# On the Energy Efficiency of On-Off Keying Transmitters with Two Distinct Types of Batteries
## 2 Power of the Battery and Transmitter
### Transmitter Power  
Clearly, \(P_{\text{tx}}(t)\) is the sum power consumption of all transmitter components, which consists of the following parts:  
Figure 1: The battery-powered transmitter under study.  
* a digital circuit responsible for digital signal processing that transforms sampled information to digital bits, and controlling other circuits in the transmitter. The digital circuit remains active and consumes a fixed power denoted by \(P_{\text{d}}\) independently of emitted symbols.
* an analog circuit which usually comprises baseband or radio frequency analog circuits such as a voltage controlled oscillator and filters. Specifically, the analog circuit is controlled by the digital circuit to generate a sinusoidal waveform and delivers it to the power amplifier when emitting a symbol '1', but to be switched off when emitting a symbol '0'. When emitting '1', we assume the analog circuit consumes a fixed power denoted by \(P_{\text{a}}\).
* a power amplifier (PA) which feeds the antenna with OOK-modulated signal expressed as \[s_{\text{o}}(t)=\left\{\begin{array}{ll}\sqrt{2P_{1}}sin(2\pi f_{\text{c}}t) &\text{when \ emitting symbol \ '1',}\\ 0&\text{when \ emitting symbol \ '0',}\end{array}\right.\] (7) where \(P_{1}\) denotes the average signal power for emitting symbol '1'.  
A diagram of commercially available low-cost PA is shown in Figure 2a, where a transistor is used. Its operation principle and power consumption is explained in details in the following.  
When symbol '0' is emitted, the PA can be turned off by the digital circuit. When symbol '1' is emitted, a sinusoidal waveform is generated by the analog circuit and input to the PA, and this waveform can be added with a quiescent bias, so as to make the transistor conducted within a whole or part of each sinusoidal waveform's cycle. As elaborated by pages 39-41 in [16], the degree of conduction is characterized by the so-called conduction angle \(\psi\), and the PA can be classified into class A (\(\psi=\pi\)), B(\(\psi=\frac{\pi}{2}\)), C(\(\psi<\frac{\pi}{2}\)), and AB(\(\frac{\pi}{2}<\psi<\pi\)) according to the specific value for \(\psi\) as illustrated in Figure 2b,c.  
To establish a power consumption model for the PA, note that the current \(I_{\text{PA}}\) drawn from the power supply is stable and equal to the DC component of the total current \(i_{d}(t)\) flowing into  
Figure 2: Illustrations for the low-cost power amplifier. (**a**) a diagram of the power amplifier; (**b**) class A; (**c**) class B; (**d**) class C; (**e**) class AB.  
the transistor, while the current \(s_{o}(t)\) driving the antenna load is equal to the first-order harmonic component of \(i_{d}(t)\), while other higher-order harmonic components are eliminated by the filter. According to the derivations made on pages 39-41 of [16], the power loss factor, defined as the ratio of the PA's power dissipation to the radiated signal power, is expressed as  
\[\alpha=\frac{1}{\eta_{\rm PA}(\psi)}-1 \tag{8}\]  
where  
\[\eta_{\rm PA}(\psi)=\frac{2\psi-sin(2\psi)}{4(sin(\psi)-\psi cos(\psi))}. \tag{9}\]  
In summary, the PA's power consumption is  
\[P_{\rm PA}(t)=\left\{\begin{array}{ll}(1+\alpha)P_{1}&\mbox{when emitting symbol `1'}\\ 0&\mbox{when emitting symbol `0'}\end{array}\right. \tag{10}\]  
Summarizing the above power models for all components, the discharge power from the battery is  
\[P_{\rm out}(t)=\frac{P_{\rm R}(t)}{\eta}=\left\{\begin{array}{ll}\frac{(1+ \alpha)P_{1}+P_{\rm a}+P_{\rm d}}{\eta}&\mbox{when emitting `1'}\\ \frac{P_{\rm d}}{\eta}&\mbox{when emitting `0'}\end{array}\right. \tag{11}\]