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Fundamentals of RF Circuit Design with Low Noise Oscillators. Jeremy Everard Copyright © 2001 John Wiley & Sons Ltd
ISBNs: 0-471-49793-2 (Hardback); 0-470-84175-3 (Electronic)
5
Mixers
5.1 Introduction
Mixers are used to translate a signal spectrum from one frequency to another. Most modern RF/microwave transmitters, receivers and instruments require many of these devices for this frequency translation. The typical symbol for a mixer is shown in Figure 5.1.
Figure 5.1 Typical symbol for a mixer
An ideal mixer should multiply the RF and LO signals to produce the IF signal. It should therefore translate the input spectrum from one frequency to another with no distortion and no degradation in noise performance. Most of these requirements can be met by the perfect multiplication of two signals as illustrated in equation (5.1):
V1 cosω1t)V2 cosω2t)= V1 2 (cos( 1 +ω2 ) + cos ω1 −ω2 t) (5.1)
Here it can be seen that the output product of two input frequencies consists of the sum and difference frequencies. The unwanted sideband is usually fairly easy to remove by filtering. Note that no other frequency terms other than these two are
236 Fundamentals of RF Circuit Design
generated. In real mixers there are a number of compromises to be made and these will be discussed later.
Mixing is often achieved by applying the two signals to a non linear device as shown in Figure 5.2.
Vi = V1 + V2
Figure 5.2 Mixing using a non-linear device
The non linearity can be expressed as a Taylor series:
Iout = I0 + aVi (t)] bVi (t)]2 +cVi (t)]3 +..... (5.2)
Taking the squared term:
bV1 +V2 ) =bV12 + 2V1V2 +V22 ) (5.3)
It can immediately be seen that the square law term includes a product term and therefore this can be used for mixing. This is illustrated in Figure 5.3 where the square law term and the exponential term of the diode characteristics are shown.
Figure 5.3 Diode characteristic showing exponential and square law terms
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Note of course that there are other terms in the equation which will produce unwanted frequency products, many of them being in band. Further, as the signal voltages are increased the difference between the two curves increases showing that there will be increasing power in these other unwanted terms.To achieve this non-linear function a diode can be used as shown in Figure 5.4:
Figure 5.4 Diode operating as a non-linear device
If two small signals are applied then multiplication will occur with rather high conversion loss. The load resistor could also include filtering. If the LO is large enough to forward-bias the diode then it will act as a switch. This is a single ended mixer which produces the wanted signal and both LO and RF breakthrough. It can be extended to a single balanced switching action as shown in figure 5.5.
5.2 Single balanced mixer (SBM)
Figure 5.5 Switching single balanced mixer
The waveform and therefore operation of this switching mixer are now shown to illustrate this slightly different form of operation which is the mode of operation
238 Fundamentals of RF Circuit Design
used in many single balanced transistor and diode mixers. The LO switching waveform has a response as shown in Figure 5.6.
Figure 5.6 LO waveform for SBM
The spectrum of this is shown in equation (5.4) and consists of a DC term and the odd harmonics whose amplitude decreases proportional to 1/n.
1 ¥ sin(nπ /2)
2 n=1 nπ /2 0
(5.4)
If this LO signal switches the RF signal shown in figure 5.7 then the waveform shown in Figure 5.8 is produced.
Figure 5.7 RF signal for DBM
Figure 5.8 Output waveform for SBM
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The spectrum of this can be seen to produce the multiplication of the LO (including the odd harmonics, with the RF signal. This produces the sum and difference frequencies required as well as the sum and difference frequencies with each of the odd harmonics. The output voltage is therefore:
V0 t)= VRF t)´S t)
= VRF cosωRF t.1 + n=1 sin π / /2)cos(nωLO t) (5.5)
It is important to note that there is no LO component. There is however an RF term due to the product of VRF with the DC component of the switching term. This shows the properties of an SBM in that the LO term is rejected. It is often useful to suppress both the LO and RF signal and therefore the DBM was developed.
5.3 Double Balanced Mixer (DBM)
If the switch is now fed with the RF signal for half the cycle and an inverted RF signal for the other half then a double balanced mixer (DBM) is produced. This is most easily illustrated in Figure 5.9.
Figure 5.9 Switching double balanced mixer
The output voltage is given by:
V0 t) = 2VRF cosωRF t ¥1 sin(n / /2)cos(nωLOt (5.6)
The waveform is shown in Figure 5.10.
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