# Design and implementation of a novel frequency modulation circuit using a phase-locked loop synthesizer.

A phase-locked loop (PLL) is widely used in frequency synthesizers
for modern communication systems. However, frequency modulation (FM) of
a PLL for the modulation rates used inside the loop bandwidth, is
generally difficult. In this article, a novel but simple FM circuit of a
PLL is described, based on the compensation of the amount of feedback of
the PLL. The designed circuit was successfully implemented and showed a
flat frequency deviation as expected in the design. In addition, a novel
method for the measurement of the FM modulation index with a spectrum
analyzer is suggested and used to verify the proposed circuit.

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A phase-locked loop (PLL) is a popular method used in frequency synthesizers for most communication systems. (10) The block diagram of a common PLL synthesizer is shown in Figure 1 and includes a charge pump phase detector (PD), a temperature-compensated crystal oscillator (TCXO), a voltage-controlled oscillator (VCO), an R divider and an N divider. [K.sub.v] is the VCO gain constant (MHz/V), [K.sub.p] is the PD gain constant (mA/rad) and Z(s) represents the loop filter transfer function. The R divider divides down the TCXO frequency to form a channel and the N divider divides down the VCO frequency. The PD compares the two divided-down signals and generates current pulses proportional to the phase difference. The PD's output current pulses are filtered by the loop filter and a slow varying voltage tunes the VCO frequency to ultimately drive the PD's average output to zero.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

A PLL frequency synthesizer can be simply frequency modulated by injecting the modulation signal [V.sub.m] into the tuning input of the VCO. However, the modulation response for [V.sub.m] has a high pass characteristic as

[[omega].sub.o]/[V.sub.m] = [K.sub.v]/[1 + [[[K.sub.p][K.sub.v]Z(s)]/[sN]]] (1)

From Equation 1, it can be found that for modulation rates greater than the PLL loop bandwidth the VCO can be modulated successfully. However, for rates less than the loop bandwidth, there is no modulated output. Narrowing the loop bandwidth will result in a longer lock time that the system may not tolerate. This article reviews some of the techniques used to overcome this problem and introduces a practical low cost modulation circuit.

FREQUENCY MODULATION USING A PLL

Figure 2 shows methods used by others to circumvent this problem. In the method using a phase modulator, (2) the signal [V.sub.m1] injected into the VCO generates the high pass modulated signal given in Equation 1 and the [V.sub.m2] injected into the phase modulator through the integrator yields the low pass modulated signal as

[[omega].sub.o]/[V.sub.m2] = [[[K.sub.i][K.sub.p][K.sub.v]Z(s)]/s]/[1 + [[[K.sub.p][K.sub.v]Z(s)]/[sN]]] (2)

If [V.sub.m] = [V.sub.m1] + [V.sub.m2] (the modulation signal [V.sub.m] is split into two paths as [V.sub.m1] and [V.sub.m2]) and [K.sub.i] = [K.sub.v]/N, the VCO output for [V.sub.m] is the sum of Equations 1 and 2 and results in a flat peak frequency deviation [DELTA]f, independent of the modulation rates. In the method using a VCXO, (3) for a low frequency modulation, [V.sub.m2], split from [V.sub.m], is injected to the voltage-controlled crystal oscillator (VCXO) instead of the TCXO. A flat [DELTA]f, independent of the modulation rates, can be obtained if

[K.sub.ref] = [R/N][K.sub.v]

Although the previous two-point modulation methods (one injection point is the VCO tuning input for high pass characteristic modulation and the other is the potential point providing a low pass characteristic modulation) provide a modulation response extending down to DC, they require specific highly priced components such as a phase modulator and VCXO.

The block diagram of an FM PLL using compensation of the amount of feedback is also shown. The amount of feedback in the denominator of Equation 1 is compensated by the modulation block D. For complete compensation, the modulation block should have a transfer function given by

[FIGURE 3 OMITTED]

D = [V.sub.o]/[V.sub.m] = [[[DELTA]f]/[K.sub.v]](1 + [[[K.sub.p][K.sub.v]Z(s)]/[sN]]) (3)

By multiplying Equations 1 and 2, the VCO may then generate an FM modulated signal with a desired peak frequency deviation [DELTA]f.

The modulation block can be implemented by various circuits for the given PLL loop filter. For the second-order charge pump loop filter Z(s) of [R.sub.1], [C.sub.1] and [C.sub.2], Rosemarin (4) has suggested the circuit shown in Figure 3.

[FIGURE 4 OMITTED]

In this circuit, the output voltage across [C.sub.b] will be developed by two currents, [i.sub.R] and [i.sub.C]. Assuming that R is large enough (R/2 [much greater than] |Z(s)|) to act as a current source and using superposition, [V.sub.o] can then be approximately written as

[V.sub.o] = [[C.sub.a]/[[C.sub.a] + [C.sub.b]]][V.sub.m] + [[Z(s)]/[s([C.sub.a] + [C.sub.b])[R.sup.2]]][V.sub.m] (4)

From R/2 [much greater than] |Z(s)|, the values of R can be consequently determined by

R = 20/[2[pi][f.sub.min]([C.sub.1] + [C.sub.2])] (5)

where

[f.sub.min] = minimum frequency of the modulation signal

Comparing Equation 3 with Equation 4, [C.sub.a] should satisfy the relation

[[K.sub.p][K.sub.v]]/N = 1/[[C.sub.a][R.sup.2]] (6)

Finally, [C.sub.b] can be determined to give the desired [DELTA]f as

[DELTA]f = [[K.sub.v][C.sub.a]]/[[C.sub.a] + [C.sub.b]] (7)

Although the above circuit is designed for a one-port, second-order charge pump loop filter, the circuit can be applied successfully for a two-port higher order loop filter frequently employed for the sufficient suppression of spurious. However, the circuit is easily influenced by the load (load impedance is open in the derivation). As a result, the proper separation or the compensation of the loading effect is required in the application.

Figure 4 shows the proposed overall FM PLL synthesizer including the modulation block determined as in the previous case. [R.sub.c] and [C.sub.c] are inserted for the compensation of the PLL loop filter loading. The [C.sub.c] capacitor also plays a role in blocking the leakage of the DC voltage at the VCO input to the modulation block. The third-order loop filter is chosen for the sufficient attenuation of the spurious. [R.sub.4] with the VCO input parasitic capacitance [C.sub.v] = 550 pF may increase the order of the loop filter; however, its time constant is selected far above the channel frequency to yield a negligible effect on the loop filter. [R.sub.4] is inserted to prevent the direct loading of [C.sub.v] to the designed [C.sub.3].

[FIGURE 5 OMITTED]

The FM PLL synthesizer has a center frequency of 164 MHz, a channel spacing of 12.5 kHz and the desired peak frequency deviation [DELTA]f of 2.4 kHz. The VCO and PD gain constants are [K.sub.v] = 17 MHz/V and [K.sub.p] = 5 mA/rad, respectively. The third-order loop filter is calculated by following Baberjee's method (5) with a phase margin of 45[degrees] and a loop bandwidth of 1 kHz, taking the lock time specification into consideration. The calculated element values are [C.sub.1] = 68nF, [R.sub.2] = 1.2 k[ohm], [C.sub.2] = 680 nF, [R.sub.3] = 33 k[ohm], [C.sub.4] = 560 pF and [R.sub.4] = 1.8 k[ohm].

The element values of the modulation block are [C.sub.a] = 3.75 pF, [C.sub.b] = 6.8 nF and R = 200 k[ohm].

The computed transfer function of the modulation block is actually affected by the connected loop filter. In order to determine [R.sub.C] and [C.sub.C], an approximated FM circuit is drawn in Figure 5, assuming modulation rates lower than the channel frequency. The output of the modulation block is denoted by [V.sub.i]. The transfer function then becomes

[V.sub.i]/[V.sub.o] = [Z(s) + [R.sub.3] + [R.sub.4] + [R.sub.C] + [1/[s[C.sub.C]]]]/[Z(s) + [R.sub.3] + [R.sub.4]] (8)

The second-order loop filter impedance Z(s) can be approximated as the parallel connection of [C.sub.1] and [C.sub.2] at a low frequency. In that case, Equation 8 becomes frequency independent of [R.sub.C] and [C.sub.C] if they are selected to satisfy the relation

1 + [[R.sub.C]/[[R.sub.3] + [R.sub.4]]] = 1 + [[[C.sub.1] + [C.sub.2]]/[C.sub.C]] (9)

The modulation block output is then scaled down by Equation 8 and transferred to the VCO without distorting its frequency response. However, Equation 8 actually shows weak frequency dependence to some degree. To get the optimum [R.sub.c] and [C.sub.c] from the many possible choices given by Equation 9, the circuit in Figure 4 was simulated and tuned using ADS for a flat peak frequency deviation. The best performance is obtained at [C.sub.c] = 680 nF and [R.sub.c] = 34.8 k[ohm].

To investigate the loading effect, the higher order loop filter elements ([R.sub.3], [C.sub.3] and [R.sub.4]) and the coupling circuit ([R.sub.4] and [C.sub.c]) are removed. The values of the remaining second-order loop filter are slightly adjusted for the phase margin of 45[degrees] and the loop bandwidth of 1 kHz. The modulation block is also tuned again according to the second-order loop filter change. The modulation signal, with a peak-to-peak voltage of 1 V, is applied through the corresponding second-order modulation circuit. Figure 6 shows the spectra of the second-order FM PLL synthesizer for a modulation signal of 4.8 kHz (modulation index [beta] = 0.5) and of 400 Hz ([beta] = 6).

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

At 4.8 kHz, the spurious is found at the channel frequency of 12.5 kHz, resulting from an insufficient attenuation. For 400 Hz, the un-suppressed spurious of 12.5 kHz grows bigger and is also modulated. It should be noted that [DELTA]f is decreased. The modulated spectrum of the desired [DELTA]f, by Carson's rule, has a 99 percent BW of 5.6 kHz (BW[approximately]([DELTA]f + [f.sub.m]) = 2(2.4 + 0.4) = 5.6 kHz. (6) But the estimated BW from the modulated spectrum is approximately 4.8 kHz < 5.6 kHz. The decrease of [DELTA]f at a low modulation frequency occurs because the effective [C.sub.b] is increased by the loading of the parallel connection of [C.sub.1] and [C.sub.2], which results in the decrease of [DELTA]f according to Equation 7.

Figure 7 shows the measured results of the third-order FM PLL circuit. Comparing these results with the last case, the spurious is suppressed sufficiently and does not show in the spectrum. The bandwidth of 5.6 kHz for the desired peak frequency deviation is estimated in the spectrum.

NOVEL [DELTA]f MEASUREMENT WITH A SPECTRUM ANALYZER

The FM modulated carrier [x.sub.c](t) by a sinusoidal signal of [[omega].sub.m] = 2[pi][f.sub.m] can be expressed as

[x.sub.c](t) = [A.sub.c][[infinity].summation over (n = -[infinity])][J.sub.n]([beta])cos([[omega].sub.c]t + n[[omega].sub.m]t) (10)

Here, [J.sub.n]([beta]) represents an n-th order Bessel function of the first kind. Based on Equation 10, the modulation index ([beta]) can be measured with a spectrum analyzer by three known methods. The first method is to change the modulation rate and to find the rate which yields the first null of the carrier amplitude [J.sub.o]([beta]). From this, the value of [DELTA]f can be found using the modulation rate and the known first zero of [J.sub.o]([beta]). However, for the method of the carrier zeros, the modulation index should be independent of the modulation rate. The method of using the 99 percent power bandwidth is useful to estimate [DELTA]f for a wide band FM signal as shown in the previous section. It is not adequate, however, to determine the precise value of [DELTA]f. In order to measure the [DELTA]f for the modulation rate change, a useful method is based on the recurrence formulas for Bessel function as (7)

[beta] = [2[J.sub.1]([beta])]/[[J.sub.0]([beta]) + [J.sub.2]([beta])] (11)

The application of Equation 11 is successful for narrowband FM, considering that the three tones in Equation 10 are dominant. However, the error in the measurement by three tones becomes critical in the determination of the modulation index, especially in wideband FM, although Equation 11 is an exact one. To show this, the ratio [[beta].sub.m,2]/[beta] is plotted in Figure 8 assuming a random measurement error of 0.1 dB of the spectrum analyzer 8564E. The [[beta].sub.m,2] represents the maximum and minimum of the measured [beta] determined by Equation 11 with the random measurement error of the spectrum analyzer. The error becomes extremely large near the zeroes of [J.sub.1]([beta]) because both the denominator and nominator in Equation 11 approach zero simultaneously.

To reduce the errors, Equation 11 is modified to contain a large number of tones in the determination of the modulation index. Extending the recurrence formulas up to n+1 tones,

[beta]([J.sub.2]([beta]) + [J.sub.o]([beta])) = 2 * 1[J.sub.1]([beta])

[beta]([J.sub.3]([beta]) + [J.sub.1]([beta])) = 2 * 2[J.sub.2]([beta])

...

[beta]([J.sub.n]([beta]) + [J.sub.n-2]([beta])) = 2 * (n - 1)[J.sub.n-1]([beta])

[beta]([J.sub.n+1]([beta]) + [J.sub.n-1]([beta])) = 2 * n[J.sub.n]([beta]) (12)

Summing up the left and the right sides of Equation 12, [beta] can be written as

[beta] = [2([J.sub.1] + 2[J.sub.2] + ... + n[J.sub.n])]/[[J.sub.0] + [J.sub.1] + 2([J.sub.3] + [J.sub.4] + ... [J.sub.n-1]) + [J.sub.n] + [J.sub.n+1]] (13)

The [[beta].sub.m,30], that is the modulation index determined by Equation 13 using 30 tone measurements, is also plotted and shows an almost flat error and does not show the error peaks compared to [[beta].sub.m,2]. From these results, the number of the measured tones is increased according to [beta] for the proposed FM PLL measurement.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Figure 9 shows the simulated and measured [DELTA]f with the suggested method for the proposed FM modulator for the different modulation rates. The results show that a flat frequency deviation, down to 200 Hz, is obtained and shows good agreement with the design. Below 200 Hz, the frequency of the function generator used as the modulation source is unstable, and reliable measurements cannot be obtained.

CONCLUSION

In this article, a low cost and practical FM PLL synthesizer circuit for a flat peak frequency deviation is suggested based on the amount of feedback compensation. The modulation rate can be extended down to below the PLL loop bandwidth. Rosemarin's amount of feedback compensation circuit is modified to include a two-port loop filter and consequently suppress the spurious sufficiently. The influence of the loop filter deteriorates the modulation circuit and the interaction should also be compensated. An approximate compensation of the influence is suggested. The possible error in the modulation index measurement is reduced by including a sufficient number of tones. With the suggested measurement, it can be successfully verified that the designed circuit has the expected frequency deviation.

ACKNOWLEDGMENT

This work has been supported by the MINT (Millimeter-wave Innovation Technology) Research Center of KOSEF.

References

1. L. Rohde, Microwave and Wireless Synthesizers, Wiley-Interscience, Somerset, NJ, 1997, pp. 1-78.

2. S. Grimmett, "Frequency Modulation in a Phase-lock Loop by Control of the Phase Inside the Loop," RF Design, June 1991, pp. 31-38.

3. R. Hunter and F. Kostedt, "Using Tow-point Modulation to Reduce Synthesizer Problems When Designing DC-coupled GMSK Modulation," MX-COM, 2000.

4. D. Rosemarin, "Wide Bandwidth Frequency Modulation of Phase-lock Loops," RF Design, February 2000.

5. D. Baberjee, PLL Performance, Simulation and Design, National Semiconductor, 2001, pp. 13-21 or 77-104.

6. M.S. Roden, Analog and Digital Communication Systems, Prentice-Hall, Upper Saddle River, NJ, 1996, pp. 272-315.

7. "Spectrum Analysis Amplitude and Frequency Modulation," Application Note 150-1, Agilent Technologies.

SEONG-SIK YANG, JONG-HWAN LEE AND KYUNG-WHAN YEOM

Chungnam National University

Daejeon, Korea

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A phase-locked loop (PLL) is a popular method used in frequency synthesizers for most communication systems. (10) The block diagram of a common PLL synthesizer is shown in Figure 1 and includes a charge pump phase detector (PD), a temperature-compensated crystal oscillator (TCXO), a voltage-controlled oscillator (VCO), an R divider and an N divider. [K.sub.v] is the VCO gain constant (MHz/V), [K.sub.p] is the PD gain constant (mA/rad) and Z(s) represents the loop filter transfer function. The R divider divides down the TCXO frequency to form a channel and the N divider divides down the VCO frequency. The PD compares the two divided-down signals and generates current pulses proportional to the phase difference. The PD's output current pulses are filtered by the loop filter and a slow varying voltage tunes the VCO frequency to ultimately drive the PD's average output to zero.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

A PLL frequency synthesizer can be simply frequency modulated by injecting the modulation signal [V.sub.m] into the tuning input of the VCO. However, the modulation response for [V.sub.m] has a high pass characteristic as

[[omega].sub.o]/[V.sub.m] = [K.sub.v]/[1 + [[[K.sub.p][K.sub.v]Z(s)]/[sN]]] (1)

From Equation 1, it can be found that for modulation rates greater than the PLL loop bandwidth the VCO can be modulated successfully. However, for rates less than the loop bandwidth, there is no modulated output. Narrowing the loop bandwidth will result in a longer lock time that the system may not tolerate. This article reviews some of the techniques used to overcome this problem and introduces a practical low cost modulation circuit.

FREQUENCY MODULATION USING A PLL

Figure 2 shows methods used by others to circumvent this problem. In the method using a phase modulator, (2) the signal [V.sub.m1] injected into the VCO generates the high pass modulated signal given in Equation 1 and the [V.sub.m2] injected into the phase modulator through the integrator yields the low pass modulated signal as

[[omega].sub.o]/[V.sub.m2] = [[[K.sub.i][K.sub.p][K.sub.v]Z(s)]/s]/[1 + [[[K.sub.p][K.sub.v]Z(s)]/[sN]]] (2)

If [V.sub.m] = [V.sub.m1] + [V.sub.m2] (the modulation signal [V.sub.m] is split into two paths as [V.sub.m1] and [V.sub.m2]) and [K.sub.i] = [K.sub.v]/N, the VCO output for [V.sub.m] is the sum of Equations 1 and 2 and results in a flat peak frequency deviation [DELTA]f, independent of the modulation rates. In the method using a VCXO, (3) for a low frequency modulation, [V.sub.m2], split from [V.sub.m], is injected to the voltage-controlled crystal oscillator (VCXO) instead of the TCXO. A flat [DELTA]f, independent of the modulation rates, can be obtained if

[K.sub.ref] = [R/N][K.sub.v]

Although the previous two-point modulation methods (one injection point is the VCO tuning input for high pass characteristic modulation and the other is the potential point providing a low pass characteristic modulation) provide a modulation response extending down to DC, they require specific highly priced components such as a phase modulator and VCXO.

The block diagram of an FM PLL using compensation of the amount of feedback is also shown. The amount of feedback in the denominator of Equation 1 is compensated by the modulation block D. For complete compensation, the modulation block should have a transfer function given by

[FIGURE 3 OMITTED]

D = [V.sub.o]/[V.sub.m] = [[[DELTA]f]/[K.sub.v]](1 + [[[K.sub.p][K.sub.v]Z(s)]/[sN]]) (3)

By multiplying Equations 1 and 2, the VCO may then generate an FM modulated signal with a desired peak frequency deviation [DELTA]f.

The modulation block can be implemented by various circuits for the given PLL loop filter. For the second-order charge pump loop filter Z(s) of [R.sub.1], [C.sub.1] and [C.sub.2], Rosemarin (4) has suggested the circuit shown in Figure 3.

[FIGURE 4 OMITTED]

In this circuit, the output voltage across [C.sub.b] will be developed by two currents, [i.sub.R] and [i.sub.C]. Assuming that R is large enough (R/2 [much greater than] |Z(s)|) to act as a current source and using superposition, [V.sub.o] can then be approximately written as

[V.sub.o] = [[C.sub.a]/[[C.sub.a] + [C.sub.b]]][V.sub.m] + [[Z(s)]/[s([C.sub.a] + [C.sub.b])[R.sup.2]]][V.sub.m] (4)

From R/2 [much greater than] |Z(s)|, the values of R can be consequently determined by

R = 20/[2[pi][f.sub.min]([C.sub.1] + [C.sub.2])] (5)

where

[f.sub.min] = minimum frequency of the modulation signal

Comparing Equation 3 with Equation 4, [C.sub.a] should satisfy the relation

[[K.sub.p][K.sub.v]]/N = 1/[[C.sub.a][R.sup.2]] (6)

Finally, [C.sub.b] can be determined to give the desired [DELTA]f as

[DELTA]f = [[K.sub.v][C.sub.a]]/[[C.sub.a] + [C.sub.b]] (7)

Although the above circuit is designed for a one-port, second-order charge pump loop filter, the circuit can be applied successfully for a two-port higher order loop filter frequently employed for the sufficient suppression of spurious. However, the circuit is easily influenced by the load (load impedance is open in the derivation). As a result, the proper separation or the compensation of the loading effect is required in the application.

Figure 4 shows the proposed overall FM PLL synthesizer including the modulation block determined as in the previous case. [R.sub.c] and [C.sub.c] are inserted for the compensation of the PLL loop filter loading. The [C.sub.c] capacitor also plays a role in blocking the leakage of the DC voltage at the VCO input to the modulation block. The third-order loop filter is chosen for the sufficient attenuation of the spurious. [R.sub.4] with the VCO input parasitic capacitance [C.sub.v] = 550 pF may increase the order of the loop filter; however, its time constant is selected far above the channel frequency to yield a negligible effect on the loop filter. [R.sub.4] is inserted to prevent the direct loading of [C.sub.v] to the designed [C.sub.3].

[FIGURE 5 OMITTED]

The FM PLL synthesizer has a center frequency of 164 MHz, a channel spacing of 12.5 kHz and the desired peak frequency deviation [DELTA]f of 2.4 kHz. The VCO and PD gain constants are [K.sub.v] = 17 MHz/V and [K.sub.p] = 5 mA/rad, respectively. The third-order loop filter is calculated by following Baberjee's method (5) with a phase margin of 45[degrees] and a loop bandwidth of 1 kHz, taking the lock time specification into consideration. The calculated element values are [C.sub.1] = 68nF, [R.sub.2] = 1.2 k[ohm], [C.sub.2] = 680 nF, [R.sub.3] = 33 k[ohm], [C.sub.4] = 560 pF and [R.sub.4] = 1.8 k[ohm].

The element values of the modulation block are [C.sub.a] = 3.75 pF, [C.sub.b] = 6.8 nF and R = 200 k[ohm].

The computed transfer function of the modulation block is actually affected by the connected loop filter. In order to determine [R.sub.C] and [C.sub.C], an approximated FM circuit is drawn in Figure 5, assuming modulation rates lower than the channel frequency. The output of the modulation block is denoted by [V.sub.i]. The transfer function then becomes

[V.sub.i]/[V.sub.o] = [Z(s) + [R.sub.3] + [R.sub.4] + [R.sub.C] + [1/[s[C.sub.C]]]]/[Z(s) + [R.sub.3] + [R.sub.4]] (8)

The second-order loop filter impedance Z(s) can be approximated as the parallel connection of [C.sub.1] and [C.sub.2] at a low frequency. In that case, Equation 8 becomes frequency independent of [R.sub.C] and [C.sub.C] if they are selected to satisfy the relation

1 + [[R.sub.C]/[[R.sub.3] + [R.sub.4]]] = 1 + [[[C.sub.1] + [C.sub.2]]/[C.sub.C]] (9)

The modulation block output is then scaled down by Equation 8 and transferred to the VCO without distorting its frequency response. However, Equation 8 actually shows weak frequency dependence to some degree. To get the optimum [R.sub.c] and [C.sub.c] from the many possible choices given by Equation 9, the circuit in Figure 4 was simulated and tuned using ADS for a flat peak frequency deviation. The best performance is obtained at [C.sub.c] = 680 nF and [R.sub.c] = 34.8 k[ohm].

To investigate the loading effect, the higher order loop filter elements ([R.sub.3], [C.sub.3] and [R.sub.4]) and the coupling circuit ([R.sub.4] and [C.sub.c]) are removed. The values of the remaining second-order loop filter are slightly adjusted for the phase margin of 45[degrees] and the loop bandwidth of 1 kHz. The modulation block is also tuned again according to the second-order loop filter change. The modulation signal, with a peak-to-peak voltage of 1 V, is applied through the corresponding second-order modulation circuit. Figure 6 shows the spectra of the second-order FM PLL synthesizer for a modulation signal of 4.8 kHz (modulation index [beta] = 0.5) and of 400 Hz ([beta] = 6).

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

At 4.8 kHz, the spurious is found at the channel frequency of 12.5 kHz, resulting from an insufficient attenuation. For 400 Hz, the un-suppressed spurious of 12.5 kHz grows bigger and is also modulated. It should be noted that [DELTA]f is decreased. The modulated spectrum of the desired [DELTA]f, by Carson's rule, has a 99 percent BW of 5.6 kHz (BW[approximately]([DELTA]f + [f.sub.m]) = 2(2.4 + 0.4) = 5.6 kHz. (6) But the estimated BW from the modulated spectrum is approximately 4.8 kHz < 5.6 kHz. The decrease of [DELTA]f at a low modulation frequency occurs because the effective [C.sub.b] is increased by the loading of the parallel connection of [C.sub.1] and [C.sub.2], which results in the decrease of [DELTA]f according to Equation 7.

Figure 7 shows the measured results of the third-order FM PLL circuit. Comparing these results with the last case, the spurious is suppressed sufficiently and does not show in the spectrum. The bandwidth of 5.6 kHz for the desired peak frequency deviation is estimated in the spectrum.

NOVEL [DELTA]f MEASUREMENT WITH A SPECTRUM ANALYZER

The FM modulated carrier [x.sub.c](t) by a sinusoidal signal of [[omega].sub.m] = 2[pi][f.sub.m] can be expressed as

[x.sub.c](t) = [A.sub.c][[infinity].summation over (n = -[infinity])][J.sub.n]([beta])cos([[omega].sub.c]t + n[[omega].sub.m]t) (10)

Here, [J.sub.n]([beta]) represents an n-th order Bessel function of the first kind. Based on Equation 10, the modulation index ([beta]) can be measured with a spectrum analyzer by three known methods. The first method is to change the modulation rate and to find the rate which yields the first null of the carrier amplitude [J.sub.o]([beta]). From this, the value of [DELTA]f can be found using the modulation rate and the known first zero of [J.sub.o]([beta]). However, for the method of the carrier zeros, the modulation index should be independent of the modulation rate. The method of using the 99 percent power bandwidth is useful to estimate [DELTA]f for a wide band FM signal as shown in the previous section. It is not adequate, however, to determine the precise value of [DELTA]f. In order to measure the [DELTA]f for the modulation rate change, a useful method is based on the recurrence formulas for Bessel function as (7)

[beta] = [2[J.sub.1]([beta])]/[[J.sub.0]([beta]) + [J.sub.2]([beta])] (11)

The application of Equation 11 is successful for narrowband FM, considering that the three tones in Equation 10 are dominant. However, the error in the measurement by three tones becomes critical in the determination of the modulation index, especially in wideband FM, although Equation 11 is an exact one. To show this, the ratio [[beta].sub.m,2]/[beta] is plotted in Figure 8 assuming a random measurement error of 0.1 dB of the spectrum analyzer 8564E. The [[beta].sub.m,2] represents the maximum and minimum of the measured [beta] determined by Equation 11 with the random measurement error of the spectrum analyzer. The error becomes extremely large near the zeroes of [J.sub.1]([beta]) because both the denominator and nominator in Equation 11 approach zero simultaneously.

To reduce the errors, Equation 11 is modified to contain a large number of tones in the determination of the modulation index. Extending the recurrence formulas up to n+1 tones,

[beta]([J.sub.2]([beta]) + [J.sub.o]([beta])) = 2 * 1[J.sub.1]([beta])

[beta]([J.sub.3]([beta]) + [J.sub.1]([beta])) = 2 * 2[J.sub.2]([beta])

...

[beta]([J.sub.n]([beta]) + [J.sub.n-2]([beta])) = 2 * (n - 1)[J.sub.n-1]([beta])

[beta]([J.sub.n+1]([beta]) + [J.sub.n-1]([beta])) = 2 * n[J.sub.n]([beta]) (12)

Summing up the left and the right sides of Equation 12, [beta] can be written as

[beta] = [2([J.sub.1] + 2[J.sub.2] + ... + n[J.sub.n])]/[[J.sub.0] + [J.sub.1] + 2([J.sub.3] + [J.sub.4] + ... [J.sub.n-1]) + [J.sub.n] + [J.sub.n+1]] (13)

The [[beta].sub.m,30], that is the modulation index determined by Equation 13 using 30 tone measurements, is also plotted and shows an almost flat error and does not show the error peaks compared to [[beta].sub.m,2]. From these results, the number of the measured tones is increased according to [beta] for the proposed FM PLL measurement.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Figure 9 shows the simulated and measured [DELTA]f with the suggested method for the proposed FM modulator for the different modulation rates. The results show that a flat frequency deviation, down to 200 Hz, is obtained and shows good agreement with the design. Below 200 Hz, the frequency of the function generator used as the modulation source is unstable, and reliable measurements cannot be obtained.

CONCLUSION

In this article, a low cost and practical FM PLL synthesizer circuit for a flat peak frequency deviation is suggested based on the amount of feedback compensation. The modulation rate can be extended down to below the PLL loop bandwidth. Rosemarin's amount of feedback compensation circuit is modified to include a two-port loop filter and consequently suppress the spurious sufficiently. The influence of the loop filter deteriorates the modulation circuit and the interaction should also be compensated. An approximate compensation of the influence is suggested. The possible error in the modulation index measurement is reduced by including a sufficient number of tones. With the suggested measurement, it can be successfully verified that the designed circuit has the expected frequency deviation.

ACKNOWLEDGMENT

This work has been supported by the MINT (Millimeter-wave Innovation Technology) Research Center of KOSEF.

References

1. L. Rohde, Microwave and Wireless Synthesizers, Wiley-Interscience, Somerset, NJ, 1997, pp. 1-78.

2. S. Grimmett, "Frequency Modulation in a Phase-lock Loop by Control of the Phase Inside the Loop," RF Design, June 1991, pp. 31-38.

3. R. Hunter and F. Kostedt, "Using Tow-point Modulation to Reduce Synthesizer Problems When Designing DC-coupled GMSK Modulation," MX-COM, 2000.

4. D. Rosemarin, "Wide Bandwidth Frequency Modulation of Phase-lock Loops," RF Design, February 2000.

5. D. Baberjee, PLL Performance, Simulation and Design, National Semiconductor, 2001, pp. 13-21 or 77-104.

6. M.S. Roden, Analog and Digital Communication Systems, Prentice-Hall, Upper Saddle River, NJ, 1996, pp. 272-315.

7. "Spectrum Analysis Amplitude and Frequency Modulation," Application Note 150-1, Agilent Technologies.

SEONG-SIK YANG, JONG-HWAN LEE AND KYUNG-WHAN YEOM

Chungnam National University

Daejeon, Korea

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Title Annotation: | TECHNICAL FEATURE |
---|---|

Author: | Yang, Seong-Sik; Lee, Jong-Hwan; Yeom, Kyung-Whan |

Publication: | Microwave Journal |

Geographic Code: | 1USA |

Date: | Feb 1, 2005 |

Words: | 2824 |

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