# Components of Digital Computers

Processor hardware is built out of three main conceptual pieces: wires, gates, and registers. These behave in some ways much like variables, operators, and data structures but in other ways they are distinct. Understanding their semantics is key to understanding hardware design.

## Current, Voltage, Power, and Heat

Ignoring many of the nuances of electricity and speaking only in broad generalities,

Current
is the rate at which electrons flow. In practice it is not always electrons that are the conceptual charge carriers and amperage does not map directly to electrons/second, but that is the general idea.

If electrical state changes, current is involved.

Voltage
is the desire electrons have to flow; the electrical pressure, if you will. Voltage does not always cause current because it contends against resistance: not all substances conduct electricity very easily.

Most elements of computer hardware are sensitive to voltage. The “1”s are “high voltage” and “0”s are “low voltage”. How high depends on the underlying design of the components, but chips with highs of around 1 volt are common today (2018). Low voltage is generally close to 0 volts.

Power
is current × voltage. It does not require power to maintain voltage without current, nor to maintain current without voltage.

Since chip state is voltage and change in state requires current, every time that a processor changes state it uses power. In practice some power is used even between state changes too, since it is hard to maintain voltage in a changeable system without some current “leakage”.

Heat
is (by the law of entropy1) an inevitable byproduct of power. It is also a major concern for chip manufacturers. Chips are so small and the power they use so concentrated that even a small amount of heat can result in a large temperature change2. Electrical properties tend to change with temperature, and notably we have trouble making microscopic components that function like logical gates at higher temperatures.

It is thus vital that chip temperature be kept low enough that the chip continues to function. Heat sinks, fans, and other cooling systems can help, but their impact is limited. To reduce heat we must reduce power; and to reduce power we either need to reduce voltage (meaning more sensitive and harder-to-engineer components) or reduce current (meaning less state change and thus a reduction in chip speed).

In this course we will not explore designing for limited power usage, but the outline given above is useful to be conversant in hardware design issues. It also explains why tablets tend to be slower than servers: they have less power to draw from (a battery instead of a connection to the power plant) and less space to devote to heat dissipation (heat sinks, fans, and so on) so they cannot change as much voltage as quickly as the servers can.

## Multi-bit “wire”s and “gate”s

We discussed gates in Boolean Algebra and Gates; these are connected by wires to make logic chips. Note that each wire needs to have one source (either an input signal or the exit point of a logic gate) and may have any number of sinks (output signals and the entry points of gates). There are many other nuances to consider if you are designing an actual physical chip, but they will not be necessary for this course.

Usually, we don’t want to work on single bits at a time but rather on multi-bit signals.

A multi-bit extension of a wire is just a cluster or wires run next to each other. Eight wires running together can transmit eight bits of information in their voltages. Such a group of wires is sometimes called just a “wire” or sometimes “an 8-bit-wide wire,” “a wire carrying an 8-bit value,” or “an 8-bit wire.”

A multi-bit mux has multi-bit inputs and output, all the bits of which share a single selector. They can be implemented as a separate 1-bit mux for each bit of output:

out_0 = selector ? in1_0 : in0_0;
out_1 = selector ? in1_1 : in0_1;
out_2 = selector ? in1_2 : in0_2;
...


In most programming languages, the ?: operator acts like a multi-bit mux. Similarly, the operators ~, &, |, and ^ act like multi-bit logic gates (so 0b0011 ^ 0b0110 == 0b0101), + like a multi-bit adder, and so on.

### Transitioning multi-bit values

It is almost never the case that all of the gates on all of the individual wires of a multi-bit component produce output at exactly the same speed. This means that during the transition time, the outputs of logical components cannot be trusted to be any valid value.

Suppose I have an adder with inputs 0b0001 and 0b0110; the adder is currently producing output 0b0111. Now I change the second input to 0b0111. Eventually the output will change to 0b1000, meaning all four bits change. However, I cannot in general predict the order in which the bits will change; if (for example) the two low order bits change first (expected for the kind of adder presented earlier in this text) then there will be a moment in time when the adder is outputting 0b0100. But 0b0100 is not the correct output for either the old or the new inputs.

Not all orders of changes are possible, but it is usually the case that the outputs of a logical component goes through a period of incorrect values during each transition. It is thus important to design circuits such that these transitional outputs are ignored and not committed to the programmer-visible state of the processor.

## Registers

Consider the following circuit3:

As we stated in the previous section, the different bits of the output of the adder appear at different times. This means that if we start with $$x = 0$$ we might see a sequence of x values like 0, 1, 0, 2, 1, …: in other words, not the sequence we want to see. To fix this we add a register4 to the circuit:

Registers have one input and one output, as well as a second single-bit input called the “clock” (drawn at the top of $$R$$ in the image above; the triangle is a traditional marker for a clock input). They store a value inside them, and most of the time they completely ignore their input and output that internal value. However, at the moment when the clock input transitions from low- to high-voltage (and only at that moment) they read their input into their internal stored value.

The clock inputs of all registers across the entire chip are (conceptually) attached to the same “clock”, a oscillator that outputs low-voltage, then high-voltage, then low-voltage, then high-voltage, …, at predictable intervals.

Adding the register solves the unpredictable output problem. The current value of $$R$$ and 1 are sent into the adder; its output jumps about a bit but stabilizes at the value $$R+1$$; then the clock changes from 0 to 1 (called the “rising clock edge”) and the new value is put into $$R$$. Once the new value is in it is output and the adder output starts jumping around again, but that is fine because the register will ignore it until the next rising clock edge.

### Clock Speed

The clock is important enough to have become one of the most noted stats about chips: a 2GHz clock speed means the clock oscillator completes 2 billion ($$2×10^{9}$$ not $$2×2^{30}$$) low-high cycles per second. Because virtually every part of a chip contains registers, and because all registers are attached to the clock, clock speed is a direct indicator of overall processor speed.

So, why not just add a faster oscillator to make the chip faster? There are two main reasons:

• Faster clock → more changes per second → more current → more power → more heat. Some chips are designed to reduce their own clock speed either to conserve battery life or to manage their temperature.

Access to plenty of power and a more efficient cooling system can reduce this constraint.

• It takes time for electrons to move across wires and through transistors. If the clock runs too fast it might grab a pre-stabilized value and cause computations to be incorrect. A faster computer that sometimes decides that 2 + 3 = 1 is not as useful as a slower one that always knows it is 5.

There are two main ways to overcome this limit on effective clock speed:

• Make the chip smaller. Smaller transistors and shorter wires → less distance for charge carriers to travel → faster operation. For many years this tactic yielded impressive gains every year, but components around than 30 atoms across5 entered mass production in 2019; it is not clear how long this trend can continue.

• Reduce the work done between each clock cycle. Less work → less logic to wait for → faster clock; we’ll learn more about this as we discuss pipelining. Once chip shrinkage stopped yielding big speedups, pipelining was seen as the next thing to increase; but it also has limitations and more-or-less optimal-depth pipelines have been determined.

Some people chose to overclock their chips. Roughly, they take a chip that was designed to run at one speed and they attach a faster clock to it. Since the shipped clock speed is generally a conservative estimate, sometimes this just works and the processor becomes faster at no extra cost. But sometimes the new clock speed is too fast and the computer overheats and/or fails to compute correctly. Worse yet, often the problem is not immediately manifest, perhaps only overheating after months of use or only making an error in computation on one out of every billion computations.

## Memory

Conceptually, memory acts like a huge array or list of bytes, or like a giant mux in front of a huge bank of registers. It is implemented much more cheaply than registers are, though also in a way that runs more slowly.

In general, processors interact with memory via loads and stores. In a load, the processor sends it an address and a number of bytes and it sends back that many bytes from that address and the next few. In a store, the processor sends it an address and one or more bytes and it changes the contents of the memory at those addresses. There is not particular magic, from the processor’s perspective, to this: the address and values are all just a bunch of wires running between gates within the processor on one end and the similar gates within the memory chip on the other.

Modern computers add significant complexity to this basic design, with address translation to achieve virtual memory and a hierarchy of different caches to make memory seem faster, but the core interface from the heart of the processor’s point of view is just some address and value wires. We’ll discuss more about these topics in CSO2.

## Code-to-Hardware Compilation

There are several code-to-hardware compilers; that is, programs that, given source code as input, output hardware layout that can be used to make physical computer chips. The best known of hardware description languages (which are designed for this code-to-hardware conversion) are Verilog and VHDL. The full design of these languages is out of scope for this course, but it is instructive to understand how they can work.

### Variables and assignment

A (multi-bit collection of) wire(s) can be named, like a variable. However, these have special constraints. Because these need to be wired into a physical circuit they can only have one source6; in code, this means each variable can appear on the left side of a = only once. The wires can fork and be used in several places, but only set in one place.

Example: The following is valid:

x = 3;
y = x ^ (1 + x);
z = 1 - y;


but the following is not legal, as it defines x twice:

x = 3;
y = x ^ (1 + x);
x = 1 - y;


### Operators

Most of the operators that common programming languages allow on integers can be implemented fairly easily in hardware. Just how simple “easily” means varies widely: x & y only needs one gate per bit, x + y about five times that many; x * y can easily approach a thousand; and x / y many more still. At least as important as how many gates are involved is how many gates are involved in the longest path between input and output, because that directly impacts how long it takes for all of the gates in the operation to stabilize on a new value after their input changes.

Comparison operators, like x < y, are much like subtraction in complexity and implementation, but are relatively uncommon to perform directly in hardware with two variables. More common is to compare variables to constants. In almost every case, having a fixed values as one input of an operator allows it to be implemented in far fewer gates than if both are variable.

In addition to operators common in other languages, hardware description languages make heavy use of the trinary operator. Code b ? x : y can be made in hardware with a mux, which can be just three gates per bit:

### Registers and memory

Registers are used to implement any kind of circular dependency or change over time. Each register has an input and an output, and with very few exceptions all computations compute register inputs from register outputs. Like variables, register inputs can only be specified once; register outputs can be used multiple times.

Example: Syntax for interacting with registers varies by language; for this example, assume R is a register with input R_in and output R_out.

The following implements a simple counter:

R_in = R_out + 1;


Note that R_out = ... is not permitted, as R_out is set by the register itself.

It is common to have a special “register bank”, a set of registers that can be accessed like a fixed-sized list or array. The full implementation of these is slightly more nuanced than is worth our attention here, but a simplified version is described in the following example.

Example: Suppose we want to implement a simplified bank of 4 registers.

The code R[i] = x can be implemented as

R0_in = (i==0) ? x : R0_out;
R1_in = (i==1) ? x : R1_out;
R2_in = (i==2) ? x : R2_out;
R3_in = (i==3) ? x : R3_out;


The code x = R[i] can be implemented as

x = (i==0) ? R0_out :
(i==1) ? R1_out :
(i==2) ? R2_out :
R3_out;


A fully functional register bank would have somewhat more complicated logic so that, for example, two different indices could be assigned to without violating the single-source rule; but the general principle applies.

Memory tends to be a lot slower than registers, and is implemented with somewhat different logic to handle its very large size, but the interface is conceptually similar to a register bank and can be considered similar to an array for our purposes.

### Register-Transfer Level Coding

One way of thinking about the combined programing components listed above is to separate out the logic and the registers. The logic is unclocked gates connected by wires, in an acyclic single-source way; the registers are treated like a separate set of clocked storage units.

Each clock cycle runs as follows:

1. The clock signal goes from low- to high-voltage (called the “rising edge” of the clock), causing the registers to take their current inputs into their internal storage.
2. The registers start outputting their new values.
3. The logic, with new inputs, begins to adjust, new values rippling through the gates until eventually a steady state is reached (the existence of a steady state is guaranteed by the acyclic nature of the logic).
4. The outputs of the logic are inputs to the registers. Although these may have changed erratically before the logic reached steady state, they are ignored by the registers during the clock’s high- and low-voltage periods and its high-to-low transition. The clock frequency was selected to ensure all inputs stabilize before the end of the clock’s low-voltage period.
5. The next rising edge of the clock causes these set of states to repeat.

In pseudo-code, this might be represented as

repeat forever:
new_register_values = logic(stored_register_values)
stored_register_values = new_register_values


This loop is implied by the constraints that all registers are on the same clock and all logic is acyclic. Neither of these constraints needed to be followed, and some digital circuits chose not to follow them; but by following them, we arrive at a “register-transfer level” programming paradigm, where the job of the chip designer is primarily to write logical expressions for how register outputs are transferred into register inputs. This paradigm will allow us to create full general purpose computers, which is the subject of our next chapter.

1. Newton’s entropy, not Shannon’s entropy.

2. Recall that heat is energy that changes temperature; temperature is the average speed of molecules within matter.

3. x + 1 can be done with a simplified circuit:

4. The details of register design is beyond the scope of this course, but one common version is the rising-edge gated D flip-flop, which looks like
See an example: D flip-flop

5. Chip component sizes are measured in nanometers, not atoms, but most of us have no concept of how small a nanometer is. The atomic radius of silicon (the principle element in most chips today) is about .11 nm; silicon bonds with itself at about 4.3 atoms per nm. AMD released 7nm-scale chips in 2019.

6. This is an over-simplification, but a reasonable rule of thumb.