# A Wideband Hybrid Envelope Tracking Supply Modulator with Slew-Rate-Enhanced Linear Amplifier
## 2 Circuit Design  
### Principle of the HETSM System  
As depicted in Figure 2, the recommended HETSM is composed of a high-efficiency SWC parallel with an LA, which has large unity-gain bandwidth (UGB). The high-efficiency SWC is responsible for delivering the primary power, catering to the low-frequency current demands of the PA [16]. Simultaneously, a high-speed, high-slew-rate LA precisely tracks the wideband envelope signal, ensuring the linearity of the PA load. The current of the LA's output stage is mirrored out to generate the SW control (SW Ctrl) signal. Additionally, an external clock is used to control the frequency of the SWC, therefore controlling the efficiency of SWC. In this paper, the proposed HETSM is designed for GaAs Pas, which are powered by fixed drain voltages ranging from 4 V to 6 V. The paralleled 5 \(\Omega\) resistor (R\({}_{\text{PA}}\)) and 100 pF capacitor (C\({}_{\text{PA}}\)) are utilized as the equivalent PA load for the LA [19; 21].  
Figure 1: Power dissipation comparison between the ET PA and the traditional PA.  
Assuming that the HETSM system is to track a sinusoidal signal with amplitude \(A_{0}\) and frequency \(f_{0}\), its expression is given in Equation (1):  
\[f\ (t)=A_{0}sin(2\pi f_{0}t) \tag{1}\]  
Its derivative is shown as Equation (2). Therefore, theoretically, to track the signal \(f(t)\), the slew rate needs to reach \(2\pi f_{0}A_{0}\).  
\[f^{\prime}(t)=2\pi f_{0}A_{0}\cos(2\pi f_{0}t) \tag{2}\]  
For a GaAs PA load, it is assumed that the maximum supply voltage of the adopted envelope tracking system is 6 V, and the minimum supply is 2 V. Theoretically, according to Equation (2), to track a sinusoidal signal with a peak-to-peak value of 4 V and a frequency of 100 MHz, the slew rate should ideally reach 1257 V/\(\upmu\)s. This value is not a practical guiding metric but a theoretical calculation because actual envelope signals are not pure sinusoids. However, it serves as a reference to estimate the required magnitude of the required slew rate.  
In this work, a 100 MHz, 16 QAM, and 8.49 dB PAPR signal is used for the test. The slew rate distribution of the corresponding envelope signal over a certain period is illustrated in Figure 3. It can be observed that for precise tracking of the envelope signal by the LA, its slew rate should at least exceed \(\pm\)800 V/\(\upmu\)s and preferably surpass \(\pm\)1000 V/\(\upmu\)s.  
Figure 3: Slew-rate distribution of a 100 MHz, 16 QAM envelope signal.  
Figure 2: The proposed HETSM diagram.  
The consumption of the LA \(P_{Lloss}\) and the SWC \(P_{SWloss}\) forms the consumption of the HETSM. The efficiency of the HETSM can be calculated as:  
\[\eta=\frac{P_{out}}{P_{out}+P_{Lloss}+P_{SWloss}}=\frac{1}{1+\frac{P_{Lloss}+P_{ SWloss}}{P_{out}}} \tag{3}\]  
For both \(P_{Lloss}\) and \(P_{SWloss}\), there is an intrinsic loss, no matter how small the output power is. For the LA, it is called quiescent dissipation, and for the SWC, it is called switching loss. Therefore, it is important to reduce the quiescent current of the LA to improve the efficiency of the HETSM.