332 Week 9: Alternating Current Circuits
-1 0 2 4 6 8 10-0.500.51tcarrier(^00246810)
1
2
3
4
t
signal
-4 0 2 4 6 8 10
-2
0
2
4
t
AM carrier
Figure 134: (a) The unencoded carrier with an arbitrary normalization voltageVc= 1 volt and
angular frequencyω 0. (b) The signal to be encoded. ADC biashas been added to the AM signal
so that the voltage is always positive. This DC bias can be removed at the far end with a simple
high-pass filter; (c) The AM encoded carrier used as (for example)the power supply to the antenna
of a radio station. Note that for real AM signals the carrier frequency is much higher compared to
the highest frequencies in the signal, which improves the averaging that takes place in the decoding
rectifier.
motors, many things create radio waves) at various frequencies.
To tune injustthe carrier (plus enough bandwidth to allow its amplitude modulation tomake
it through the receiver circuit) we build a circuit that effectivelyshorts outall of the signals but the
desired carrier atω 0 by providing them with an easy path to ground through either an inductor (for
lower frequencies) or a capacitor (for higher frequencies). Thesimplestcircuit that accomplishes
this is our parallelLRCcircuit above.
However, we have to add two features in order to make it a tunable AM radio. First is a way
to tune it! We note that we do the best possible job of filtering out unwanted frequencies when the
conditionω 02 = 1/LCand whenR=r, so our receiver resistance/impedance matches the internal
resistance of the voltage source. We therefore have to be able toadjustL,C, or both in order to
tune in our AM encoded carrier.
It is beyond our scope in this work to discuss all the various aspectsof this decision. The
antenna, diode (crystal), headphones or amplifier input all have some impedance – characteristics
of resistance, inductance and capacitance – and have to be corrected for. Also, we need to be able
to tune theQof the circuit so that the receiver bandwidth is adequate to pick up all of the encoded
signal while still being narrow enough to reject nearby AM encoded stations. Many simple crystal
radio designs that use wire wrapped around e.g. a simple tube of somesort allow one to varyL
across a range (which adjustsω 0 andQsimultaneously) – this is especially wise if one’s headphones
and/or antenna have enough capacitance already to make it difficultto add a tuning capacitor “in
range” to permit tuning. Others use fixedL(and hence fixedQ) and a variable capacitor to tune.
Still others may do both – allow one to varyL(possibly to one of a small set of discrete values) and
then use a continuously tunableCto find the signal.
In an idealized circuit for the simplest of crystal radios in figure 135,I arbitrarily show a variable
C(that’s the arrow symbol) and also introduce the symbol for an antenna and ground. The resistance