# Hardware Design of a High Dynamic Range Radio Frequency (RF) Harmonic Measurement System
## 2 Nonlinear Device Modeling and Characterization  
To accomplish the goal of characterizing nonlinearities generated by RF circuit components, a nonlinear system model is needed. The model must characterize the nonlinear properties of the devices and predict the behavior of the devices under different conditions. Having the ability to predict the nonlinear response of weakly nonlinear devices enables a system designer to better predict how much distortion will be present in the system. Therefore, the input-output relationship for nonlinear devices needs to be defined.  
In general, nonlinear systems do not satisfy the linearity property, i.e., superposition and scaling properties. The output of nonlinear systems contains signals that cannot be represented as a linear combination of the input signals. These signals can be at different frequencies than the input frequencies. In this paper, nonlinear systems are modeled using the power series [7; 8; 41; 42; 43], which is represented as  
\[y=\sum_{n=0}^{\infty}y_{n}=\sum_{n=0}^{\infty}a_{n}x^{n}, \tag{1}\]  
where \(x\) is the input signal, \(y\) is the output signal, and \(a_{n}\) is the scale factor coefficient of the \(n\)th order nonlinearity.  
When the input to a nonlinear system is a single frequency, the output consists of harmonics of that frequency occurring at integer multiples of the fundamental frequency. With a single frequency input to the power series given by \(x=x_{0}\cos(\omega_{0}t+\theta)\), the output can be shown to be [44]  
\[y=\sum_{n=0}^{\infty}c_{n}x_{0}^{n}\cos(n\omega_{0}t+n\theta). \tag{2}\]  
The amplitude of the frequency domain representations of Equation (2) is illustrated graphically in Figure 1. Generally, the output amplitude drops as the harmonic order \(n\) increases.  
Equation (2) expresses the nonlinear system input-output relationship in terms of voltage. Engineers typically prefer to use power units; therefore, the voltage expression is converted to power. The average power of a sinusoidal signal is given by \(P_{\text{ave}}=V_{\text{peak}}^{2}/2Z_{0}\), where \(V_{\text{peak}}\) is the peak voltage and \(Z_{0}\) is the system characteristic impedance. Thus, the power in the harmonics can be represented in a decibel (dB) scale as:  
\[P_{\text{out},n}[\text{dBm}]=nP_{\text{In}}[\text{dBm}]+D_{n}, \tag{3}\]  
where \(P_{\text{out},n}\) is the output power at the \(n\)th harmonic, \(P_{\text{in}}\) is the input power at the fundamental frequency (\(n=1\)), \(D_{n}\) is a coefficient expressed in dBm\({}^{(1-n)}\), and dBm represents power expressed in decibels relative to one milliwatt.  
Figure 1: Illustration of the output due to a single frequency input into a nonlinear system.  
The implications of Equation (3) are that the output-to-input power ratio for the \(n\)th harmonic follows an \(n:1\) relationship in a dB scale. Therefore, a 1-dB increase in the input creates a 2- and 3-dB increase in output power for the second and third harmonics, respectively. In general, the \(D_{n}\) values are frequency-dependent and characterize the \(n\)th order nonlinear properties of a device. A more commonly found characterization of a device's nonlinear properties is its output intersect point OIP\({}_{n}\), which is the intersection point at which the 1:1 slope of the linear response intersects the \(n:1\) slope of the \(n\)th order nonlinearity. This is shown in Figure 2 for the second harmonic, i.e., \(n=2\).  
The relationship between \(D_{n}\) and OIP\({}_{n}\) can be found by manipulating Equation (3) to yield:  
\[D_{n}=(1-n)\text{OIP}_{n}+nD_{1}. \tag{4}\]