Intuitively, the total power consumption of the cascaded system of the two devices is the sum of power consumed by each device alone and the power input to the system and this is shown in Equation 20.
Also, the output signal power of the cascade is shown in Equation 21:
It follows from Equation 18 and Equation 19 that the total power consumption of the cascade can be shown in Equation 22:
Since Equation 22 is of the same form as Equation 13 and Equation 8, the power waste factor for the cascaded system of Figure 2 is given by Equation 23:
From Equation 23, W for a cascaded system with N devices may be generalized in Equation 24:
Note the approach and mathematics are strikingly similar to the cascaded noise factor in Equation 2,1 except W is referred to the output and is generally most impacted by the component closest to the Nth component closest to the sink.5-7
The wasted power of the cascade when there is a continual signal flow can be determined based on either the output power as in Equations 6, 8, 12 and 15 or based on the input source power and the gain of each stage as in Equation 25, but a signal flow to the output is not required to define W.
In Equation 25, W is the waste factor for the entire cascade and Gi is the gain of ith stage, with i = 1 denoting the stage closest to the source and N is the number of cascaded components.
HANDLING TIME-VARYING POWER STATES WHEN USING WASTE FACTOR
In a cascaded system, the components often exhibit dynamic behavior of varying power efficiency, leading to temporal variations in the waste factor. Amplifier efficiency levels may change above their quiescent power consumption levels over time when a signal of varying magnitude is applied or different devices, systems or chains are turned on and off at different times or for different functions to conserve power, etc. To account for these fluctuations, it is practical to calculate the average waste factor of each device over a designated time interval or averaged over all the finite power states. This can be mathematically described using the time average, which is simply the integral of the time-varying waste factor over an observation interval. Just like with the noise factor, a time-averaged waste factor may be used, denoted as __W in Equation 26:
Where:
T represents the selected period over which the average is taken, presumably over all operating states
Wt denotes the instantaneous waste factor at time t
Wi represents each of the finite number of power states
In this calculation, there are N states and the duration of each state is considered in the average. This formulation provides a time-averaged measure of the waste factor to capture an average value of the dynamic nature of the power consumption and efficiency. For on/off components (e.g., amplifiers) with static/quiescent power, when a state is “off” during an observed epoch, the static power of such components may be treated as non-signal power (Pnon-signal), although it may be more useful to consider these “turned-off” devices as contributing to non-path power rather than using Equation 26 since the “turned off” components do not carry signal power across a cascade when in the off state. This is an open research area ripe for discovery and definition of convention.
ANALOGIES BETWEEN WASTE FACTOR AND NOISE FACTOR
The analogous mathematical formulation of F and W is immediately apparent from the above text. However, there are important characteristics of each metric to keep in mind. Since noise figure is a measure of the additive noise from a source that results in the degradation of SNR caused by the components in a cascaded system, it quantifies the amount of total noise added to the signal referred to the input of the cascade. Therefore, F increases when observed from device 1 (closest to the source) to N (closest to the sink). On the other hand, W is a measure of the additive wasted power consumed by a cascaded system and quantifies the amount of power consumed and wasted by the cascade compared to the total signal power that is transported to the output. Since W is defined as the ratio of the total power consumed by the cascade to the signal output power, W is referred to the output and increases (e.g., efficiency decreases) when observed from device N (closest to the sink) to device 1 (closest to the source).
A larger value of W signifies more power wasted. The value of W is always equal to or greater than 1, with W = 1 signifying that all power supplied to a component or cascaded network is fully utilized in the signal output. This is the optimal condition with no power wasted. W equal to infinity indicates that no power is contributed to the signal output and all power is squandered (e.g., a perfect dummy load or a completely lossy channel). Some comparisons in the utility and scope of F and W in communication systems are summarized in Table 2.
In conclusion, both F and W are useful in the analysis of communication systems. F is a well-established metric that provides a measure of the additive noise of a system and the degradation of the SNR. W is a new metric that provides a measure of the additive wasted power of a system and the energy efficiency of a system. Both metrics are important for circuits and communication systems, but as shown later in this article, W has further applications for understanding power efficiency in computing and processing, as well as communication systems. With the increasing importance of energy efficiency for our planet, W can be a useful metric for enabling the design of a wide range of greener systems.
APPLICATIONS OF WASTE FACTOR IN DEVICES AND SYSTEMS
Waste Factor for a Passive Device
Figure 3 Passive attenuator circuit diagram.
Consider applying the waste factor to an attenuating stage like the passive attenuator shown in Figure 3.
The gain of the attenuator is defined as the output power divided by the input power as shown in Equation 27:
Where:
Vi-1 is the input voltage of the attenuator
Vi is the output voltage of this attenuator
L is the loss of this component
(L > 1)
This assumes that the impedance at the input and the output of the attenuator are equal, which directly relates the power ratio to the square of the voltage ratio. Treating this attenuator as the ith stage of the cascade, the output power is shown in Equation 28 as:
The power wasted by the attenuator (e.g., not provided in the signal output) is defined in Equation 29:
Where:
Psigi is the total signal power delivered by the ith stage to the (i +1)th stage
Pnon-sigi is the signal power used by the ith stage component but not delivered as signal power
Based on the definition of W in Equations 4, 10 or 14, W for a passive attenuator is computed in Equation 30 as:
Note the duality between attenuators using the waste factor and the noise factor. Waste factor, represented by Watten = Latten, quantifies the wasted power expended in the attenuator, while noise factor Fatten = Latten quantifies the degradation of the SNR due to additive noise.
Waste Factor for a Communication Channel
In a communication system, the concept of waste factor (W) provides valuable insights into comprehensively understanding the power efficiency of the data transmission process through a lossy channel.5-7 Any type of channel, whether wireless, wired, optical, etc., plays a vital role in the overall power efficiency of an end-to-end communication link. Consider a scenario where the power transmitted from the source is denoted as PTX and the power received at the receiver is PRX. Consider a lossy channel with a loss (e.g., L = 1/attenuation) given by Equation 31:
