11.RF mixers

242 RF MIXERS

RF signals that are multiplied together because of a quadratic nonlinearity:

[A cos ω1at Ð B cos ω1bt] cos ωpt

The resulting amplitude proportional to AB will increase 2 dB when A and B each increase by 1 dB. The third-order intermodulation product is a result of a

cubic nonlinearity:

[A2 cos2 ω1at Ð B cos ω1bt] cos ωpt

The resulting amplitude proportional to A2B will increase by 3 dB for every 1 dB rise in A and B. Thus, when the RF signal rises by 1 dB, the desired IF term will rise by 1 dB, but the undesired third-order intermodulation term rises by 3 dB (Fig. 11.18). The interception of the extrapolation of these two lines in the output power relative to the input power coordinates is called the third-order intercept point. The input power level where this intersection occurs is called the input intercept point. The actual third-order intermodulation point cannot be directly measured, since that point must be found by extrapolation from lowerpower levels. It nevertheless can give a single-valued criterion for determining the upper end of the dynamic range of a mixer (or power amplifier). The conversion compression on the desired output curve is the point where the desired IF output drops by 1 dB below the linear extrapolation of the low level values.

The range of mixer LO frequencies and RF signal frequencies should be chosen so as to reduce to a minimum the possibility of producing intermodulation products that will end up in the IF bandwidth. When dealing with multiple bands of frequencies, keeping track of all the possibilities that may cause problems is often done with the aid of computer software. Such programs are available free of charge off the internet, and other programs that are not so free.

244 RF MIXERS

The noise figure depends on the temperature of the generator. This ambiguity in noise figure is removed by choosing by convention that the generator is at room

such as a mixer (the LO port being conceptually ignored) can be characterized with either noise figure or noise temperature. Because of the greater expansion of the temperature scale over that of noise figure in dB, noise temperature is preferred when describing very low noise systems and noise figure for highernoise systems. However, the concept of noise temperature becomes increasingly convenient when describing mixers with their multiple frequency bands.

The noise figure of a mixer can be described in terms of single-sideband (SSB) noise figure or double-sideband (DSB) noise figure. If the IF term, ω0 in Fig. 11.2 comes solely from the signal ω1 and the image frequency ω 1 is entirely noise free, then the system is described in terms of its single-sideband noise figure, FSSB (Fig. 11.20a). Double-sideband noise figure comes from considering both the noise contributions of the signal and the image frequencies (Fig. 11.20b). In general, the output noise of the mixer will be the sum of the noise generated within the mixer itself and the noise power coming into the mixer multiplied by the mixer conversion gain. The noise power from inside the mixer itself can be referred to either the output port or the input port as described by Eq. (11.32). If all the internal mixer noise is referred back to the input RF signal port, then this will designated as NSSB. The total noise power delivered to the load is found by multiplying NSSB by the RF port conversion gain, Grf, and adding to this the power entering from the signal source, NG, at both the RF signal and image frequencies:

The gains at the RF signal and image frequencies, Grf and Gim, are typically very close to being the same since these two frequencies are close together. The terms in this definition are readily measurable, but Eq. (11.36) is at variance with the

FIGURE 11.20 Mixer noise specification using (a) single-sideband noise, and (b) doublesideband noise.

way the IEEE standards define single-sideband noise figure. For further discussion on this point, see [3]. The single-sideband noise figure is conventionally defined as the ratio of the total noise power delivered to the load to the noise power entering at the RF signal frequency from a generator whose temperature is T0 and when the mixer itself is considered to be noise free:

Since NSSB is referred to the mixer input, so its associated noise temperature, TSSB, is also referred to the input side.

If the internal mixer noise power is referred back to both the RF frequency band and the image frequency band, then this power will be designated as the double-sideband power, NDSB. For the double-sideband analysis, both the RF signal and image frequencies are considered as inputs to the mixer. In this case

The double-sideband noise figure is determined by taking the ratio of the power delivered to the load and the power from both of these frequency bands if the mixer were considered noise free:

Substituting Eq. (11.39) into Eq. (11.40) and again assuming that Grf D Gim,

In the single-sideband case, all mixer noise power is referred to the mixer input at the RF signal frequency. In the double-sideband case, all the mixer noise is referred to the mixer input at both the RF signal and image frequencies. Since the internal mixer power is split between the two frequency bands,

246 RF MIXERS

so that

This illustrates the of-stated difference between singleand double-sideband noise figures. Noise figure specification of a mixer should always state which of these is being used.

PROBLEMS

11.1Using the Fourier transform pair, show that F e jωa D 2 υ ω ωa .

11.2Two closely separated frequencies are delivered to the input signal port of a mixer of a receiver. The center frequency of the receiver is 400 MHz, and the two input frequencies are at 399.5 and 400.5 MHz. The mixer has a conversion loss of 6 dB and the local oscillator is at 350 MHz. The power level of these two input frequencies is 14 dBm (dB below a milliwatt). At this input power, the third-order modulation products are at 70 dBm.

(a)What are the numerical values for the output frequencies of most concern to the receiver designer?

(b)What is the output third-order intercept point?

REFERENCES

1.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Wiley, eq. 1972, 9.6.33–9.6.35.

2.H. L. Krauss, C. W. Bostian, and F. H. Raab, Solid State Radio Engineering, New York: Wiley, 1980.

3.S. A. Maas, Microwave Mixers, 2nd ed., Norwood, MA: Artech House, 1993.

4.G. D. Vendelin, A. M. Pavio, and U. L. Rohde, Microwave Circuit Design, New York: Wiley , 1990, ch. 7.

5.Microwave Designer’s Handbook, Watkins-Johnson Co., 1997–98 Catalog.

6.B. Gilbert, “A Precise Four-Quadrant Multiplier with Subnanosecond Response,” IEEE J. Solid State Circuits, pp. 365–373, 1968.

7.J. M. Moniz and B. Maoz, “Improving the Dynamic Range of Si MMIC Gilbert Cell Mixers for Homodyne Receivers,” IEEE 1994 Microwave and Millimeter-Wave Monolithic Circuits Symp., pp. 103–106, 1994.

8.D. G. Tucker, “Intermodulation Distortion in Rectifier Modulators,” Wireless Engineer, pp. 145–152, 1954.