# Calculating LC Values of RF and Local Oscillator Stages of a Single Conversion MW/SW Receiver

## Motivation

Traditional shortwave receivers use variable capacitor tuned LC circuits in preselector or RF amplifier stages as well as the local oscillator. The value of minimum and maximum capacitance of a variable capacitor is predermined, as well as the ratio between them, so it is cosidered as initial condition of the design proces. The capacitance ratio in conjunction with other factors, like oscillation conditions of a given local oscillator construction, reduces the achiebable tuning ratio with a given inductance value – typically somewhere around a 1:2.5 … 1:3.5 range. Because of this, the 0.5…30 MHz tuning range of a MW/SW receiver is broken down into several tuning bands. The number of bands is a compromise between build cost, easy tuning and frequency stability.

To match exactly the the frequency ranges of RF and LO stages, L and C values are built trimmable. The inductance has an adjustable coil, and additional fixed and trimmer capacitors are used in parallel to the main variable capacitor.

## Objectives

With this little experiment, my goal was are to find decent band breakdown, and to calculate ideal L and C values for each band.

## Determining L and C Values for a Given Band

Let $C_l, C_h$ the minimum and maximum capacitance of the variable capacitor (same for all bands), and $\omega_{min}, \omega_{max}$ the angular frequencies for the band.

We are looking for the $L$ inductance value and $C_t$ parallel (fixed + trimmer) capacitor value of each LC circuit for each band.

From the Thomson-formula, we can express L, the inductance for our LC circuit:

$f_0={1 \over 2 \pi \sqrt{LC}} \rightarrow L = {1 \over \omega^2 C}$

For a given band, when we tune to the bottom of the band, the following should hold:

$L = { 1 \over \omega_{min}^2 (C_l + C_t)}$

Similarly, tuning to the top of the band, the following should also hold:

$L = { 1 \over \omega_{max}^2 (C_h + C_t)}$

From this, we can deduct $C_t$:

\begin{aligned} { 1 \over\ \omega_{min}^2(C_l + C_t)} & = { 1 \over \omega_{max}^2(C_h + C_t)} \\ \omega_{min}^2 C_t - \omega_{max}^2 C_t & = \omega_{max}^2 C_h - \omega_{min}^2 C_l \\ C_t & = { \omega_{max}^2 C_h - \omega_{min}^2 C_l \over \omega_{min}^2 - \omega_{max}^2 } \end{aligned} .

Just like that, we can also deduct L by expressing C for both ends of the band:

$\begin{cases} C_l + C_t= { 1 \over \omega_{min}^2 L } \\ C_h + C_t = { 1 \over \omega_{max}^2 L } \end{cases}$

Deducing L:

\begin{aligned} { 1 \over \omega_{min}^2 L } - C_l & = { 1 \over \omega_{max}^2 L } - C_h \\ { 1 \over L }( {1 \over \omega_{min}^2} - { 1 \over \omega_{max}^2}) & = C_l - C_h \\ L &= { { 1 \over \omega_{min}^2} - { 1 \over \omega_{max}^2 } \over C_l - C_h} \end{aligned}

### Example

Let’s say our variable capacitor has $C_l = 320pF$ and $C_h=7pF$.

The MW band has $f_{min} = 560$ kHz, and $f_{max}=1.6$MHz. For the RF amplifier, we will get $C_t = 33.2pF, L=246 \mu H$. The tuning ratio will be 2.96. $C_t$ can be constructed using a 22pF fixed capacitor and a trimmer capacitor with around 22pF maximal value. L and C for the local oscillator can be calculated by shifting the frequency range by the desired intermediate fequency.

To be continued…..