j
A1,1j
e j ϕð A1,1Þ
: ð
6:
5Þ
All mismatch terms are neglected, for simplicity, and therefore only the X pð F ,k Þ terms are considered. The time variation of the input envelope is speci ed by either an explicit time-dependent I – Q signal or one of several modulated source components available in the simulator. The settings for the envelope simulation are chosen to allow adequate time sampling of the envelope corresp onding to the input signal and to allow a suitably long period for the simulation.In the quasi-static approximation, the DUT response, at time t , is computed to be the value of the static X-parameter mapping, as described in (6.5), applied to the value of the input envelope at the same time instant, t . This results in (6.6):
B p,k
ð
tÞ ¼
X pð F Þ,k
A1,1ð
tÞ e j ϕð A1,1ðt ÞÞ: ð
6:
6Þ
This procedure produces time-varying envelopes for each of the harmonics, indexed by integer k , produced by the DUT in response to the modulated amplitude around the carrier at the fundamental frequency.
The quasi-static approximation is an extrapolation from steady-state conditions to a dynamic (time-varying) condition. It is certainly valid for suf ciently slowly varying An(t ), since it reduces to the static mapping as the instantaneous envelope amplitudes become constant. It is an accurate approximation to the actual time-dependent response provided that any underlying system dynamics are suf ciently fast that, at any instant of
time, the DUT is nearly in its steady-state condition determined by the value of the input amplitude at that same time. That is, the system adiabatically tracks the input from one steady state to the next, parameterized by the time. The quasi-static approximation breaks down as the signal modulation rate increases and becomes comparable to or
faster than timescales for which other dynamical effects become observable. In particu-lar, electro-thermal effects, bias-line interactions, and other phenomena come into play for which a more elaborate dynamical description of the DUT is required. This will be discussed in more detail in the following.
6.3
6.3.1 .1 Qua Quasi- si-sta static tic two two-to -tone ne int intermo ermodul dulati ation on dis distort tortion ion fro from m a a sta static tic one one-to -tone ne X-parameter model
X-parameter model
In the following, the theory is illustrated using an actual ampli er, namely a Mini-Circuits ZFL-11AD
þ
. This ampli er is characterized by performing a one-tone X-parameter model at a carrier frequency of 1750 MHz. The two-tone experiments are perfor med with a tone spacing of 1200 Hz, whereb y the tone s are symmet rically placed around the carrier frequency. A time-domain representation of the two-tone input 152 MemoryMemorysignal is shown in Figure 6.1. The carrier oscil lation period is so small compared to the millisecond range of the time axis of Figure 6.1that the signal looks like a solid black area.
The individual oscillations can only be seen on a higher-resolution timescale expressed in nanoseconds rather than milliseconds. This is illustrated inFigure 6.2, which represents a zoom on the rst 20 nanoseconds of the 2.5 milliseconds shown in Figure 6.1.
This two-tone sinusoidal signal can be represented as a single carrier with a modu-lated complex amplitude A(t ) according to (6.7). Equal magnitudes, A1, for the two tones, are considered for simplicity:
A1 cos
ð
ω1tÞ þ
A1 cosð
ω2tÞ ¼
2 A1 cos
Δω2 t
cos ω0t
ð Þ
¼
Re 2 A1 cos
Δω2 t
e j ω0t8 <
:
9 =
;
¼
Ref
Að
tÞ
e j ω0tg:
ð
6:
7Þ
0 500 1000 1500 2000 2500
–0.2 –0.1 0.0 0.1 0.2
time (µs) i n p u
t ( V )
Figure 6.1
Figure 6.1 Two-tone input signal in the time domain.
0.000 0.005 0.010 0.015 0.020
–0.2 –0.1 0.0 0.1 0.2
time (µs) i n
p u t ( V )
Figure 6.2
Figure 6.2 Two-tone input signal in the time domain (zoomed in).
6.3 153
6.3 Quasi-static X-parameQuasi-static X-parameter evaluatioter evaluationn
Here, Δω
¼
ω2
ω1 and ω0¼
(ω1þ
ω2)/2. For our example, ω1 and ω2 correspond to (1750 MHz
600 Hz) and (1750 MHzþ
600 Hz), respective ly.From (6.7), it follows that the envelope representation A(t ) of any two-tone signal, using as carrier frequency the middle of the two frequencies, ω0, is simply given by
A t
ð Þ ¼
2 A1cos Δω 2 t : ð6:
8Þ
The real and imaginary parts of A(t ) are shown in Figure 6.3.
Consider now the trans mission of this two-tone signal throu gh a system that is described by a simple X-parameter model that only takes into account the transmission of the fundamental signal. In other words, only the presence of a term X ð2 F Þ
,1
ð Þ :
, which is here more simply denoted by X F (.), is considered in the follow ing. The ampli er response B(t ) using the quasi-static approximation is calculated according to (6.6), whereby A(t ) is given by (6.8). The result is given below:B t
ð Þ ¼
X F j A t ð Þj
e j ϕ A t ð ð ÞÞ ð
6:
9Þ
withj
Að
tÞj ¼
2 A1
cos Δω 2 t
,
ð
6:
10Þ
ϕ
ð
A tð Þ Þ ¼
arccos sgn cos Δω 2 t
: ð6:
11Þ
The amplitude
j
A(t )j
and phase ϕ( A(t )) are shown in Figure 6.4 and Figure 6.5, respectively.Note that (6.11) is simply a way to express that ϕ( A(t )) toggles between 0and 180, whereby e j ϕ( A(t ))
toggles between
þ
1 and
1, depending on the sign of the function0 500 1000 1500 2000 2500
–0.2 –0.1 0.0 0.1 0.2
time (µs) r
& e i m i n p u t ( V )
Figure 6.3
Figure 6.3 Two-tone input signal in the envelope domain: real part (solid line) and imaginary part (dashed line).
154 MemoryMemory
cos(Δωt /2). The amplitude and phase of the X-parameter function X ( F )(.) of our ampli er example are shown in Figure 6.6 and Figure 6.7, respectively.
Note that Figure 6.6 corresponds to a classic AM-to-AM characteristic, whereas Figure 6.7 corresponds to a classic AM-to-PM characteristic. The corresponding pression characteristic is shown in Figure 6.8.
Equations (6.9) – (6.11) are used to evaluate and analyze the DUT response. For the ampli er example considered, the real and imaginary parts of the two-tone response B(t ) are shown in Figure 6.9.
Note that the output signal B(t ), regardless of the shape of the nonlinear function X ( F )(.), has exactly the same period as the function cos(Δωt /2), namely a period equal to 4π / Δω.
The periodic output envelope B(t ) can therefore be expanded into a complex Fourier series, with harmonic frequencies equal to an integer times Δω/2, as shown in (6.12):
0 500 1000 1500 2000 2500
0.00 0.05 0.10
| A ( t ) | ( V )
0.15 0.20
time (mms) Figure 6.4
Figure 6.4 Input signal envelope amplitude.
0 500 1000 1500 2000 2500
0 50 100 150
time (µs)
φ
( A ( t ) ) (
)
Figure 6.5
Figure 6.5 Input signal envelope phase.
6.3 155
6.3 Quasi-static X-parameQuasi-static X-parameter evaluatioter evaluationn
0.00 0.05 0.10 0.15 0.20 0.25 0.0
0.1 0.2 0.3 0.4
input amplitude (V) o u
t p u t a m p l i t u d e ( V )
Figure 6.6
Figure 6.6 Amplitude of X ( F )(.) versus input amplitude (AM-to-AM).
0.00 0.05 0.10 0.15 0.20 0.25
0.0 0.5 1.0 1.5 2.0 2.5 3.0
input amplitude (V)
o u t p u t p h a s e (
)
Figure 6.7
Figure 6.7 Phase of X ( F )(.) versus input amplitude (AM-to-PM).
–30 –25 –20 –15 –10 –5
5 6 7 8 9 10
input amplitude (V) g a
i n ( d B )
Figure 6.8
Figure 6.8 Compression characteristic.
156 MemoryMemory
B t
ð Þ ¼ Xþ
∞
k ¼∞
Bk e jk Δ2ω t
: ð
6:
12Þ
The coef cients of this series, Bk , have some interesting properties. Based on the symmetry conditions
B
2π Δω
t ¼ B t ð Þ ð
6:
13Þ
and
B
ð Þ ¼
t B tð Þ
,ð
6:
14Þ
which can readily be veri ed using (6.9) – (6.11), the resulting series Bk will always be a symmetric odd harmonic series. In other words, Bk¼
0 if k is an even integer, andBk
¼
Bk: ð
6:
15Þ
These odd components Bk , having harmonic indices 1, 3, 5,
. . .
or
1,
3,
5,. . .
, correspond to the upper and lower intermodulation products, respectively, generated by the two-tone signal. Note that the spacing between the harmonic frequencies, as one would expect for a two-tone signal, equals Δω. The lower third-order intermodulation (IM3) product correspond s to k¼
3, the lower fth-order intermodulation (IM5) corresponds to k¼
5, the upper IM3 corresponds to k¼ þ
3, the upper IM5 corresponds to k¼ þ
5, etc.By making use of the symmetry conditions (6.13) and (6.14) and the de nition of the complex Fourier series, the following simple relationship between the one-tone X-parameter function X ( F )(.) and the quasi-statically generated intermodulation products Bk results:
Bk
¼
2 πð
π =2
0
X ð F Þ
2 A1 cos θð Þ cos k ð Þ
θ d θ , ð
6:
16Þ
only valid, of course, for k equal to an odd integer.
0 500 1000 1500 2000 2500
–0.4 –0.2 0.0 0.2 0.4
time (µs) r
& e i m o u t p u t ( V )
Figure 6.9
Figure 6.9 Calculated output signal in the envelope domain.
6.3 157
6.3 Quasi-static X-parameQuasi-static X-parameter evaluatioter evaluationn
Experimental validation of the described quasi-static approach is provided by paring amplitude values of the intermodulation products Bk as calculated by using
(6.16) with actual measured amplitude values, for several different input power levels.
Figure 6.10 shows the calculated output power, per tone, as a function of the input power, per tone (solid line), together with actual measured data acquired from the
example two-tone experiments (dots).
Figure 6.11 shows the calculated values of B3 and B5, which correspond to IM3 and IM5, respectively, together with the measured data points. The measured and calculated values correspond very well, thereby proving the validity of the quasi-static approach.
Equation (6.16) reveals another important characteristic of quasi-statically generated intermodulation products, namely that Bk only depends on A1 and X ( F )(.), but is
–22 –20 –18 –16 –14 –12 –10 –8 –12
–10 –8 –6 –4 –2
tone input power (dBm) t o n e o
t u p u t p o w e r ( d B m )
Figure 6.10
Figure 6.10 Measured and calculated tone output power.
–22 –20 –18 –16 –14 –12 –10 –8 –60
–50 –40 –30 –20
tone input power (dBm) I M
3 &
I M 5 ( d B m )
Figure 6.11
Figure 6.11 Measured (symbols) and calculated IM3 (solid line) and IM5 (dashed line).
158 MemoryMemory
independent of Δω. In other words, the intermodulation products are independent of the spacing between the two tones of the input signal.
6.
6.3. 3.2 2 AC ACPR PR es esti tima mati tion ons s us usin ing g qu quas asi- i-st stat atic ic ap appr proa oach ch
In a similar manner to the case of the sinusoidal amplitude modulation considered above, it is possible to estimate other nonlinear gures of merit (FOMs), such as the adjacent channel power ratio (ACPR) and others common to digital communications circuits, from a simple static X-parameter model in an envelope analysis. For digitally modulated signals, it is easiest to use built-in modulated sources available in the simulator. The simulator simply evaluates the static X-parameter model at each sampled time according to (6.6). This is illustrated using an actual device, name ly a 25 W GaN monolithic microwave integrated circuit (MMIC) power ampli er (CREE CMPA2560025F). The device is rst characterized by a static X-parameter model. This is equivalent to measuring the AM-to-AM AM-to-PM characteristic of the device. Next, a long-term-evolution (LTE) input signal with a 2.5 GHz carrier frequency is applied to the device. The spectrum of the input signal is shown in Figure 6.12.
The output signa l is predicted by applying the stati c X-parameter model to the measured input envelope. Then the spectrum of the modeled output signal is compared to the actual measured output spectrum. The result is shown in Figure 6.13. Note the signi cant spectral re-growth in both the measured and the modeled output spectra.
The relevant FOM is computed from the simulated output spectrum according to the speci c protocol appropriate to the modulation format. Examples of quasi-statically estimated ACPR from X-parameters for a real ampli er and independent experimental validation are provided in Table 6.1.
Note that there is a good correspondence between the measurements and the values derived from the static X-parameter model.
–30 –20 –10 0 10 20 30
–110 –100 –90 –80 –70 –60
frequency (MHz) s p
e c t r a l d e n s i t y ( d B m / H z )
Figure 6.12
Figure 6.12 Input signal spectrum.
6.3 159
6.3 Quasi-static X-parameQuasi-static X-parameter evaluatioter evaluationn
6.
6.3. 3.3 3 Li Limi mita tati tion ons s of of qu quas asi- i-st stat atic ic ap appr proa oach ch
As explained in Section 6.3.1 the quasi-static evaluation of a static X-parameter model will produce an intermodulation spectrum with identical levels for upper and lower sidebands from a two-tone input signal. Moreover, the simulated levels are independent of the modulation rate – the frequency separation, Δ f , between the two incident tones.
That is, the distortion spectrum shows no “ bandwidth dependence.” In the envelope domain, it can be shown that the time-varying output envelopes are symmetric with respect to their peaks.
In the limit of slowly varying input envelopes, (6.6) reduces to evaluating a set of independent steady-state mappings at different power levels. That is, (6.6) reduces to (6.5), provided the latter was characterized over the full range of amplitudes covered by the time-varying input signal of (6.6). Simulations for modulated signals become
exact as the modulation rate, or signal bandwidth (BW), approaches zero (the band limit).
For the two-tone case, it is possible actually to measure true steady-state X-parameters as functions of an LSOP that depends on both large input tones, using the approach discussed inChapter 5. Two-tone X-parameters provide exact intermodulation spectra, in magnitude, phase, and their dependence on the frequency separation of the tones.
–30 –20 –10 0 10 20 30
–80 –70 –60 –50 –40 –30
frequency (MHz) s p
e c t r a l d e n s i t y ( d B m / H z )
Figure 6.13
Figure 6.13 Output spectra: static X-parameter model (dotted line) and measurements (solid line).
Table 6.1
Table 6.1 ACPR estimated from static X-parameters versus independent measurements ACPR estimated from static X-parameters versus independent measurements
Output power (dBm)
Lower sideband ACPR (dB)
Upper sideband ACPR (dB)
X-parameter model 37.9 18.6 18.7
Measurements 38.0 19.0 19.1
160 MemoryMemory
It should not be surprising that full information about the DUT response to two steady-state tones cannot be obtained from measurements taken with an LSOP set by only a single incident CW signal. It is clear there is more information in a set of two-tone measurements than in a single-two-tone measurement.
At high modulation bandwidths, the actual time-dependent scattered waves are no longer accurately computable from (6.6). That is, the quasi-static approximation of going from (6.5) to (6.6) becomes less valid. Any FOMs, such as third-order intercept (IP3) or ACPR, derived from the scattered waves simulated under the quasi-static approximation will therefore not be in complete quantitative agreement with the actual performance characteristics of the DUT.
6.3
6.3.4 .4 Adv Advant antage ages s of of qua quasi- si-sta static tic X-p X-para aramet meters ers for for dig digita ital l mod modula ulatio tion n
Key nonlinear FOMs are usually scalars (numbers). For example, IP3, ACPR, and (error vector magnitude) EVM are numbers, typically computed from measurements made with a spectrum analyzer. The FOMs of particular components, such as individual stages of a multi-stage ampli er, cannot generally be used to infer the overall FOM for the composite system. That is, just knowing the FOMs of individual parts is not enough to design a nonlinear RF system for lowest ACPR.
X-parameters, on the other hand, enable at least a quantitative estimate of any nonlinear FOM for the full system just from knowledge of the X-parameters of the constitutive components. The X-parameters of the composite system follow from the nonlinear algebraic composition of the component X-parameters according to the general treatment inChapter 2. The quasi-static approximation can be used easily in the envelope domain to evaluate the overall system response to modulated signals of various kinds.
Improved estimates of DUT behavior in response to wide-band modulated signals require a more careful treatment of dynamical behavior that begins in the following section.