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Chapter 2

REALIZATION OF FPGA BASED Q-FORMAT ARITHMETIC LOGIC

UNIT FOR POWER ELECTRONIC CONVERTER APPLICATIONS

In this chapter, analysis of FPGA resource utilization using QALU,

and is compared with the existing implementations of integer fixed point

format for data representation is presented. The analysis is carried out

through simulation of QALU using ModelSim 5.7 and synthesis using

Xilinx 9.2i. Further, it also includes the validation of possibility of single

chip implementation in FPGA for generating PWM pulses using VSI fed

induction motor.

2.1. Introduction

The power electronic converter control realization using Q-Format

through FPGA requires the following concepts:

i. Digital data representation using fixed point formats [129-130].

ii. FPGAs and the developments tools [131-139].

In the hardware implementation point of view, fixed point

representations have been used in the computation which yields the

simple hardware, there by the area as well as cost of FPGA is reduced.

Due to the flexibility of simple hardware, the fixed point processors are

commonly used in applications where the accuracy is not a constraint. In

such applications, the designer has to use the proper signal processing

to achieve the required accuracy.

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The signal processing is a key issue in the performance of a digital

system. In the existing FPGA based PWM implementation, integer fixed

point format is used in the signal processing which requires an

additional processor to implement the arithmetic functions [64-70].

Therefore, an efficient VLSI signal processing is necessary to improve the

performances of FPGA in terms of resource utilizations, reduction of

hardware cost and improved accuracy through single chip solution. To

overcome the above limitation, an alternate solution, called QALU using

Q-Format representation has been implemented [28-32, 95,107,129-

131]. The preliminary concepts like FPGA architecture [126-128],

resources and its domain use [5], development tools [132-137], and

design flow [138-139] are discussed in the literature.

2.2. Digital Data Representation by different Formats

The binary representations of numbers used in digital systems are

sign magnitude, one’s compliment, and two’s compliment. In general, 2’s

compliment data representation is used in processors and it is an

efficient way of representing signed numbers [121-122]. The most

commonly used type of data formats are: (i) Fixed Point Representation

(FXPR) [129-130], and (ii) Floating Point Representation (FLPR) [34]. In

case of fixed point processors, the application specific functions are

performed and the final result is stored in accumulator every time at the

end of each arithmetic operation. At last, the Most Significant Bit (MSB)

of accumulator is left shifted by one and is given as an output. In FPGA

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based implementations, integer fixed point is commonly used for data

representation [27-54, 64-70], whereas FLPR is used in few FPGA based

applications [33]. The FXPR and arithmetic computation is described in

[130].

The arithmetic calculations for the equation (α × β) + are presented

for fixed point format with integer part (Qm), fixed point format with

fractional part (Qn) and fixed point format with integer + fractional part

(Qm.n) in [26,130] and the corresponding error is shown in Table 2.1.

The procedure is presented in Appendix-1.

Table 2.1. Error in arithmetic computation using different formats

Type of data representation error (8 bitarithmetic operation

Fixed point format with integer part (Qm) 0.34975Fixed point format with fractional part (Qn) 0.20525Fixed point format with integer and fractional part (Qm+Qn) -0.02525

In general, the fixed point format data are scaled to integer number,

after completion of arithmetic computation, the Least Significant Bit

(LSB) in the result is truncated. The MSB is taken as the final result. In

case of integer fixed point format Qm, all 16 bit will give the correct

result. Therefore, the entire result (both MSB and LSB) has to be stored

in memory and considered for the successive computation. Hence, this

kind of representation takes more FPGA resources for an acceptable

accuracy and also this is not suitable for single chip implementations. In

fractional fixed Qn point and Qm.n arithmetic functions, the accuracy of

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final the result is not affected even after truncation of LSB. Finally, Qn

and Qm.n representation need one WL and Qm needs two WLs which

results, increased in FPGA resource utilization.

2.2.1. Error and Area Analysis in Q-Format Representation

In Table 2.1, the error in 8 bit arithmetic computation in fixed point

formats, compared with floating point format is shown. From the

analysis, Q-Format arithmetic gives improved accuracy in arithmetic

computation compared to traditional fixed integer and fractional

arithmetic. In 16 bit format with proper scaling, the error is still reduced

to negligible level. Therefore, the fixed point representation Qm+Qn (Q-

Format) is used to implement PEC control algorithm in a single FPGA.

In integer multiplication, the multiplication of two 8 bit data will give

result in 16 bit. Therefore a 16 bit register is needed to store the result

where as in fractional / Qm.n Format with proper scaling, the result is in

8 bit format and therefore, it needs only an 8 bit register to store the

results. Hence, the Qm.n Format based signal processing will reduce the

FPGA resource or area utilization [28-32, 95, 107].

2.2.2. Advantages of Q- Format Representation

Arithmetic computations using Q-Format representation in FPGA

gives the following advantages:

i. Occupies less FPGA resources i.e. less area in the chip [28-32, 95,

107].

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ii. From the analysis of multiplication, the Q-Format gives improved

accuracy for the same number of bits in integer approximation and

fixed fractional Format. In FPGA implementation, Q-Format with

representation and proper dynamic range gives improved accuracy

[28-32, 95, 107].

iii. Multiplications of two 8 bit numbers with proper scaling requires one

8 bit register to store the result [29, 95], whereas, in integer format it

requires a 16 bit register which results in increased FPGA area for

arithmetic operations.

2.3. Realization of Q-Format ALU in FPGA

In most of FPGA based PEC control implementations, FXPR is used to

compute a part of algorithm while the major computation carried by a

DSP. In this Thesis, fixed point implementation with both integer and

fractional part Qm.n is considered to implement PWM algorithm in a

single chip. The data is converted to its equivalent Q- Format in the

representation as well in arithmetic calculations. This computation is

carried out by QALU which consists of arithmetic computation functions

such as adder, subtractor and multiplier. The QALU is developed as a

library function and it is included in the project directory using the

reported literature [28, 95, 107].

The block diagram and the functional flow of QALU is shown in Fig.

2.1 and Fig. 2.2 respectively. The functional flow for addition and its

pseudo code are shown in Fig. 2.3 and Fig. 2.4 respectively.

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Fig. 2. 1. Functional block diagram of QALU

QADD/QSUB QSQRT

QMULT

QMOD

QDIVIDE

QSQR

QALUCONTROLLER

Start

Find the length of the integer part

Find the length of the fractional part

Represent the number in Qm.n format

Stop

Fig. 2. 2. Functional flow of QALU

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Pseudo code:

Functional unit name: QADDInput: two numbersOutput: addition result of two numbersSteps:

1. Convert the given tow numbers into Qm.n format

2. If both the numbers are in same Qm.n format then perform the signed

addition else Shift and sign extend to align the decimal places.

3. Setting the overflow bit, if any overflow occurs

End

Fig. 2.4. Pseudo code for addition in QALU

Start

Read the numbers

Is both thenumbers are in

same Qm.n format

Perform signed addition

Shift and sign extend to align the decimal places

Setting overflow bit if any overflow occurs

No

Yes

Stop

Fig. 2. 3. Functional flow chart of Addition in QALU

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2.3.1. Q-Format multiplication

Since multiplication plays a major role in arithmetic computations, its

basics are discussed in this section and the detailed explanation is given

in Appendix- I. Signal processing in fractional form has the specific

properties. The multiplication of data in the range of -0.999 to +0.999,

the resultant product will not have overflow. The common way for

lowering the operand is called scaling, i.e. operand is divided by the

adequate constant [29, 95].

The functional flow for multiplication and its pseudo code are shown

in Fig. 2.5 and Fig. 2.6 respectively

Fig. 2.5. Functional flow chart of Multiplication in QALU

Start

Read the numbers

Perform signed multiplication

Stop

Integer Part: QIproduct=QImultiplier + QImultiplicandFractional Part: QFproduct=QFmultiplier + QFmultiplicand

Adjust the resultant Qm.n format to the requirement

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2.3.2. Simulation Results of QALU

The proposed QALU using Q-Format and its application to power

electronic converter control has been simulated using ModelSim 5.7 and

implemented using Xilinx 9.2i. The Q-Format data representation and

the arithmetic operations of addition and multiplication has been

developed and simulated.

The simulation results for addition and multiplication using integer

fixed and Q-Format has been obtained for unsigned and signed data

inputs. The Q-Format implementation has been validated by the

simulation and implementation of an arithmetic operation

(multiplication). The implementation report of multiplication using

integer fixed arithmetic and Q-format is shown in Table 2. 2 and Table

2. 3 respectively. The simulation result for multiplication is shown in

Fig. 2. 7.

Pseudo code:

Functional unit name: QMULTInput: two numbersOutput: multiplication of two numbersSteps:

1. Convert the given tow numbers into Qm.n format

2. Perform the signed multiplication

3. Adjust the resultant Qm.n format according to the requirement(either

16/32-bit)

End

Fig. 2.6. Pseudo code for multiplication in QALU

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Table 2. 2. Implementation report of multiplication using integer fixed

arithmetic

Logic Utilization Used Available Utilization

Number of 4 input LUTs 94 4, 896 2%

Number of occupied Slices 50 2, 448 2%

Number of Slices containing only related logic 50 50 100%

Total Number of 4 input LUTs 96 4, 896 2%

Number of bonded IOBs 64 158 40%

Number of MULT18X18SIOs 3 12 25%

Total equivalent gate count for design 975

Table 2. 3. Implementation report of multiplication using Q-Format

arithmetic

Logic Utilization Used Available Utilization

Number of 4 input LUTs 30 4, 896

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From the simulations and synthesis report, the multiplication of two

unsigned data 0.25 and 0.75 is 0.1875 which is the actual result. From

the results shown in Fig. 2.31, the result achieved using Q- Format is

0.1875 which is the actual result. For the same data the result achieved

from integer fixed point format is 0D00H. The actual result evaluated by

the digital controller is 0.203125.

In Q- Format, the result is 0.1875.

In integer fixed point format, the result is 0.203125.

The error is evaluate as

0.1875 -0.203125 = -0.015625 (-1.56% for one multiplication operation)

This procedure utilizes more resources in the FPGA.

2.4. Realization of QALU based PWM control through FPGA

In order to analyze resource utilization in FPGA and to validate the

possibility of single chip implementation of PWM, QALU based SVPWM is

simulation using ModelSim 5.7 and synthesis using Xilinx 9.2i are

considered.

The practical feasibility of QALU for PEC control is verified by

applying to SVPWM generation. The SVPWM patterns have been obtained

for different fs and fo. The result for SVPWM pulse patterns in 6 sectors

with fs=20 kHz is shown in Fig. 2.8. The SVPWM patterns have been

generated using the proposed Q-Format based signal processing.

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2.4.1. Experimental Results

The Experimental setup for QALU based SVPWM controller fed three

phase VSI is shown in Fig. 2.9. The experimentation has been carried out

with the FPGA hardware SPARTAN XC3S400PQ208 from Xilinx. Inc. The

hardware setup consists of FPGA, three phase bridge inverter with

IGBTs, bridge rectifier to supply DC voltage, opto isolator, pulse driver

and induction motor. In the experiments, the fo has been varied from 0.3

Hz to 50 Hz and the fs is varied from 1 kHz to 15 kHz, and the PWM

switching patterns are achieved. The results for PWM output in the

different channels are obtained and the PWM patterns in the channels

P1-P4 are shown in the Fig. 2. 10. The three phase inverter output wave

forms with varying fo of 15 Hz to 50 Hz and 1.157 kHz to 12.28 kHz

varying fs is obtained. The result of the inverter output line to line voltage

is shown in Fig. 2. 11.

Fig. 2. 8. SVPWM waveforms in 6 sector with fs =20 kHz

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Fig. 2.9. Experimental setup of the Q-Format ALU applied to FPGA-SVPWM controller fed induction motor

Induction motorFPGA

Power module

Driver circuit

Displacement between two voltage waves

50 V

/ d

iv

20 ms / div

Fig. 2. 11. Inverter output voltage, U Y-B and U B-R Line to linevoltage: f0 = 20 Hz, fs =12.28 kHz

Pulse P1

Pulse P4

Fig. 2. 10. SVPWM wave form in channels P1 and P4with fs =1.157 kHz

5 V

/ d

iv

25 ms /div

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2.4.2. Discussions

The existing PWM control in power electronic converters is

implemented using DSP and FPGA technologies. In the conventional

PWM control implementations, the integer fixed point based signal

processing is used which takes more FPGA resources for a processing a

part of algorithm and also it requires an additional controller.

The first feature of QALU is, it performs the arithmetic computations

in the algorithm, and it is possible to implement the PWM controller in a

single FPGA. And also the host processor such as DSP is not required.

Therefore, the system complexity is reduced.

From the simulation and experiments, the practical possibility of

QALU applied to a PEC control is verified. The results show that the Q-

Format implementation takes total gate count of 348 and integer fixed

format implementation takes 975. The Q-Format takes less chip

resources and due to this feature the speed of execution will also be

increased. The simulation result for 8 bit multiplication using Q-format

and integer fixed point format is presented in Fig. 2. 7. It is proved that,

the error in Q-format based operation is zero and in integer fixed point

format, it is -1.56%. Therefore, the accuracy has been increased by Q-

Format when compared to integer fixed format. The PWM modulator with

QALU is implemented and the possibility of practical implementation is

verified.

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2.5. Conclusion

The preliminary requirements to implement the QALU design such as

digital data representation in different data formats and their arithmetic

computation has been discussed. The advantages of QALU are verified

with simulation and synthesis. The QALU is designed and developed

using VHDL which performs the arithmetic and logic functions in the

algorithm. To facilitate the digital implementation of QALU, the single

chip FPGA implementation has been developed. By employing the QALU,

the algorithm accuracy has been improved as well as the FPGA resource

utilization is tremendously reduced. Due to this superiority, the IP core

designed on FPGA requires less resource and provides improved

accuracy in single chip FPGA implementation compared to conventional

integer fixed point for signal processing. The proposed signal processing

concept and QALU can be used for PEC control applications to generate

the accurate PWM pattern in real time. Moreover, the proposed signal

processing method is more suitable for single chip FPGA and SoC

implementations. The main objectives such as implementation of entire

PWM control in a single FPGA and area efficient digital PWM control is

verified.