Electrical Memory Effects
The main origins of electrical memory effects are the transistor terminations, including intrinsic and extrinsic parasitic elements, and matching networks. In order to better analyze the electrical memory effects, it is important to understand the impedance termination in transistor amplifiers [19, 20].
Figure 2.4 shows a block diagram of a common source MESFET (Metal Semiconductor Field Effect Transistor) amplifier. ZGmatch is the impedance presented by the input matching network, excluding the biasing network, to the source of the gate of the transistor, ZG bias is the impedance presented by the biasing network to the gate of the transistor, and ZG in is the impedance presented by looking at the gate of the transistor. Similarly, at the output of the transistor, the impedance presented by the drain
Figure 2.4 Block diagram of a common source MESFET amplifier showing the definition of the different impedances is ZD_in, the impedance presented by the biasing network is ZD_bias, and the impedance presented by the loading or output matching network to the drain of the transistor is ZDn L. The impedances of gate and drain nodes can then be obtained by:
Given that the transistor impedances ZG_in and ZD_in vary as a function of the driving signal power level and operating conditions of the transistor, the transistor will exhibit a nonlinear behavior at the gate and drain levels. It can then be concluded that nonlinear power amplifiers may include more than one nonlinear element. The simplest model of a nonlinear power amplifier will include:
- 1. A nonlinear block representing the gate voltage as a function of the input signal to the power amplifier.
- 2. A nonlinear block representing the relationship between the gate voltage and drain voltage.
Each of these two blocks also includes a frequency response that is due to the transistor behavior variation versus frequency and matching network response versus frequency at the fundamental frequency and each of the harmonic frequencies. The cascade of these nonlinear elements results in mixing the linear memory effects (frequency responses around a carrier frequency or its harmonic) along with the nonlinear behaviors of the nonlinear elements. This mixing of linear memory effects and nonlinear response will result in an output signal that includes nonlinearity along with nonlinear memory effects around the fundamental carrier of the signal. These nonlinear memory effects include products that are function of the frequency response of the matching network and transistor not only at the fundamental frequency but also around the different harmonic, which are translated to the fundamental frequency via the nonlinear elements [14, 21, 22].
In order to understand this concept, one can simplify the modeling of the transistor to a cascade of two nonlinear systems, G and H, each having linear memory in the form of a frequency response at each of the fundamental and harmonic frequencies. Figure 2.5 shows a block diagram of this cascade and illustrates the origins of the intermodulation products at the output and how they are affected by the frequency response of the system at the fundamental and harmonic frequencies. Each of the two systems is modeled by a nonlinearity in the order of three and a set of frequency responses around each of the fundamental and carrier frequencies (G0, G1, G2, and G3 are the frequency responses of G around the envelope, the fundamental, second and third harmonics, respectively; and H0, H1, H2, and H3 are the frequency responses of H around the envelope, the fundamental, second, and third harmonics, respectively). The
Figure 2.5 Modeling of nonlinear electrical memory effects in a cascade of two nonlinear systems
third order intermodulation products at the output of the system are the combination of different products including products generated by:
- • The third order nonlinearity of the first system, G, passed through the frequency response H1 around the fundamental carrier frequency, of the second system, H.
- • The second order mixing product of the fundamental output, and the envelope and the second harmonic outputs of the first system, G, which also passes through the frequency response, H2, around the second harmonic in the second system, H.
- • The third order mixing product of the fundamental output of the first block, G, which also passes through the frequency response, H3, around the second harmonic in the second system, H.
This third intermodulation product at the output of the power amplifier is a function of different nonlinearity orders including even nonlinearity orders and frequency responses at the envelope frequency, the fundamental frequency, and different harmonic frequencies.
If the signal bandwidth is W, and by only considering nonlinearities up to the third order, the nonlinear memory effect is a transistor is affected by: 
For practical considerations, on one hand, the frequency responses around the fundamental, second harmonic, and third harmonic frequencies are considered to occur around the same fractional bandwidth and are generally insignificant for single carrier and relatively narrowband applications. Their effect may be of importance if multi-carrier and significantly wideband signals are considered. On the other hand, the frequency response around DC frequency will have significant effect on the memory even for relatively narrowband applications if no careful design of the biasing circuit is carried out in order to maintain constant gate node impedance in this frequency band.
-  The frequency response along a band of W around DC frequency, • The frequency response along a band of W around the fundamental frequency, • The frequency response along a band of 2 W around the second harmonic frequency,and • The frequency response along a band of 3W around the third harmonicfrequency.