Harmonics
( z a
^ 1 12F2I (2F1+F2) (2F2+F1)
-120
-+
5 6f M H z
p .p S M A L L S IG N A LI
D IST O R T IO N out D IST O R T IO NI
S A T U R A T IO N 10 11 2 0 ~ U I P 2 _ I 1 P 3 ; - - 2 0 - P - l d B IIP2 IIP3-100
-30 -25-20
-15-10
50
510
Pin dBm
Fig 2.3.1 (a) H arm on ic and In term o d u la tio n D istortion P rod u cts to 3rd
o rd er, (b) Sm all Sign al D isto rtio n , In tercep t p oints and D istortion
S a tu ra tio n E ffects
2 . 3 . 1. 2
D isto rtio n S a tu ra tio nThe am plitudes o f distortion products changes as the amplitude o f the excitation signals are
dB
f
dB
f
F ig 2.3.2 In a b an d lim ited m o d u la ted sig n a l, th e n o n lin ea rity o f a pow er a m p lifier has th e effect o f regen era tin g th e sid e lo b es, ca u sin g ad ja cen t ch a n n el in ter fe ren ce
products are related to the excitation amplitude by power law relationship defined by the order o f the product (i.e. 2nd order have a power o f 2 etc). At higher amplitudes the distortion products depart from this simple power rule and can rise sharply or even null. This effect is attributed to the contribution o f higher order nonlinear terms to lower order terms. At these levels, the gain o f the system begins to fall (although some times regions o f gain enhancement are oberved such as that presented in Fig 2.3.1b).
2 . 3 . 1 . 3 S p e ctra l S p rea d in g
One o f the effects o f intermodulation distortion is spectral spreading. If the two input tones F i and F2 o f Fig 2.3.1 were the result of, say, double sideband amplitude modulation with suppressed carrier, the intermodulation products 2F1+F2 and 2F2+ F1 can be considered as spectral spreading o f the effective baseband signal (Fig 2.3.2). This could lead to adjacent channel interference.
2 . 3 . 1 . 4 A M /P M C o n version
In a power amplifier, the distortion from higher order products can contribute to the fundamental frequency. These contribution have an arbitrary phase relationship to the input and their amplitude depend on the magnitude o f the input power. This leads to a phase modulation that varies with any amplitude modulation present.
2 . 3 . 1 . 5 C ro ss M o d u la tio n
If an unmodulated carrier and a modulated canier pass through a nonlinearity, there is a tendancy for the unmodulated carrier to become modulated. This effect is particularly serious close to a strong signal source whilst receiving a weak signal where cross modulation can obliterate the wanted signal.
2 . 3 . 1 . 6 In ter-S y m b o l In terferen ce and B it E rror R a te
In complex modulation schemes such as QPSK, the amplitude and phase shifts associated with distortion saturation can cause the resulting signal to malligned with its original constellation diagram, leading to intersymbol interference and bit errors. This problem is viewed a serious enough problem in some mobile communication systems for the use o f less power efficient amplifiers (Class A) with very high linearity.
2 . 3 . 1 . 7 L in e a r ity S p e c ific a tio n s
The nonlinearity o f an amplifier is usually specified by three parameters. The IdB compression point (P-idfi)? and the 2nd and 3rd order intermodulation intercept points (denoted by IIP2 & IIP3). These are illustrated in Fig 2.3.1b. The IdB compression point is where the power gain drops below the small signal gain by IdB. The 2nd and 3rd order intercept points are defined as the intersection o f the extrapolated small signal distortion curves for the 2nd and 3rd order intermodulation products with the extrapolated fundamnetal under the small signal gain. The more positive the intercept point, the more linear the system is at small signals. These parameters can be related to either the input or output power at which they occur. The output referred parameters are useful for giving a measure o f relative distortion product level for a fixed output power. The input referred parameters are useful for giving a measure o f the nonlinearity o f the transfer function. The two are related by the small signal gain.
At Audio frequencies, the Total Harmonic Distortion is used to give a measure o f system performance. This expresses the distortion o f the signal as the square root o f the ratio o f the sum o f power o f the distortion components compared to that o f the output at the
fundamental frequency. This measurement is o f limited value as it does not attempt to separate 2nd and 3rd order contributions. In RF systems, it is usually the 3rd order distortion that is the most harmful. For completeness w e include the mathematical definition o f THD
V n
n=2
where is the amplitude o f harmonic x.
2 . 3 . 2 D isto rtio n M ea su rem en t
When measuring distortion, it is normal to measure intermodulation distortion instead o f harmonic distortion. This is due to the difficulty o f obtaining synthesised signal sources with low harmonic distortion. In general harmonic and intermodulation distortion follow the same behaviour, but on occasions may differ due to subtle differences in phase and magnitude relationships. In a narrowband system, the signals have to be close together so that the excitation tones are in the working bandwidth o f the system. In wideband systems, it is convenient to u sa the Total Difference Frequency Distortion Technique [3 ,4 ,5 ,6 ], where the excitations are just o ff o f a 2:3 spacing. This brings a pair o f 2nd and 3rd order difference products close together, whilst maximising their separation from the excitations giving enhanced dynamic range in the measurement process (due to the limit o f bandwidth o f the Spectrum analyser).
In very accurate distortion measurements, care has to be taken to prevent distortion being generat^' by the signal from one source entering another source through the input combiner. This is best achieved by using a combination o f wideband isolating combiners (hybrids) and notch filters. Another source o f error, is intermodulation being generated within the spectrum analyser. Some workers use diplexing filters [7,8] to separate the input signal from the distortion products, allowing the use o f more sensitive settings on the Spectrum Analyser. Some workers follow the diplexing filters with a LNA to further boost further the dynamic range.
The author has found that these precaution not always necessary, because the distortion being generated in the circuit under test is usually significantly greater than these residual distortion products. It is virtually impossible to implement these precautions when measuring frequency dependant distortion over a range o f frequencies.
2 . 3 . 3 D isto rtio n A nalysis with H an d and C o m p u ter A id ed T ech n iq u es
Several computer aided techniques have emerged for the study o f the nonlinear behaviour o f systems. These techniques include Power Series, Volterra Series, Perturbation Method and Harmonic Balance. Each o f the techniques has its own advantages and limits, and no single technique offers a universal solution and this is reflected in the availability o f different methods in commercial simulator packages. W e give only a brief description o f their most important properties.
2 . 3 . 3 . 1 P o w er S eries
This technique develops algebraic relationships o f circuit behaviour by assuming polynomial approximations o f device nonlinear behaviour. The technique is only really suitable for low frequency systems with an isolated non-linearity. This technique is frequently encountered in the literature o f low frequency distortion, e.g. [9 ,1 0 ,1 1 ]. The
technique may be coupled with mathematical simplifications that may restrict the amplitude,
frequency and the number o f simultaneous tones studied. A formal definition is given in
[
12]
2 . 3 . 3 . 2 V o lterra A n a ly sis
This technique is algebraic, deriving complex number transfer functions o f linear, 2nd order nonlinearity, 3rd order nonlinearity etc. Not only does it allow for a non-linear transconductance, but it can also handle nonlinear capacitances. The technique is only really suitable for weakly non-linear systems at low levels o f excitation. (Where weakly non-linear is defined as not having discontinuities in the transfer function!). This technique is widely used for studying medium and microwave frequency distortion, e.g. [ 13 ,14, 15]. This technique is very useful in studying intermodulation in systems under non-
commensurate excitation (i.e. two or more tones not harmonically related to each other). The technique does lead to equations, but these are frequently formidably complicated even for relatively simple systems. A formal definition is given in [16]. A more advanced discussion o f the topic can be found in [17]. A generalised matrix form has been developed which has been implemented into a circuit simulator by S.A.Maas offering a significant speed advantage over Harmonic Balance based simulators.
2 . 3 . 3 . 3 P ertu rb a tio n M eth od
This is an iterative numerical method used in SPICE and other circuit simulators. A numerical solution is calculated for each successive time p oin t All normal components can be handled, although transmission lines cause some difficulty. For reliable convergence, nonhnear models must have continuous 1st derivatives. A formal description might be given in [18,19]. This technique is generally fairly slow and memory intensive for studying intermodulation distortion.
2 . 3 . 3 . 4 H a rm o n ic B a la n ce
This is an iterative numerical method that solves the linear part(s) o f the system in the frequency domain (computationally fast and requires little memory) and the non-hnear part(s) in the time domain. Normally some kind o f initial solution (estimate) is required to find a solution, but these can be fairly vague and convergence will still occur. It is restricted to commensurate excitation (i.e. all the frequency components have a common factor). It does allow solution o f large signal excitation. Harmonic balance can handle nonlinear transfer functions and nonlinear capacitors. Sometimes it is possible to express the non-linearity in the frequency domain, removing the need o f using Fourier transforms to swap between the frequency and time domains. A formal definition is given in [20]. A more advanced discussion o f the topic can be found in [21]. This technque has a significant speed advantage over the Perturbation method but is slow compared to a Volterra based simulator.