Davis W.A.Radio frequency circuit design.2001

240 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:
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

SPURIOUS RESPONSE

241

In the first case, the output is near 3fp, and therefore well outside the IF pass band. The second case presents an output frequency slightly above or below the local oscillator frequency, fp, which again is well outside the IF pass band. However, the third-order intermodulation products do present a special problem:

š2f1a Ý 1f1b š fp

š1f1a Ý 2f1b š fp

A numerical example illustrates what occurs with the third-order intermodulation products. If fp D 500 MHz, the desired RF input signal is f1a D 410 MHz, and a second signal of the same amplitude is at f1b D 400 MHz. The first-order products would give the desired output IF frequencies and a high frequency that could be easily filtered out:

jf1a š fpj D 90, 910			MHz
jf1b š fpj D 100, 900			MHz
The third-order intermodulation products would be
j2f1a f1b š fpj D j820	400	š 500j D 80, 920		MHz
j2f1b f1a š fpj D j800	410	š 500j D 110, 890		MHz

As shown if Fig. 11.17, the undesired 80 and 110 MHz third-order intermodulation products could lie inside the IF pass band and thus distort the signal. The surest defense against this is to keep the amplitudes of the third-order intermodulation products small.

The measure of the size of the third-order intermodulation product is the intersection of third-order term with the desired first-order term, f0 D fp f1, (Fig. 11.17). The second-order intermodulation product is a result of having two

					LO
			IF					RF

f 1a –f p	f p		f 1b	f 1a
	f 1b –f p
2f 1a –f 1b–f p	2f 1b –f 1a–f p	2f 1b –f 1a		2f 1a –f 1b

FIGURE 11.17 Third-order intermodulation distortion.

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.

	Conversion	Intercept Point
	Compression
, dBm
out		P in, dBm
P
FIGURE 11.18	Two-tone third-order intermodulation intercept point.

SINGLE-SIDEBAND NOISE FIGURE AND NOISE TEMPERATURE

243

11.8SINGLE-SIDEBAND NOISE FIGURE AND NOISE TEMPERATURE

The frequency independent noise power from a resistor is to a good approximation kT where k is Boltzmann’s constant, and T is the absolute temperature. In the two-port circuit shown in Fig. 11.19, a generator resistance, RG, produces noise with an equivalent noise temperature of TG. The network itself is characterized as having a certain transducer power gain, GT, and noise temperature. When describing the noise temperature of a two-port, it must be decided if the noise is measured at the input or the output. The noise power at the output is presumably

Tout D GTTin

11.32

where Tin is the noise temperature referred to the input port and GT is the transducer power gain. For mixers, this is the conversion gain between the signal and IF ports. In the land where amplifiers are broadband, linear and have wide dynamic range, Eq. (11.32) is accurate. However, low-level random noise voltages may not necessarily be amplified the same way a clean sinusoid would. But to wander from this idealistic world would complicate things beyond their basic usefulness for the present discussion. So the noise power delivered to the load, ZL, is

NL D k GTTG C Tout	11.33
or
TL D GTTG C Tout	11.34
D GT TG C Tin

While the load will generate its own noise, this is defined out of the equation. What is described here is the noise delivered to the load.

The noise figure is sometimes defined in terms of the signal-to-noise ratio at the input to the signal-to-noise ratio at the output of a two-port:

F D

Si/Ni

Si

No

D

Ð

So/No

So

Ni

1 GT TG C Tn

D

GT

Ð

TG

D

1 C

Tn

11.35

TG

T in

T G

G T

T L

FIGURE 11.19 Noise within the circuit is referred to the input side.

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,		. 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

		N SSB		N DSB
	N G	ω 1	N G	ω 1
		ω 0		ω 0
				N DSB
		ω–1		ω–1
	N G	+	N G	+
		ω p		ω p
		–		–
		(a )		(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:

	FSSB D			NL		11.37
				NGGrf
Making the assumption Grf D Gim,
F	SSB D	NSSBGrf C 2GrfNG
				GrfNG
	D	TSSB	C 2			11.38
		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
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.

Radio Frequency Circuit Design. W. Alan Davis, Krishna Agarwal

Copyright 2001 John Wiley & Sons, Inc.

Print ISBN 0-471-35052-4 Electronic ISBN 0-471-20068-9

CHAPTER TWELVE

Phase Lock Loops

12.1INTRODUCTION

Phase lock loops (PLL) are not a recent invention. Their use became widespread with the availability of high-quality integrated circuit operational amplifiers (opamps) in the 1960s. This versatile circuit has found applications across the frequency spectrum in consumer, commercial, deep space, and military projects. Tracking Voyager through the solar system and tuning a car radio are made to order uses for a PLL. To understand a PLL, a good working knowledge of RF techniques, oscillator design, closed loop control theory, analog circuit design, and digital circuit design is required. A comprehension of each of the components and its place in the system is essential. Fortunately not all of this knowledge is required at once. The books listed at the end of this chapter can each provide an in-depth insight into areas beyond the present scope [1–5]. This discussion will begin with the basic concepts and rapidly expand these ideas into practical considerations.

12.2PLL DESIGN BACKGROUND

Discussion of the PLL draws heavily on many other areas of analysis, which includes an understanding of the principles of closed loop control theory. From control theory comes the concept of negative feedback to tailor the performance of closed loop systems. Response time, transient performances, bandwidth, damping ratio, and phase margin are used to describe PLL operation. The type and order of a closed loop system define the complexity and response to a stimulus.

In most PLL’s, at least two of the components, the voltage-controlled oscillator (VCO) and phase detector, are high-frequency components. There may also be amplifiers, mixers, frequency multipliers, and other oscillators. To use these items, a familiarity with RF design practices and terminology is important.

Frequency multiplication may require digital integrated circuits (ICs) within the PLL. These ICs require digital control words to set the desired frequency.

247

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

PLL BASICS

249

decide to rapidly change the correction to tightly match the desired speed, or he may choose to compensate slowly so his speed averages out to the correct value. His actions coupled with the car’s controls form a system closely analogous to a phase lock loop. Replace the human with an electrical circuit that senses the speed error, include another circuit that tempers the response time, and couple it to the accelerator controls. This is the typical cruise control system. The elements of understanding the operation of a phase lock loop are all available here. The next step is to apply the concepts of this example to the classical elements that make up a PLL.

12.4PLL BASICS

A PLL is a closed loop system used for frequency control. Several building blocks are common to most PLL designs:

1.The phase detector

2.The loop filter

3.The voltage-controlled oscillator

Figure 12.1 illustrates the connection of these blocks to make a complete phase lock loop. The phase detector has two inputs and one output. This block can be realized by a specialized mixer described in Chapter 11 where the IF port passband goes down to dc. If the two input signals are very close in frequency, then the output will contain a term at twice the input frequency and a term that is almost zero frequency. The loop error signal in the PLL is the near-zero term. This signal contains everything that is needed to control the VCO. This error signal goes to the loop filter for amplification and frequency limiting. The loop filter may be as simple as a capacitor and a resistor, or it can be one or more operational amplifiers with many resistors and capacitors. The loop filter is generally a simple circuit that requires an in depth analysis. It is the “glue” that holds all the other parts together and makes the PLL work the way it is supposed to work. The VCO is the control element of this loop. The input is a control voltage from the loop filter, while the output is the required frequency.

Phase

Detector

	fref	Loop	VCO	fout
		Filter

FIGURE 12.1 Basic phase lock loop.