Prototype System
S- Ddata(t)
10.4 Noise Calculations
reference frequency, 1/T , of 20 MHz is assumed, which leads to Nnom = 92 to achieve an output carrier frequency of 1.84 GHz. The value of Kv is 30 MHz/V since a Z-COMM V602MC06 part was used to implement the VCO. A large value of C3 is desirable to obtain good noise performance, but its value cannot be made much higher than 30 pF due to area constraints on the die of the custom IC.
Parameter Value Parameter Value
C1 3 pF C2 120 fF
C3 30 pF C4 3 pF
I 1.5 uA T 1/(20 MHz)
Kv 30 MHz/V Nnom 92
Table 10.3
Component values for prototype system.
The final step in design of the frequency synthesizer is to insure that nonideal effects of OP1 and OP2 do not significantly affect the modulation data as it passes through the PLL transfer function. Aside from issues described in Chapter 9, the primary objective in designing OP1 and OP2 is to insure a high unity gain bandwidth relative to the modulation bandwidth. (Note that the feedback ratio of each opamp is very close to unity.) Table 10.4 lists the relevant specifications in the prototype. In the case of OP1, a bandwidth of 7 MHz is adequately fast to allow excellent tracking between the tail current of the charge pump and its top currents. The choice of 6 MHz for OP2 allows data rates as high as 2.85 Mbit/s to be achieved without distortion by its dynamics. The limited bandwidth of OP2 has the benefit of increasing the attenuation of noise produced by the Σ-∆ modulator and charge pump/loop filter circuitry at high frequencies.
Parameter OP1 Value OP2 Value
Open Loop Gain 83 dB 86 dB
Unity Gain Bandwidth 7 MHz 6 MHz Phase Margin 80 degrees 72 degrees
Table 10.4
Simulated opamp specifications.
10.4 Noise Calculations
Table 10.5 displays the value of each noise source shown in Figure 10.5. Many of these values were obtained through AC simulation of the relevant circuits in HSPICE. Note
that all noise sources other than ˆq(t) are assumed to be white, so that the values of their variance suffice for their description. This assumption holds for the VCO provided that its output phase noise rolls off at -20 dB/dec as predicted by Leeson’s model [62, 63]; the -20 dB/dec rolloff is achieved in the model since vvco,in(t), which has a flat spectral density, passes through the integrating action of the VCO. As will be seen in measured data in Chapter 11, the actual VCO doesn’t quite follow Leeson’s model, but is close enough that calculations and measured results agree well at the intermediate frequency offsets of interest.
Noise Source Origin Nature Calculation Value
i2ch1 Ch. Pump, OP1 CT HSPICE 1.2E-24 A2/Hz i2ch2, i2ch3 Ch. Pump CT HSPICE 1.8E-25 A2/Hz i2sw Switched Cap DT Equation 10.8 1.0E-26 A2/Hz
vop2 OP2 CT HSPICE 1.85E-16 V2/Hz
v2vco,in VCO CT Equation 10.7 1.4E-16 V2/Hz
ˆ
q(t) Σ-∆ DT Equation 10.9 —
Table 10.5
Values of noise sources within PLL.
The input referred noise of the VCO was calculated from open loop phase noise measurements of the VCO and the expression
10 log(vvco,in2 |Kv/(jf )|2) = -143 dBc/Hz at f = 5 MHz, (10.7) where Kvis 30 MHz/V. The value of the kT/C noise current produced by the switched capacitor operation, i2sw, was calculated as
i2sw = (1/T )kTKC2, (10.8)
where k is Boltzmann’s constant, and TK is temperature in degrees Kelvin. Finally, the spectral density of the Σ-∆ quantization noise was calculated as:
Sˆq(f ) = 1
12(2sin(πf T ))2n, (10.9)
where n = 2 is the order of the Σ-∆ modulator.
10.4.1 A Simplified Model for Noise Analysis
To analyze the effect of the noise sources in Table 10.5 on the transmitter output, it is convenient to lump them into three dominant components. Figure 10.6 depicts
10.4. NOISE CALCULATIONS 161
such a model, and denotes the relevant noise sources as isum(t), vsum(t), and ˆq(t).
Comparison of Figures 10.6 and Figure 10.5 reveals that i2sum = fi(i2ch1, i2ch2, i2ch3, i2sw)
vsum2 = fv(v2vco,in, vop2 ),
(10.10)
where fi(·) and fv(·) are appropriately defined functions. Rather than explicitly determining fi(·) and fv(·), we make the following argument:
i2sum ≈ i2ch1/2, vsum2 ≈ vvco,in2 + vop2 .
(10.11)
1 + jf/fz j2pfC3(1 + jf/fp)
Tp I
FDIV(t)
FREF(t) KV
jf
FOUT(t)
N1nom
2p 1 - D1 n(t)
T 1
Fn(t) Fe(t)
T
insd(t) 1
q(t)
S-D data(t)
Pc(f)
isum(t) vsum(t)
Figure 10.6: Model of test system with lumped noise sources added.
The validity of i2sum ≈ i2ch1/2 is argued in two steps, the first of which is presented in Figure 10.7. In the drawing, superposition is used to break up the response of the opamp into the transfer functions associated with current coming into each of its terminals. The negative terminal is connected to a noise source with variance i2a, and the positive terminal to a noise source with variance i2b. (It is assumed that i2a= i2b.) For frequencies greater than fz, the magnitude of the transfer function from the positive terminal is greater than that associated with the negative terminal. The opposite is true for frequencies less than fz. Thus, the output of OP2 is influenced by either the i2a source or the i2b source, but not both (except when the frequency is close
to fz). Therefore, we achieve approximately the same spectral density at the output of OP2 by connecting the same noise source to each of the terminals.
Examination of Table 10.5 reveals that i2ch1 is an order of magnitude larger than i2ch2, i2ch3, and i2sw. Since the i2ch1 noise source is switched alternately between the positive and negative terminals of OP2, its contribution to isum(t) will be pulsed in nature. At the nominal duty cycle of fifty percent, we would expect the energy of i2ch1 to be split equally between the positive and negative terminals of OP2. As such, i2sum is then i2ch1/2. This intuitive argument was verified using a detailed C simulation of the PLL with no modulation signal applied (i.e., the duty cycle of the charge pump remained constant except for variations due to Σ-∆ quantization noise). Note that i2sum should have a different value than i2ch1/2 if the nominal duty cycle is offset from its desired value of fifty percent.
1 + C4/C3 1 + j2pfC1T/C2 ia2
1 jf2pC3
1 + C4/C3 1 + j2pfC1T/C2
fz
1 jf2pC3 ib2
1 + C4/C3 1 + j2pfC1T/C2 isum2
1 jf2pC3 ia2= ib2
isum2 =ia2 =ib2
Figure 10.7: The lumping together of current noise sources entering OP2.
Since Table 10.5 reveals that v2vco is of the same order of vop2 , we simply add these components to obtain vsum2 ≈ v2vco+vop2 . This expression is accurate at frequencies less than the unity gain bandwidth of OP2; the v2opnoise source is passed to its output with a gain of approximately one in this region. At frequencies beyond OP2’s bandwidth, the expression is conservatively high since v2op is attenuated in this frequency range.
10.4.2 Resulting Transmitter Output Noise
The effect of isum(t), vsum(t), and ˆq(t) on the transmitter output is now examined with the aid of Figure 10.6. All calculations will assume that G(f ) is described by Equation 10.3 with the parameter values specified in Equation 10.5.
10.4. NOISE CALCULATIONS 163
Assuming that each of the three noise sources are independent of each other, we can express the overall noise spectral density at the transmitter output, Stn(f ), as
Stn(f ) = SΦv(f ) + SΦi(f ) + SΦq(f ), (10.12) where SΦv(f ), SΦi(f ), and SΦq(f ) are the noise contributions from vsum(t), isum(t), and ˆq(t), respectively. If we further assume that isum(t) and vsum(t) are white, each of the above spectral density components is calculated from Figure 10.6 as
SΦv(f ) = i2sum(πNnom/I)2|G(f)|2, SΦi(f ) = vsum2 |Kv/(jf )|2|1 − G(f)|2,
SΦq(f ) = (1/T )121 (2π)2(2sin(πf T ))2(n−1) |T · G(f)|2.
(10.13)
Plots of these spectra are shown in Figure 10.8. As shown in the figure, the influence of isum(t) dominates at low frequencies, and the influence of vsum(t) and ˆq(t) dominates at high frequencies.
105 106 107
−150
−140
−130
−120
−110
−100
−90
−80
−70
−60
Calculated Noise Spectra for Synthesizer
L(f) dBc/Hz
Frequency Offset From Carrier (Hz) (1)
(2)
(3)
(4)
Figure 10.8: Calculated noise spectra of synthesizer: (1) charge pump induced, SΦi(f ), (2) VCO and opamp induced, SΦv(f ), (3) Σ-∆ induced, SΦq(f ), (4) overall, SΦ(f ).