Mentor Graphics Tutorial: Schematic Capture, Simulation, & Placement/Routing 1.0 Introduction This tutorial demonstrates a simple VLSI circuit design process from concept to chip layout of an 8-bit Modified Booth Multiplier on a 0.5µm process using software from Mentor Graphics Corp. The topics covered in this tutorial include schematic capture & design, simulation, and placement & routing. 2.0 Schematic Capture & Design The specification for the design calls for an 8-bit unsigned multiplier that accepts two 8-bit unsigned inputs and produces a 16-bit unsigned output as a result. Three additional control signals are required, including a DONE, START, and CLOCK signal. The implementation presented in this tutorial consists of these baseline requirements, in addition to a RESET control signal. Also, the multiplier accepts signed inputs and therefore performs signed multiplication. We use Booth’s Modified Algorithm as the underlying architecture for the design due to its ability to produce a result quickly and reliably. In an effort to better understand the derivation of the design, a brief description of Booth’s Modified Algorithm follows. 2.1 Booth’s Modified Algorithm On average Booth’s Modified Algorithm can produce results in approximately half the time that the traditional “add and shift” multiplier can. This is because Booth’s Modified Algorithm “looks” at strings of three bits simultaneously with a one-bit overlap in each successive comparison in order to decide what to do next. Since one bit of each string of three bits overlaps with the previous triplet, two new bits are effectively considered during each clock cycle. In the case of an 8-bit multiplier, this means that calculations can optimally be performed in 4 clock cycles, excluding additional control states. The following is a more formal definition of the Modified Booth Algorithm. Let x be the multiplier and m be the multiplicand. Let two bits of x plus the last bit from the previous two bits represent the triplet xL. Assume x and m are n-bit signed binary numbers. The triplet xL can be represented by the following vector:

12212 ,, −+= yyyL xxxx (1)

where 12;2

,...,2,1,0 += yxny is the first bit of the triplet, x2y is the second bit of the triplet,

and x2y-1 is the overlapped bit from the previous triplet. Letting xi be the ith bit of x and let x-1 = 0, the two’s complement of x can be written as

( )∑−

=

−− ⋅+⋅−=

2

0

11 22

n

i

ii

nn xxx (2)

( )( )∑−

=−+ ++⋅−⋅=

1

012212

22

22n

yyyy

y xxx (3)

The product ( )xmp , can be expressed as

( ) ( )( )∑−

=−+ ++⋅−⋅⋅=

1

012212

22

22,n

yyyy

y xxxmxmp (4)

( )∑−

=

⋅=1

0

22

,2n

yy

y mxδ (5)

where ( )mxy ,δ represents the Modified Booth recoding function and is defined by the piecewise function:

(6)

In order to implement this algorithm in hardware, several conclusions must be drawn from the relationships above. From equation 5, it follows that an 2

n -operand adder is necessary to cumulatively sum the 2

n terms during each successive multiplication. For the Modified Booth recoding function defined in equation 6, a combinational logic network is necessary in order to produce the recoded multipliers 0, ±m, and ±2m from m, the multiplicand. This algorithm reduces the total number of additions from n to 2

n at the cost of extra logic to generate and select the necessary recoded multipliers [1,2]. 2.2 Design & Methodology The Mentor Graphics Design Architect tool is used in this tutorial for schematic capture and design. Due to the nature of schematic capture, a hierarchy consisting of encapsulation and abstraction is used to make the design more modular and comprehensible. The ADK libraries consist of all the necessary standard cells needed to build each functional unit comprising the circuit, and therefore they will be used extensively. The top level circuit schematic, presented in figure 1, consists of 7 primary functional units. In total, there are 11 functional units that encompass the design.

( )=mxy ,δ

1,0,0, =yxm0,1,0, =yxm1,1,0,2 =⋅ yxm0,0,1,2 =⋅− yxm1,0,1, =− yxm0,1,1, =− yxm

0,0,0,0 =yx

1,1,1,0 =yx

Figure 1. Top-Level Circuit Schematic

Some components contain other custom symbols, and therefore these symbols cannot be found via the ADK libraries, rather they must be imported by clicking on the CHOOSE SYMBOL button in the ADD/ROUTE schematic palette window. To import a symbol, you must navigate to the directory containing the symbol. For each functional unit, a symbol must be generated in order for the component to be used elsewhere. As stated earlier, the use of symbols make the schematic more comprehensible. In addition, the design makes extensive use of data buses. Buses allow bits to be grouped so as to avoid having separate wires for each bit running all over the place. Grouping bits into a bus simplifies the overall look of the design and is therefore recommended. A bus can be instantiated by clicking on the ADD BUS/BUNDLE button on the ADD/ROUTE schematic palette window. After a bus route has been placed, the name and size must be specified. To do this, select the bus by clicking on it with the LMB. Choose the Net→Name Nets popup menu item with the RMB. Enter the name that defines the bus in the Property Value text box. Click the OK button. Move the cursor in the schematic window. Drag the bus name to the location you want it displayed. Click the LMB to fix the text position. The naming of the bus should conform to the following format: bus_net_name(msb:lsb), where bus_net_name is any name of your choice, msb is the most significant bit, and lsb is the least significant bit. Next, add a wire to the bus by clicking on the ADD WIRE button on the ADD/ROUTE schematic palette window and attach it to the bus. A pop-up menu will appear asking you which bit you would like to

use. A bus does not have to be named in the event that two symbols with common bus sizes are to be connected. Ports (inputs/outputs), GND & VDD, basic logic gates, flip-flops, transistors, pads, etc. can be added by navigating to the ADK libraries under the Libraries pull-down menu. This will display the ADK libraries palette menu to the right of the screen. This concludes the basic foreknowledge needed in order to reproduce the multiplier presented in this tutorial. The tutorial will proceed as follows: Each functional unit will be presented and a brief description of each will be given. 2.2.1 The Control Unit Figure 2 presents the Design Architect circuit schematic for the control unit. The main control unit is implemented as a finite state machine consisting of eight states.

Figure 2. Control Unit Circuit Schematic (control)

Figure 3 presents the state diagram flow chart and the corresponding state table for the control unit. From the diagram, the reset and ready signals control which state the multiplier is in. The reset signal has the effect of clearing all storage units in the multiplier in addition to reinitializing the current state to zero, while the ready signal only has an effect in states zero and seven.

Figure 3. Control Unit State Definitions

The primary function of the control unit is to manage the state of the multiplier. The inputs to the control unit consist of a ready signal, which signals the start of a multiplication cycle, a clock signal; and a clear signal, which resets the state of the entire multiplier. The output signals of the control unit consist of a 3-bit bus, which defines the current state of the multiplier; a done signal, which signifies that the multiplication has finished; and an enable signal, which enables storage of the multiplicand into an 8-bit register for recoded multiplier generation. The control unit was designed using D-Flip-Flops, and the equations were derived from the state table in figure 4 below.

QS2 QS1 QS0 RDY RST Q'S2 Q'S1 Q'S0

0 0 0 0 0 0 0 00 0 0 1 0 0 0 10 0 1 0 0 0 1 00 0 1 1 0 0 1 00 1 0 0 0 0 1 10 1 0 1 0 0 1 10 1 1 0 0 1 0 00 1 1 1 0 1 0 01 0 0 0 0 1 0 11 0 0 1 0 1 0 11 0 1 0 0 1 1 01 0 1 1 0 1 1 01 1 0 0 0 1 1 11 1 0 1 0 1 1 11 1 1 0 0 1 1 11 1 1 1 0 0 0 1X X X X 1 0 0 0

PRESENT INPUT NEXT

Figure 4. Control Unit State Table

In an effort to minimize the amount of logic gates required to implement the state machine, the above table was placed into three 4-bit Karnough maps. The following figures present the Karnough maps obtained from the state table shown in figure 4.

00

11 77

44

33

22

55

66

XX11,, 0000

1100

XX00

XX11

XX00

XX11

XX00

XX11

XX00

XX00

XX00

XX11

XX11

0000

XX11

1100

STATE DEFINITION0 0 0 CLEAR / WAIT FOR RDY SIGNAL0 0 1 RDY ASSERTED / LOAD MULTIPLICAND0 1 0 LOAD MULTIPLIER0 1 1 ADD RECODED MULTIPLIER AND SHIFT BY 21 0 0 ADD RECODED MULTIPLIER AND SHIFT BY 21 0 1 ADD RECODED MULTIPLIER AND SHIFT BY 21 1 0 ADD RECODED MULTIPLIER AND SHIFT BY 21 1 1 WAIT FOR RDY SIGNAL AND ASSERT DONE

CONTROL STATES

00 01 11 1000 0 0 0 001 0 0 1 111 1 1 0 110 1 1 1 1

DS2: QS2,QS1 / QS0,RDY

00 01 11 1000 0 0 1 101 1 1 0 011 1 1 0 110 0 0 1 1

DS1: QS2,QS1 / QS0,RDY

00 01 11 1000 0 1 0 001 1 1 0 011 1 1 1 110 1 1 0 0

DS0: QS2,QS1 / QS0,RDY

Figure 5. Control Unit Karnough Maps

Following the process of map simplification, the equations listed below are obtained.

The following is a brief explanation of the derivation of the state table listed in figure 3. Upon initialization, the state of the multiplier is unknown. Therefore, to ensure that the multiplier is in a known state, the external reset signal must be asserted to clear the contents of the state machine as well as the result register. When this is done, the multiplier enters state 0 and stays in this state until the ready signal is asserted. When the ready signal is asserted, the multiplier enters state 1, and it loads the multiplicand into an 8-bit storage register for recoded multiplier calculations. At this time, the state machine asserts the enable signal to allow the multiplicand to be loaded and it then de-asserts the enable signal on the next clock edge. The state machine enters state 2 during the next clock cycle, where it loads the multiplier into the lower byte of the result register from the input. Due to external pin limitations, the two 8-bit input operands (the multiplicand and the multiplier) are loaded sequentially with the same 8-bit input bus. The following four states are required to produce the product. The product is calculated by adding the proper recoded multiplier to the upper byte of the result register and then performing a right 2-bit shift during each clock cycle. During state 7, the done signal is asserted and the content of the result register is stable and valid. The multiplier stays in this state and the output is valid until the ready signal is asserted again. The figure below presents a timing diagram of the circuit.

RDYQQQQQQD SSSSSSS 0201011 ++=

12002010 SSSSSSSS QQRDYQQQQQD +++=

120122022 SSSSSSSSS QQQQQRDYQQQD +++=

Figure 6. Timing Diagram 2.2.2 An 8-bit Multiplicand Register An 8-bit register was designed with D-Flip-Flops with the sole purpose of storing the input multiplicand for recoded multiplier calculations. Figure 7 presents the circuit schematic for this component.

Figure 7. Multiplicand Register Circuit Schematic (8register)

Booth’s Modified algorithm works by adding these so called “recoded multipliers” to the partial sum of the upper byte of the result register. The recoded multipliers are generated according to the Modified Booth recoding function defined in equation 6. The register has an enable signal, which is asserted only during state 1 in order to load the multiplicand. The contents of the register are held constant throughout the rest of the multiplication cycle.

S0 S1 S2 S3 S4 S5 S6 S7

A (7:0)

B (7:0)

RESET

DONE

READY

CLOCK

R (15:0)

2.2.3 Multiplicand Select Decoder Unit Figure 8 presents the circuit schematic for this component. The purpose of this component is to select the appropriate recoded multiplier based on the lower 2 bits of the result register and the carry out bit. The input to this component consists of a 3-bit bus containing the result bits just described, and the 4-bit output bus contains an enable bit signal for each of the four non-zero recoded multipliers.

Figure 8. Multiplicand Select Decoder Circuit Schematic (mdecoder)

2.2.4 The Recoded Multiplier Unit As probably deduced from the title of this section, this component serves to simply generate each of the four non-zero recoded multipliers. Figure 9 presents the circuit schematic for this component. One of the four multipliers is selected based on a 4-bit input signal generated by the multiplicand select decoder. The contents of the 8-bit multiplicand register serve as the other input to this component. The outputs consist of 4 10-bit recoded multipliers, in which one of the recoded multipliers will be active, depending on the input select signal from the multiplicand select decoder.

Figure 9. Recoded Multiplier Circuit Schematic (recodemult)

2.2.5 An 8-bit 2’s Complement Unit This unit simply produces the 2’s complement of the input. The input is 8 bits wide, and the output is 8 bits wide with two additional bits for sign extension. Additionally, a sign output bit gives the polarity of the result. This component is used in generating the recoded multipliers for the recoded multiplier unit. The circuit schematic for this component is presented in figure 10.

Figure 10. An 8-bit 2’s Complement Unit Circuit Schematic (82scomplement)

2.2.6 The Addend Unit This unit simply consists of 10 4-bit OR gates. The gates are reconvergent so as to allow one of the four recoded multipliers to pass as input to the partial sum of the product register. The multiplicand select decoder enables only one of the recoded multipliers by AND-ing that particular multiplier with a logical high and AND-ing the remaining three recoded multipliers with logical low. Therefore each of the 4-bit OR gates will have three guaranteed logic “0” inputs, while the other bit will be the ith bit of the enabled recoded multiplier. The output is 10-bits wide. The circuit schematic for this component is shown in figure 11.

Figure 11. Addend Circuit Schematic (addend)

2.2.7 A 10-bit Full Adder A 10-bit full adder is used in computing the partial sum of the upper byte of the result register. The inputs to the adder consist of the upper byte of the result register including two bits for sign extension, and the other 10-bit input comes from the addend unit. The 10-bit result is re-deposited into the upper byte and sign bit of the result register. The circuit schematic for this component is shown in figure 12.

Figure 12. 10-bit Full Adder Circuit Schematic (8fadder)

2.2.8 A 16-bit Result Register This component is by far the most complex logic block in the circuit. The result register is 16-bits wide and contains three additional D-Flip-Flops, two for sign extension, and the other to store the shift-out bit. Therefore, the result register is actually 19-bits wide, but only the 16 bits are available to the user. The result register can be in one of three modes: load, hold, or shift. The load mode has two separate contexts for the upper and lower bytes of the result register. For the lower byte, the load mode loads the input multiplier. For the upper byte, the load mode simply loads “0”, effectively clearing the byte. This mode corresponds to state 2 of the finite state machine. The hold mode keeps the contents of the result register regardless of changes at the input. This mode is valid for state 7 of the finite state machine. The shift mode shifts the contents of the result register to the right by two bits during each successive clock cycle. This mode is valid for states 3, 4, 5, and 6 of the finite state machine. The circuit schematic for this component is presented in figure 13.

Figure 13. The 16-bit Result Register Circuit Schematic (16shft2reg)

In order to implement the result register, special D-Flip-Flops were designed to perform these modes of operation. The next section discusses these special D-Flip-Flops. 2.2.9 D-Flip-Flops with Load, Hold, & Shift Load, hold, & shift modes of operation were added to standard cell D-Flip-Flops in order to avoid including an additional 16-bit register to store a stable and valid result upon the completion of a multiplication cycle. The need for these special flip-flops stems from the fact that three inputs needed to be multiplexed with each D-Flip-Flop in the result register. Some control logic was needed in order to properly multiplex the inputs with the select signals. The corresponding states of the result register in response to each of the 8 states generated by the state machine are listed in the table of figure 14.

DEFINITION

0 0 0 CLEAR REGISTERS0 0 1 CLEAR MULTIPLICAND0 1 0 LOAD MULTIPLIER0 1 1 SHIFT BY 21 0 0 SHIFT BY 21 0 1 SHIFT BY 21 1 0 SHIFT BY 21 1 1 HOLD / QNEXT = QPRESENT

REGISTER CONTROL

Figure 14. Result Register State Definitions

The state table, shown in figure 15, was derived from the result register state definition table in figure 14.

QS2 QS1 QS0 L0AD HOLD SHIFT

0 0 0 0 0 00 0 1 0 0 00 1 0 1 0 00 1 1 0 0 11 0 0 0 0 11 0 1 0 0 11 1 0 0 0 11 1 1 0 1 0

Figure 15. Result Register State Table Placing the above table into three separate Karnough maps, the following diagrams listed below in figure 16 are obtained.

00 01 11 100 0 0 0 11 0 0 0 0

LOAD: QS2 / QS1,QS0

00 01 11 100 0 0 1 01 1 1 0 1

SHIFT: QS2 / QS1,QS0

00 01 11 100 0 0 0 01 0 0 1 0

HOLD: QS2 / QS1,QS0

Figure 16. Result Register Karnough Maps

The equations governing the necessary control logic needed to implement the special flip-flops were derived from Karnough maps shown in the figure above and are listed below.

This control logic is embedded within the special D-Flip-Flops, and they are in turn encapsulated in the register mode decoder, which is the subject of the next section. Figure 17 presents a circuit schematic for this component.

012 SSS QQQLOAD =

0201212 SSSSSSS QQQQQQQSHIFT ++=

012 SSS QQQHOLD =

Figure 17. D-Flip-Flops with Load, Hold, & Shift Circuit Schematic (dfflhs)

2.2.10 The Register Mode Decoder This is the last significant logic block to be discussed. This logic block decodes the current state of the control unit into select lines that are used to multiplex the D-Flip-Flops with load, hold, and shift capabilities. The input consists of the 3-bit state of the control unit (this serves as the select input) as well as the inputs for the load, shift, and hold lines. The output is one of the load, hold, or shift lines. Figure 18 shows the schematic diagram for this component.

Figure 18. The Register Mode Decoder Circuit Schematic (rdecoder) Assuming some familiarity with the Mentor Graphics Design Architect tool, and having presented each component that comprises the design, in addition to including circuit schematics, one should be able to reproduce this implementation of a Modified Booth Multiplier with little effort. 3.0 Simulation The modified booth multiplier inputs (multiplier and multiplicand) need to have the binary numbers sent to them individually. If the pattern generator is set to count from 0 to 255 with two separate sets of 8 binary digits, it will not iterate as expected. For this reason, the 8-bit multiplier and multiplicand should be combined into a 16-bit bus. If the pattern generator is set to count a 16 bit binary number from 0 to 65536, then the multiplier can count from 0 to 255 every time the multiplicand counts a single binary digit. In this implementation, we make use of buffers for the combination of the multiplier and multiplicand. A 16 bit input bus goes into the bus combine component and is split into two 8-bit buses (one goes to the multiplier and the other to the multiplicand).

Figure 19. Bus combine Circuit Schematic

3.1 Adding Forces

For this simulation, the clock is set to the maximum operating frequency of 30.3Mhz, for an equivalent period of 33ns. This is done by clicking on the clock input line and adding a clock force. The clock period should be 33ns (see figure 20).

Figure 20. Clock signal window

Next, the multiplier needs to be cleared on the first clock cycle. This is done by clicking on the reset input line and adding a force to the reset line. The reset line will have a value of 1 at time 0ns and a value of 0 at 33ns. The multiplier should be ready to multiply all the 65536 values after the reset cycle. To set the multiplier to be ready, the ready input line is clicked and a force is added to the ready line. The ready line will have a value of 0 at time 0 and a value of 1 at 33ns.

Figure 21. Force signal window

3.2 Pattern Generation

The input pattern for the multiplier and multiplicand must be created. To do this, click on the 16 bit input bus going into the bus combine component. Next, click on the PATTERN GENERATOR icon. The pattern generation should begin 33ns after the assertion of the reset signal. Since the multiplier needs 7 cycles to make a multiplication, 7*33ns (or 231ns) is needed per multiplication cycle. A total of 65536 patterns are needed, so this requires that the entire pattern sequence should be 65536*231ns or 15,138,816ns (15.1ms) long.

Figure 22. Pattern Generator window

3.3 Running the Simulation and Viewing Results To view the relevant traces, select the multiplicand, multiplier, and result lines. Next, add the selected traces in hexadecimal format. The other single bit lines can be directly added by clicking on the TRACE or LIST button. Having completed this, the inputs should be set and added to the list; however, the simulation still needs to be completed. The simulation should be allowed to run slightly longer then the total time of the pattern generation. Type “run 15139000” to run the pattern sequence.

Figure 23. Trace results

After the simulation has completed, the data in the trace list should be exported. To do this, click on the LIST icon, and then go to the file menu and export report. 4.0 Placement & Routing 4.1 Pad Placement To start the creation of an IC design the 88mult component must be connected to pads for I/O and power. Certain pads may be reserved for VDD and ground depending on your chip layout type. AMI 0.5 µ technology is used to create the pad layout for this multiplier.

Figure 24. Pad Layout

4.2 Internal Core Layout The core layout will automatically be generated, as shown in figure 25. Some design rules may be violated in the creation process and must be manually corrected. See Dr. Milenkovic’s website for some tutorial tips. Typing the command “Peek” followed by a number will reveal that amount of hidden layers so that errors can be more easily fixed.

Figure 25. Core Internal Routing

4.3 Design Rule Check Errors You might encounter one or more of the following errors when performing a design rule check. It is important that all DRC errors be corrected in order to increase the probability that the chip will perform as expected after fabrication. 4.3.1 “Via must NOT be stacked with contact” Select the via and then right click, select edit→move→unconstrained. Move the via over 2λ’s. Then add metal layer 1 (or metal 2) in between the via and trace. Use the same metal layer that the trace is made out of. Right click, select add→shape, and click options on the pop-up menu that appears. Select metal 1 (or metal layer 2) to add between the via and trace. Figure 26 shows this DRC error and the corresponding correction.

Figure 26. “Via must NOT be stacked with contact”

4.3.2. “Port must be completely covered with Metal” Add metal layer 1 (or metal 2) over the white indicator box. Use the same metal layer that the trace is made out of. Right click and select add→shape, click options on the pop-up menu, and select metal 1 (or metal layer 2) to cover the white indicator box with metal. Figure 27 shows an example of the DRC error.

Figure 27. “Port must be completely covered with Metal”

4.3.3 “Metal spacing = 4L” Add metal layer 1 (or metal 2) in between the white indicator lines. Use the same metal layer that the trace is made out of and only extend the metal coverage to the length of the shortest white line. Right click, select→shape, click options on the pop-up menu and select metal 1 (or metal layer 2) to cover the gap between the two white indicator lines. Figure 28 shows this particular DRC error and a typical fix.

Figure 28. “Metal spacing = 4L”

4.4 Core Placement and Routing

The core of the chip should be centered inside the pad frame. Traces should connect the core to the pad frame. The traces have two layers. If manual routing is required, then the route should be placed to connect the core to the frame by changing layers when necessary to avoid inadvertent connections between two separate traces. You may find the “preferred route” facility of the auto-route feature useful in performing manual routing.

Figure 29. Core Placement and Routing

5. Conclusion

In this paper, we have endeavored to present the design of an 8-bit Modified

Booth multiplier in an effort to demonstrate an approach that can be taken toward the design and verification process of a VLSI circuit with the Mentor Graphics tools. We have included circuit schematics, timing diagrams, as well as simulation procedures in this technical paper in an effort to convince the reader of the correctness of the design. Using design techniques such as those presented in this paper, any VLSI circuit, whether complex or simple, can be successfully realized.