Davis W.A.Radio frequency circuit design.2001
.pdf240 RF MIXERS
and the upper ones [7]. Distortion products produced in Q5 and Q6 are thus filtered out before the RF signal reaches the transistors being switched by the LO. A 20 dB improvement in dynamic range over the conventional Gilbert cell is reported using this filtering technique.
11.7SPURIOUS RESPONSE
The previous sections considered some representative mixer circuits. Here some of the primary mixer performance criteria for mixers are described. The first of these are the spurious frequencies generated when the mixer is excited by a single tone RF signal. A second measurement of mixer performance results from exciting it with two tones near to each other that produces two IF terms. The latter is termed two-tone intermodulation distortion.
Single-tone intermodulation is an effect of the imbalance in the transformers or the diodes used in the mixer. A distinction is made between the inherent nonlinear current–voltage curve of a diode and the nonlinearity associated with the switching action of the diode [8]. Fitting a polynomial function to an ideal diode characteristic whose current is zero when off, and whose i–V slope is a straight line when the diode is on, would yield a polynomial fitting function with many powers of the independent variable. Indeed, the switching of the diodes appears to be the predominant effect in a mixer. Analytical estimates of intermodulation distortion suppression can be made solely on the basis of the switching action of the diodes in the mixer rather than on any curvature of individual diode curves. Such an expression is presented in Appendix H. That equation has also been coded in the program IMSUP as described in Appendix H. Basically the intermodulation suppression in dBc (dB below the carrier) is Snm for a set of frequencies nfp š mf1.
Two-tone intermodulation distortion is best explained by following a simple experimental procedure. Normally one RF signal excites the RF port of the mixer, which then produces the IF output frequency along with various higher-order terms that can be easily filtered out of the IF circuit. Now consider exciting the RF port of the mixer with two RF signals, f1a and f1b, spaced close together, which thus lie within the pass band of the mixer input. The nonlinear mixer
circuit will then produce the following frequencies: |
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šm1f1a š m2f1b š nfp |
11.31 |
The order of the mixing product is m1 C m2. It would be nice if the IF output were only jf1a fpj and jf1b fpj, since that would represent the down-converted signal to the IF output. Those terms containing harmonics of fp would be far outside the band of interest and could be filtered out. There are essentially two possibilities for the second-order intermodulation products:
š1f1a š 1f1b š fp
š1f1a Ý 1f1b š fp
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.
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Conversion |
Intercept Point |
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Compression |
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, dBm |
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out |
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P in, dBm |
P |
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FIGURE 11.18 |
Two-tone third-order intermodulation intercept point. |
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
temperature, |
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. Thus the noise characteristics of a two-port |
TG D 290°K D T0 |
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:
NL D NSSB C NG Grf C NGGim |
11.36 |
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
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N SSB |
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N DSB |
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N G |
ω 1 |
N G |
ω 1 |
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ω 0 |
ω 0 |
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N DSB |
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ω–1 |
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ω–1 |
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N G |
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N G |
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ω p |
ω p |
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(b ) |
FIGURE 11.20 Mixer noise specification using (a) single-sideband noise, and (b) doublesideband noise.
SINGLE-SIDEBAND NOISE FIGURE AND NOISE TEMPERATURE |
245 |
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:
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FSSB D |
NL |
11.37 |
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NGGrf |
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Making the assumption Grf D Gim, |
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F |
SSB D |
NSSBGrf C 2GrfNG |
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GrfNG |
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D |
TSSB |
C 2 |
11.38 |
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T0 |
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 total power delivered to the load is |
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NL D NG C NDSB Grf C Gim |
11.39 |
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:
NL |
11.40 |
FDSB D Grf C Gim NG |
Substituting Eq. (11.39) into Eq. (11.40) and again assuming that Grf D Gim,
FDSB D |
TDSB |
C 1 |
11.41 |
T0 |
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,
TSSB D 2TDSB |
11.42 |
246 RF MIXERS
so that
FSSB D |
TSSB |
C 2 D |
2TDSB |
C 2 D 2FDSB |
11.43 |
T0 |
T0 |
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.
248 PHASE LOCK LOOPS
Many integrated circuits are presently available that combine many of the PLL functions on a single chip. Most of the interface control is digital.
Analog circuit design is perhaps the most demanding of the circuit areas within a PLL. Op-amps are used in many of the filtering circuits used within a loop. Inverting and noninverting circuits are required for loop filters and search circuits. Integrators, dc amplifiers, Schmitt triggers, and offset circuits are used to set the loop operation. Resistor/capacitor circuits provide phase shift for stability. The oscillator is an intrinsic part of a PLL, and its design in itself is a specialized and technically challenging area.
12.3PLL APPLICATIONS
A phase lock loop is a frequency domain device that can be used to multiply, divide, or filter different frequencies. Consider a space probe rapidly moving away from the earth. To recover data from the probe, the transmitter frequency must be known. The signal is very weak because of the distance, and the low signal-to-noise ratio requires a very small receiver filter bandwidth to recover the data. However, because of the relative motion, there is a significant and changing Doppler shift to the transmit frequency. The system requires a filter that may be only a few Hertz wide operating at a varying frequency that is centered at several GHz.
An electronic phase lock loop is one form of a closed loop system. The cruise control is another. A switching power supply, a camera’s light meter, a radio’s automatic gain control, the temperature control in a building, a car’s emission system controls, and a Touch-Tone dialing system are examples of closed loop systems. A broadcast receiver changes frequency with a button push or electronically. Each time the station is accurately centered with no manual adjustment required. Physically these PLLs are all very different working at different jobs and in different environments. However, they all must follow the same rules, and the loops must all be stable.
A clear understanding of the concept of feedback control is illustrated by an everyday situation of the simple action of controlling the speed of a car. If the desired speed is 60 mph, then this becomes the reference. Any deviation from this speed is an error. The accelerator pedal is the control element. On level terrain, a constant pressure on the pedal will maintain constant speed. As the car goes up a hill, it will slow down. The difference between the actual speed and the reference value generates an error. This error generates a command to push the accelerator pedal. Pushing the pedal will increase the speed, but there will continue to be a slight error. As the car crests the hill and starts down, the speed will increase. Releasing pedal pressure will slow the acceleration, but an error will remain until a steady state condition is again reached. For this example, the driver’s brain is the feedback path. The driver controls the sense of the feedback by knowing when to push and when to release the pedal. By his reaction time, he controls how close to the reference he maintains the car’s speed. He may