$ \displaystyle \lambda_m=1.885 \sqrt{L_{\mu H}C_{pF}}$ or $ \displaystyle f_{MHz}=\frac {159} { \sqrt{L_{\mu H}C_{pF}}}$

Example

Consider a coil made of 100 close turns of enameled copper wire with a diameter of 0.26 mm. They are wound on a circular bakelite mandrel with a diameter of 26 mm. This coil is given as L = 177µH. A 250pF mica capacitor is placed in parallel on the coil. The resonance of this circuit will occur for $f \approx 159\sqrt{(250 \times 177)} \approx 0.756MHz=756KHz $ and $\lambda \approx 300/0.756=397 m$. This is a frequency in the Medium Waves range (MW).

fig.4

The device given as an example favors signals close to $f \approx756KHz$ and rejects those that are «far away» by $\pm4.5KHz$. It is customary for signals belonging to the bandwidth to have an amplitude $\leq 0.707 \times V_{MAX}$ .

The bandwidth varies with the quality factor of the coil. This coefficient is: $Q=\dfrac{L\omega}{r}$ where $\omega$ is $2\pi f$ and $R$ is the internal resistance of the coil.

The bandwidth is $\Delta f= f_0/Q$. The listener of the vintage radio will only hear the signals located to the right (or to the left but not both) of the carrier $f_0$. The audible band will therefore be $\Delta f=4.5KHz$. The narrowness of this band characterizes the typical sound of the AM range.

Sources and references

[1] Paul BERCHE, "Pratique et théorie de la TSF", Librairie de la Radio, Paris, 1937, reviewed by Roger RAFFIN, 1958.

[2] Lucien CHRETIEN, "Théorie et Pratique de la Radioélectricité", Editions Chiron, Paris, 1933.

[3] Jean QUINET, "Théorie & pratique des circuits électroniques" T.1, Ed. DUNOD, 1968.

[4] F. E. TERMAN, "Radio Engineer's Handbook", McGraw-Hill, New York, 1943.