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Bandpass Signaling Principles and Circuits Chap. 4
where s(t) is a unipolar switching square wave that has unity peak amplitude. (This is analogous
to the PAM with natural sampling that was studied in Chapter 3.) Using the Fourier series for a
rectangular wave, we find that Eq. (4–65) becomes
v 1 (t) = v in (t)c 1 q
2 + 2 sin (np/2)
a
n=1 np
cos nv 0 t d
(4–66)
The multiplying action is obtained from the n = 1 term, which is
2
p v in(t) cos v 0 t
(4–67)
This term would generate up- and down-conversion signals at f c + f 0 and f c - f 0 if v in (t) were
a bandpass signal with a nonzero spectrum in the vicinity of f = f c . However, Eq. (4–66) shows
that other frequency bands are also present in the output signal, namely, at frequencies f =
1
|f c ± nf 0 |, n = 3, 5, 7, ... and, in addition, there is the feed-through term 2 v in(t) appearing at the
output. Of course, a filter may be used to pass either the up- or down-conversion component
appearing in Eq. (4–66).
Mixers are often classified as being unbalanced, single balanced, or double balanced.
That is, in general, we obtain
v 1 (t) = C 1 v in (t) + C 2 v 0 (t) + C 3 v in (t)v 0 (t) + other terms
(4–68)
at the output of mixer circuits. When C 1 and C 2 are not zero, the mixer is said to be
unbalanced, since v in (t) and v 0 (t) feed through to the output. An unbalanced mixer was illustrated
in Fig. 4–9, in which a nonlinear device was used to obtain mixing action. In the
Taylor’s expansion of the nonlinear device output-to-input characteristics, the linear term
would provide feed-through of both v in (t) and v 0 (t). A single-balanced mixer has feedthrough
for only one of the inputs; that is, either C 1 or C 2 of Eq. (4–68) is zero. An example
of a single-balanced mixer is given in Fig. 4–10, which uses sampling to obtain mixer action.
In this example, Eq. (4–66) demonstrates that v 0 (t) is balanced out (i.e., C 2 = 0) and v in (t)
feeds through with a gain of C 1 = 1 2 . A double-balanced mixer has no feed-through from
either input; that is, both C 1 and C 2 of Eq. (4–68) are zero. One kind of double-balanced mixer
is discussed in the next paragraph.
Figure 4–11a shows the circuit for a double-balanced mixer. This circuit is popular
because it is relatively inexpensive and has excellent performance. The third-order IMD is
typically down at least 50 dB compared with the desired output components. This mixer is
usually designed for source and load impedances of 50 Ω and has broadband input and output
ports. The RF [i.e., v in (t)] port and the LO (local oscillator) port are often usable over a
frequency range of 1,000:1, say, 1 to 1,000 MHz; and the IF (intermediate frequency) output
port, v 1 (t), is typically usable from DC to 600 MHz. The transformers are made by using
small toroidal cores, and the diodes are matched hot carrier diodes. The input signal level at
the RF port is relatively small, usually less than -5 dBm, and the local oscillator level at the
LO port is relatively large, say, +5 dBm. The LO signal is large, and, in effect, turns the diodes
on and off so that the diodes will act as switches. The LO provides the switching control