### Antenna

A quick and easy choice for an antenna is a long-wire antenna. Simply cut a length of wire 1/4 the wavelength of the signal you are trying to receive and hang it up as high as possible. Shortwave radio covers the frequency band 1.6 MHz to 30 MHz. Arbitrarily choosing 10 MHz, the length of the wire should be 300,000,000 m/s / 10 MHz = 30 m. This is a decent length of wire, so feel free to experiment with shorter lengths if you are stuck indoors. I’ve had success with only around 3-4 meters of wire.

It is important that the receiver ground has a very solid connection to Earth ground, otherwise you will not get a clear signal.

### The Tuner

One end of the antenna is fed into the tuning circuit, which is a simple parallel LC circuit (also known as a tank circuit). We will model the signal from the antenna as an AC current source.

To get some insight into how this circuit works, we begin by analyzing the impedance of the parallel combination of an inductor and capacitor as a function of frequency

$Z(\omega) = Z_L|| Z_C = \frac{j \omega L/ j \omega C}{j \omega L + 1 / j \omega C} = \frac{j \omega L}{-\omega^2 LC + 1}$

Taking the magnitude of the complex impedance yields

$|Z(\omega)| = \frac{\omega L}{\omega^2 LC - 1}$

We are interested in how the impedance changes relative to the frequency. Performing asymptotic analysis on $|Z(\omega)|$. Taking the limit as ω→0 results in zero impedance. Taking the limit as ω→∞ also results in zero impedance. Therefore, we expect at some value of ω in between 0 and infinity, the impedance will take on a maximum value. We can confirm this analysis by setting the derivative of $d|Z(\omega)|$ and setting it equal to zero.

$\frac{d|Z(\omega)|}{d\omega} = \frac{\omega^2 L^2 C - L}{(\omega^2 L C + 1)^2} = 0$

Solving this expression for ω and converting to units of Hz leaves us with an expression for the resonant frequency of the circuit

$f_{r} = \frac{1}{2 \pi \sqrt{LC}}$

At the resonant frequency (i.e. the frequency of the station we are tuned to), Ohm’s law tells us we will get a large voltage across the inductor and capacitor. At all other frequencies, the tank circuit will appear as a short to ground, allowing us to reject the signal energy from neighboring stations.

Choosing L = 50μH and C = 470pF, we compute the resonant frequency to be $1 / (2 \pi \sqrt{50 \mathrm{\mu H} \times 470 \mathrm{pF}}) = 1.038 \mathrm{MHz}$. This can be confirmed by plotting the magnitude of the |Z(jω)| as a function of frequency. By using a variable inductor and capacitor, we will be able to tune the circuit to any frequency we desire.

### Rectification

After receiving the signal and filtering out the station we want using the tank circuit, we need to recover the original audio signal via a process called demodulation. Modulation is a process by which a low frequency signal (e.g. music, voice) is embedded into a high frequency radio signal to allow efficient transmission.

While several types of modulation are used for shortwave broadcasts, the most common and easiest to build a receiver for is amplitude modulation, where the audio signal is used to modulate the amplitude of a carrier signal. In the plot below, a low frequency audio signal is multiplied with a high frequency carrier wave. You can see the original audio in the modulated signal by looking at its envelope.

To recover the audio, we use a process called rectification. This means taking the absolute value of the signal. Applying a low pass filter to the output of the rectifier leaves the recovered audio signal, as shown below. Rectification can be performed easily by placing a Schottky diode after the tank circuit. Schottky didoes are preferred due to their small forward voltage drop (~200 mV) compared with regular Si diodes. This means more signal power will reach the speaker instead of being lost as heat in the diode. Following this by a first-order RC low pass filter tuned for 10 kHz (R = 3.3kΩ, C = 4.7nF) will help to filter out the carrier frequency.

### Amplification

The final step is to amplify the signal. Two stages of amplification are required: an initial low-noise preamplifier followed by a LM386 audio amplifier. The first stage is an op-amp configured as a non-inverting amplifier with a potentiometer used for the feedback resistor. A DC blocking capacitor is placed at the input to prevent any DC signal being amplified. The LM386 is configured per the datasheet for maximum gain.

### Putting It All Together

Let’s recap the design:

1. One end of the long-wire antenna is connected to the tank circuit (antenna modeled as a current source in the schematic below)
2. The tank circuit (L1 and C1) is tuned for the frequency of interest
3. D1 rectifies the signal
4. C2 blocks any DC voltage
5. R1 and C3 form a low-pass filter to complete the demodulation process
6. U2, R2 and R3 form a non-inverting amplifier that amplifies the signal to a reasonable level. R3 can be adjusted as needed.
7. The amplified signal is fed into the LM386 which drives an 8-ohm speaker.