Radio Frequency Integrated Circuit Design - Webs

Mixers
V2 = 1
2 cos [(� LO − � RF )t + � �1] + 1
2 cos [(� IM − � LO )t − � �1]
225
(7.27)
2. Now V1 experiences an amplitude error relative to V2, and V2 experiences
a phase shift that is not exactly 90° to give V3 and V 4, respectively.
V3 = 1
2 (1 + �A) sin (� LO − � 1
RF )t +
2 sin (� IM − � LO )t (7.28)
V4 = 1
2 sin [(� LO − � RF )t + � �1 + � �2] + 1
2 sin [(� IM − � LO )t − � �1 + � �2]
(7.29)
3. Now V3 and V4 are added together. The component of the output
due to the RF signal is denoted V RF and is given by
VRF = 1
2 (1 + �A) sin (� 1
IFt) +
2 sin (� IFt + � �1 + � �2) (7.30)
V RF = 1
2 (1 + �A) sin (� 1
IFt) +
2 sin (� IFt) cos (� �1 + � �2) (7.31)
+ 1
2 cos (� IFt) sin (� �1 + � �2)
4. The component due to the image is denoted V IM and is given by
V IM =− 1
2 (1 + �A) sin (� 1
IFt) +
2 sin (� IFt) cos (� �2 − � �1)
+ 1
2 cos (� IFt) sin (� �1 − � �2) (7.32)
5. Only the ratio of the magnitudes is important. The magnitudes are
given by
|VRF| 2 = 1
4 {[sin (� �1 + � �2)] 2 + [(1 + �A) + cos (� �1 + � �2)] 2 }
(7.33)