# Power Integrity Analysis and Management for Integrated Circuits: Power, Delivering Power, and Power Integrity

- 1.1 Electromotive Force (emf)
- 1.2 Electrical Power
- 1.3 Power Delivery
- 1.4 Power Integrity (PI)
- 1.5 Exercises
- References

*Power* may be defined as the capacity to perform work, and is measured as the work done or energy transferred per unit of time.
The greater the power, the greater the capacity to move something (against forces of attraction, repulsion, or friction) or
to transfer energy (by raising the temperature of a substance, for example). With reference to electronic systems, and integrated
circuits in particular, power is the presence of voltage, or electromotive force, and current, or the flow of charge. Power
enables the performance of desired functions in the system and is delivered by devices that generate, store, or regulate electromotive
force or flow of charge. Electronic circuits and systems consume this delivered power and convert it into other desired forms
of energy or activity. The integrity of delivered power relates to its stability and invariability through variations in energy
expenditure, in transfer elements, in the generating source, or in the surrounding environment. These aspects are expounded
on in the following sections of this chapter, with a focus on integrated circuits and systems.

## 1.1 Electromotive Force (emf)

The discovery of electromotive force is attributed to physicist Alessandro Volta (1745–1827), who invented the electric battery
or voltaic pile. Electric batteries possess the ability to move electric charge. They perform work, a prerequisite of which
is the existence of force; batteries are therefore termed sources of electromotive force or *emf*.

The unit of emf is the *volt,* and as defined in the international system of units, one volt equals one *joule* per *coulomb,* where *joule* is the unit of energy and *coulomb* is the unit of electrical charge. Viewed another way, if a charge of a coulomb gains a joule of energy in passing through
a device, the emf present across the device is a volt.

Since the movement of charged particles and gain (or loss) of energy is involved, mechanical analogies are often employed
to illustrate aspects of electrical behavior such as *emf,* current, and power. The *force-voltage* analogy is a common example, wherein mechanical force is equated to electrical voltage.

### 1.1.1 Force-Voltage Analogy

The force-voltage analogy [1] is attributed to James Clerk Maxwell.^{1} Since the SI (French *Le Système International d’Unités,* or International System of Units) unit of force is the *newton,* and the unit of emf is the volt, this analogy is not altogether consistent. Nevertheless, many aspects of electrical behavior
mirror the behavior of physical objects, and such an electrical-mechanical analogy assists comprehension. An electrical “tank”
circuit, for example, behaves in much the same way as the pendulum of a mechanical clock.

Work done, whose SI unit is the joule, involves force and distance. Hence:

#### Equation 1-1

Similarly, voltage difference, or electrical potential difference, in a homogeneous medium, is the work in joules required to move a coulomb of charge from a point to another. Therefore:

#### Equation 1-2

If one associates force with voltage, following the relationships above, *distance* in mechanical terms will be equivalent to *charge* in electrical terms. Distance over time, or *velocity,* will be equivalent to charge transferred over time, or *current*.

Consider the expression for kinetic energy:

#### Equation 1-3

where *m* is the mass and *v* the velocity of the object. Also consider the expression for energy stored in a magnetic coil:

#### Equation 1-4

where *L* is the inductance of the magnetic coil, and *I* the current flowing through the coil. These expressions suggest an equivalence between *mass* and *inductance*.

The *distance* moved by an elastic spring increases the energy stored within it and linearly increases its force of resistance. Similarly,
*charge* transferred into an electrical capacitance device increases the energy stored in the capacitance, and linearly increases
emf across its terminals. Work done on a spring, or an increase in its potential energy, is given, following *Hooke’s Law,* by:

#### Equation 1-5

where *k* is the elastic constant of the spring and *x* the elongation or compression. This relationship is equivalent to that for electrical potential energy stored in a capacitor:

#### Equation 1-6

The equivalence is to an extent hidden within these expressions, since *V* does not correspond to *x*. The expression for potential energy in a spring may be derived by the integration of force (given by Hooke’s Law as *k · x*) over distance traversed (*dx*), which derivation is left to the reader. Hence, the expression for potential energy is obtained from force · distance ((*k · x*) · *x*), as in the fundamental definition of work done. Similarly, the expression of potential energy in a capacitor is derived from voltage · charge (*V* · (*C · V*)), which also follows from the definition of work done in electrical terms, in a manner consistent with the force-voltage
analogy.

It is also useful to note the equivalence of energy in static and dynamic analysis of fluid flow to electrical energy, since
electric current may be described as a fluid-like flow of electrons or charge at a macroscopic level. Gravitational potential
energy is given by *m · g · h*, where *m* is mass, *g* is acceleration due to *gravity,* and *h* is the height increase. In an analogy with liquids, it is common to equate pressure at the bottom of a liquid column with
emf*.* A conduit joining two columns of different diameter, but of the same height and fluid matter, will see no fluid flow since
fluid pressure will be the same at both ends of the conduit. This example is equivalent to two capacitors of different capacitance
value charged to the same electric potential. If the column heights are different, a flow of liquid between the two columns
will be established through the conduit, subject to its diameter. The electrical equivalent to this physical analogy is the
relationship between emf, or voltage, and charge flow, or current, as presented by *Ohm’s Law:*

#### Equation 1-7

where *R*, or *resistance,* is the property by which materials oppose or resist the flow of charged particles through their atomic structures. Resistance
is a property that leads to energy absorption. It is also a property that leads to loss of emf in an electrical circuit, since:

#### Equation 1-8

whereby some electromotive force is spent in overcoming the resistance of a connecting element to the flow of charges through it, much as force is spent overcoming friction in mechanical systems.

Another analogy sometimes used is the force-current analogy, in which an electrical current source is equated with a force generator, and voltage equated with input velocity to a mechanical system. This analogy results in capacitance being equated to mass, inductance to the inverse of the spring constant, and resistance to the inverse of friction [1].