SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

190 CHAPTER 4. JUNCTIONS IN SEMICONDUCTORS:P-NDIODES

A

B

(Signal) AB

(Local Oscillator)

(Output to load)

Figure 4.27: Mixer symbol. A represents the signal input; B the reference input.

input sinusoidal signals as shown in equation 4.8.12. One of these is the input signal (A) whose
amplitude and phase generally vary with time. The other input (B) is a reference signal, locally
generated, called the local oscillator, normally with fixed amplitude and phase. With the ideal
analog multiplication process shown in figure 4.27, no harmonics or spurious signals are pro-
duced. Also, there is no feed through of A or B to the output. But, in reality, mixers always
produce many spurious outputs that consist of harmonics of A and B and additional mixing
productsmω 1 ± nω 2 ,wheremandnare integers. A “good” mixer is designed such that it
suppresses these spurious outputs and provides a highly linear amplitude and phase relationship
between signal input (A) and the output.
The forward I-V characteristic of the diode can be represented by a series expansion. For
example, in the case of a simple exponential diode characteristic, equation 4.8.13 can represent
the current voltage characteristic. Coefficients aiwill vary with DC bias, series resistance, and
the shape of theI−Vcharacteristic.

ID=IS

[

exp

(

eVD

kBT)

)

− 1

]

IS

[

a 1 VD+

1

2

a 2 VD^2 +

1

6

a 3 VD^3 +...

]

(4.8.13)

VRF+VLO−VD−IS

(

a 1 VD+

1

2

a 2 Vo^2

)

(RS+RL)=0

Vo(t)=IS(a 1 VD+

1

2

a 2 VD^2 )RL

a 2 VD^2 sin^2 ωt=a 2 VD^2 [1−cos 2ωt] (4.8.14)

Now, suppose that two inputs are summed as shown in figure 4.27 and the diode current produces
an output Vo(t) across resistor RL. One input VRFis the signal; the other VLO is the reference
local oscillator. The diode voltage, VD, can be found using the series approximation equation
4.8.13, and the output voltage, Vo(t), is calculated from the diode current ID. If only first and
second order terms are used, a quadratic equation is easily solved.
While only the outputs shown in equation 4.8.12 are desired, the mixer output will also contain
a DC term, RF and LO feed through, and terms at all harmonics of the RF and LO frequencies.
Only the second-order product term produces the desired outputs. It can be seen in equation