Embedded Software Design

## ELECTENG 209
# Embedded Software Design 
### Digital Signal Processing

.TitleAuthor[Duleepa J Thrimawithana]

---

.logo-slide[]
.footer[[Duleepa J Thrimawithana](https://www.linkedin.com/in/duleepajt), Department of Electrical, Computer and Software Engineering (2020)]

---

# Learning Objectives

- How do the ADC samples of the voltage and current waveforms relate to the analog signals?
- What do we need to be aware of when extracting information from sampled waveforms?
- How can we calculate the peak value of a sampled waveform?
 - Using digital signal processing techniques
 - Using analog peak detector circuitry to aid digital processing
- How can we calculate the RMS value of a sampled waveform?
 - Approximating the RMS using peak value of a sinusoidal waveforms 
 - Using Riemann sum to evaluate the RMS formula 
- How can we calculate the power factor angle?
 - Using output of the zero crossing detectors to approximate power factor angle
- How can we calculate power delivered to the load?
 - Approximating power using RMSs and power factor angle
 - Using Riemann sum to evaluate the power formula

---

# Lecture Quiz

- The lecture quiz is now available on Canvas
 - Quiz is available for 3 days and allows 3 attempts
 - Best of the 3 attempts taken as the final score

---

# Digital Representation of Voltage and Current
### Understanding the Relation to Real Signals

---

.logo-slide[]
.footer[[Duleepa J Thrimawithana](https://www.linkedin.com/in/duleepajt), Department of Electrical, Computer and Software Engineering (2020)]

---

# The Voltage Measurement

- The step down in voltage achieved by the voltage divider can be quantified by a gain Gvs 
 - In your design Gvs = Rb / (Ra + Rb )
- The differential amplifier provide a gain Gvo and an offset Voff
 - In your design Gvo = R2 / R1 
- The gain of the passive RC filter, Gvf, is 1 in your design
- The analog signal at the ADC, Vvf, is therefore

\\[ V\_{vf} = G\_{vs} G\_{vo} V\_{AC} + V\_{off} \\]

---

# The Current Measurement

- The relation with IL and Vis can be quantified by a gain Gis 
 - In your design Gis = Rs 
- The differential amplifier provide a gain Gio and an offset Voff
 - In your design Gio = R2 / R1 
- The gain of the passive RC filter, Gif, is 1 in your design
- The analog signal at the ADC, Vif, is therefore

\\[ V\_{if} = G\_{is} G\_{io} I\_{L} + V\_{off} \\]

---

# ADC Data Representing Voltage

- In this lecture, lets assume a 50Hz Vvf at ADC0 is sampled and converted to an ADC value every 2ms
- Note that project specifications limits maximum sampling rate to 10kHz and therefore if repeatedly a sample of Vvf is taken followed by a sample of Vif then each Vvf sample will be 0.2ms apart
 - Alternatively Vvf can be continuously sampled over a few periods to take samples of Vvf every 0.1ms 
 - Need zero crossing detector to correctly align the Vif samples taken after this with the Vvf samples
- From the ADC data we can estimate the AC source voltage at ith sample point since 
\\[ V\_{AC}[i] = \left( ADC0Value[i] \times 5/1024 - V\_{off} \right) / \left( G\_{vs} G\_{vo} \right) \\]

---

# ADC Data Representing Current

- Similar to Vvf lets assume a 50Hz Vif at ADC1 is sampled and converted to an ADC value every 2ms
- Project specifications limits maximum sampling rate to 10kHz and therefore if repeatedly a sample of Vvf is taken followed by a sample of Vif then each Vif sample will be 0.2ms apart
 - Alternatively Vif can be continuously sampled over a few periods to take samples of Vif every 0.1ms 
 - Need zero crossing detector to correctly align the Vif samples with the Vvf samples
- From the ADC data we can estimate the AC load current at ith sample point since 
\\[ I\_{L}[i] = \left( ADC1Value[i] \times 5/1024 - V\_{off} \right) / \left( G\_{is} G\_{io} \right) \\]

---
name: S6

# Voltage & Current Samples

- When repeatedly a sample of Vvf is taken followed by a sample of Vif these samples will be 1ms apart
 - This delay can introduce a *perceived* phase-shift (i.e. phase error) 
 - With good signal processing techniques this error can be minimized
- During the project you are encouraged to investigate and improve on the techniques taught in this lecture

---

# Peak Current/Voltage
### Techniques to Obtain the Peak of a Signal

---

.logo-slide[]
.footer[[Duleepa J Thrimawithana](https://www.linkedin.com/in/duleepajt), Department of Electrical, Computer and Software Engineering (2020)]

---
name: S7

# Finding Max of ADC Samples

- With a limited sampling rate and without a method to synchronize the sampling to align with a peak we cannot measure the peak accurately by sampling just one cycle of the signal
 - Notice in example above, that while during the first cycle the ADC data does not capture the peak, during the second cycle it does
- The signal can be sampled over a large number of cycles to capture the peak as long as sampling frequency is not an integer multiplication of the signal frequency
 - This process takes time

---
name: S8

# Using a Peak Detector Circuit

- A dedicated peak detector circuit could be used to detect the peak
- A diode and a capacitor can be used to sample the peak 
 - Works like a half-wave rectifier
 - A resistor, Rbld, is needed to bleed the charge in Cpk so a peak value that changes over time can be monitored (the RC time constant needs to be larger than the period of the signal)
 - Alternatively, a switch, Srst, can be used to reset the peak detector by discharging Cpk regularly through a resistor, Rrst
- The forward voltage drop across diode significantly impacts the accuracy

---
name: S9

# An Improved Peak Detector Circuit

- To overcome the forward voltage drop of the diode a *superdiode* can be used
- When VAC is greater than Vpk, the op-amp forward biases the diode completing negative feedback loop
 - Op-amp acts as a follower and Cpk charges to peak value of VAC 
 - The maximum peak value that can be tracked is equal to VOH - VF, where VF is the forward voltage drop of the diode
- A mechanism to discharge Cpk is still required so a peak value that changes over time can be monitored

---

# RMS Current/Voltage
### Techniques to Obtain the RMS of a Signal

---

.logo-slide[]
.footer[[Duleepa J Thrimawithana](https://www.linkedin.com/in/duleepajt), Department of Electrical, Computer and Software Engineering (2020)]

---
name: S10

# Estimating RMS Using Peak

- [Recall](#S7) that we have already learnt how to obtain the peak of a signal
- Assuming this signal is sinusoidal, the peak value is related to the RMS value as given by

\\[ RMS = \frac {Peak} {\sqrt{2}} \\]

- The assumption of a sinusoidal signal is not always correct, especially in a practical design, as Vac and IL are often distorted sinusoidal signals
 - The distortions in the signals lead to significant errors in the RMS estimate when using this method
- We can improve the accuracy by developing software to implement the RMS formula given by

\\[ V\_{AC\_{rms}} = \sqrt {\frac {1} {T\_p} \int\_{0}^{T\_p} {V\_{AC}^2 dt}}  \quad \text{OR} \quad I\_{L\_{rms}} = \sqrt {\frac {1} {T\_p} \int\_{0}^{T\_p} {I\_{L}^2 dt}}\\]

---
name: S11

# Riemann Sum

- RMS formula can be converted in to a form that can be evaluated in an MCU by applying it to ADC data
- The integral term in the RMS formula can be evaluated in an MCU using the Riemann sum 
 - The area under a curve is divided in to a number of rectangles where the height of these rectangles are evaluated using the ADC data of the signal
 - The width of each rectangle is equal to time between 2 ADC samples of the signal Δtsample
 - By adding the areas of each rectangle we can evaluate the integral
 - Trapezoidal rule and a number of other more advanced methods can also be used

.center[<img src="img/Riemann_sum_(leftbox).gif" height="150"> <img src="img/Riemann_sum_(middlebox).gif" height="150"> <img src="img/Riemann_sum_(rightbox).gif" height="150">
.credits[
Riemann Sum [[1]](https://en.wikipedia.org/wiki/Riemann_sum)
]
]

---
name: S12

# Calculating the RMS Numerically

- If we have taken N ADC samples at regular Δtsample intervals over one time period of the signal, the time period Tp = NΔtsample
- We can square each VAC or IL sample and use Riemann sum to numerically evaluate the RMS as given by

\\[ V\_{AC\_{rms}}^2 = \frac {1}{N \Delta t\_{sample}} \sum\_{i=0}^{N-1} V\_{AC}^2[i] \Delta t\_{sample} \quad \text{OR} \quad I\_{L\_{rms}}^2 = \frac {1}{N \Delta t\_{sample}} \sum\_{i=0}^{N-1} I\_L^2[i] \Delta t\_{sample}\\]

---

# Power Factor 
### Techniques to Obtain the Power Factor Angle

---

.logo-slide[]
.footer[[Duleepa J Thrimawithana](https://www.linkedin.com/in/duleepajt), Department of Electrical, Computer and Software Engineering (2020)]

---
name: S13

# Time Between Zero Crossings

- Because of the limited sampling frequency, using ADC data to calculate the time between voltage and current zero-crossings introduces significant error
- Comparators can be used to detect the zero-crossings and generate rising or falling edge signals
 - To avoid noise from generating false edges, filtered signals should be fed to the comparators
- The edges can be used as interrupts to control a timer to obtain the time, Tzc, between edges
 - We can estimate power factor angle, θ, by looking at the time between the zero-crossings of the voltage and current waveforms since θ = 3600 Tzc / Tp

---

# Power 
### Techniques to Obtain the Real Power

---

.logo-slide[]
.footer[[Duleepa J Thrimawithana](https://www.linkedin.com/in/duleepajt), Department of Electrical, Computer and Software Engineering (2020)]

---
name: S14

# Estimating Power Using RMSs

- [Recall](#S10) that we have already learnt how to obtain the RMS of VAC and IL
- Assuming VAC and IL are sinusoidal, the power is related to their RMS values and the power factor angle as given by

\\[ P = V\_{AC\_{rms}} I\_{L\_{rms}} cos(\theta) \\]

- The assumption of a sinusoidal signal is not always correct, especially in a practical design, as Vac and IL are often distorted sinusoidal signals
 - The distortions in the signals lead to significant errors in the power estimate when using RMSs and the power factor angle
- We can improve the accuracy by developing software to implement the power formula given by

\\[ P =  \frac {1} {T\_p} \int\_{0}^{T\_p} {V\_{AC} I\_{L} dt}  \\]

---
name: S15

# Calculating Power Numerically

.left-column[
- Assume we have taken N ADC samples of both VAC and IL
 - Samples of VAC as well as IL are at regular Δtsample intervals
- We can multiply each VAC sample with corresponding IL sample to numerically evaluate power as given by

\\[ P = \frac {1}{N \Delta t\_{sample}} \sum\_{i=0}^{N-1} V\_{AC}[i] I\_L[i] \Delta t\_{sample} = \frac {1}{N } \sum\_{i=0}^{N-1} V\_{AC}[i] I\_L[i]\\]

- Since the two ADC channels are sampled one after the other, there is a time delay between each sample of VAC and its corresponding IL sample
 - This can lead to a significant error in power calculation in the form of a phase-shift

]

---
name: S16

# Linear Approximation of Missing Data

.left-column[
- The missing VAC and IL samples can be approximated using the average of the samples from either side (i.e. linear approximation)

\\[ \bar{V}\_{AC}[i] = \left( V\_{AC}[i] + V\_{AC}[i+1] \right) / 2 \\]
\\[ \bar{I}\_{L}[i] = \left( I\_{L}[i-1] + I\_{L}[i] \right) / 2 \\]

- How can we estimate the 11th sample of VAC and the -1th sample of IL?
- Power can now be evaluated as given by

\\[ P = \frac {1}{2N} \sum\_{i=0}^{N-1} \left[ V\_{AC}[i] \bar{I}\_L[i] + \bar{V}\_{AC}[i] I\_L[i] \right] \\]

]

---
name: S17

# The Power Calculation

- Once the missing samples are approximated, Riemann sum can be used to evaluate power
 - Can we improve the calculation accuracy?
- In the project using Excel/Matlab/Repl.it validate your power calculation algorithm 
 - Operate on VAC and IL ADC samples collected from the MCU

---
name: S18

# Demo: Power Calculation