A mixer is used for frequency conversion. For example, it is used in radio receivers to convert incoming signals at a high frequency to a lower intermediate frequency. And, in transmitters, it is used to convert low frequencies such as voice to high carrier frequencies.
A linear time invariant network cannot generate new frequencies. For frequency conversion, one needs nonlinearity or time variance. The mixer you'll build here is of the latter variety-i.e. its input signal will experience a time varying gain(controlled by another signal). In this respect, it'll be different from circuits you have studied so far, most of which have been linear and time invariant. Be sure to go through the description and this lecture before the experiment.
A message signal vRF(t) has to be shifted in frequency by an amount fLO while retaining its spectral shape. i.e. VRF(f) has to be turned into VRF(f±fLO). Frequency conversion is accomplished by the mathematical operation vout = k*vRF*vLO where vLO is a sinusoid of frequency fLO. Of the two components at f±fLO, one of them can be selected using a filter to attenuate the other. (The terms vRF and vLO are carried over from radios where they stand for radio frequency input and local oscillator signal respectively)
For practical reasons, switching mixers are used instead of multipliers. i.e. vLO is a square wave of frequency fLO instead of a sinusoid. These are described by vout = k*vRF*sgn(vLO) where sgn() denotes the signum function. sgn(vLO) where vLO is a sinusoid at fLO contains odd harmonics n*fLO, n=1,3,5,… In this case, VRF(f) is turned into VRF(f±n*fLO), n=1,3,5,… As before, the desired component can be selected using a filter to attenuate all but one of them.
The figure below shows a mixer. The combination of Q0 and RE converts the input signal vRF to a signal current iRF riding on a bias current Ibias. The differential pair Q1,2 is driven by vLO which is a square wave or a sufficiently high amplitude sinusoid such that only one of Q1,2 is completely on at a given time and the other is switched off. The current Ibias+iRF is steered to the two load resistors alternately. Consequently, the voltage at the collector will be a certain bias voltage plus k*(Ibias+iRF)*sgn(vLO).
One deviation of this circuit from the description in the previous paragraphs is the presence of Ibias in addition to iRF. Therefore, if vRF=0(i.e. iRF=0), the output will contain a component proportional to sgn(vLO). This also needs to be filtered out if only the sum or difference frequency component is to be extracted. This is known as local oscillator(LO) feedthrough and is a characteristic of this type of mixers which are known as single balanced mixers.