# 04Efficiency and calculations

Ee04PN of the Electricity and Energy topic
• ## 01 Energy stores accumulate workingEe04PNnugget01 Introduction

### The power in a pathway determines the quantity of energy shifted to a store each second

Connect up a circuit and leave it running in a steady state – lamps glow and resistors warm. How much they glow or warm depends both on the electrical current and on the electrical potential difference.

So much for the physical description; now to re-describe it in terms of pathways.

Electrical working at a constant rate occurs in an electrical loop in which there is a constant electrical current. Every device in this loop that has a potential difference across it shifts energy. The power in the electrical pathway tells you how much energy is shifted every second by that device. This power is set by the potential difference across the device and the current through the device.

### Power in a pathway here can be equal to power in a pathway over there

The power in this pathway shifts energy to or from a store, adding to or subtracting from the energy already in the store. So a constant power in a pathway leads to a steady accumulation in a store. This accumulation can be either positive or negative.

As a starting example, use a simple large-scale electrical loop that does something useful: a hydro-electric generator, deep in the Welsh countryside, lights a domestic lamp in the West Midlands. Model the circuit, simplifying as much as possible, with wires of negligible (as near to zero as makes no difference) resistance. Then there are only two devices in the circuit – places where pathways are switched. The system is designed to optimise the following switches: at the generator, from the mechanical pathway to the electrical pathway; at the lamp, from the electrical pathway to the heating by radiation pathway (again, visible radiations only – that is our particular interest because of how we evolved).

In both locations the device will not be perfect – there will be some switching to other pathways, chiefly heating by particles as the water is warmed by churning through the turbine, and the wires in the generator are warmed by the current driven through them, as the air surrounding the lamp is warmed.

Describing the process in terms of energy shifted, you end up with an even more abstract picture, as the gravitational store is depleted and the thermal store augmented. This is the view that was emphasised in the earlier SPT: Energy and SPT: Electric circuits topics.

### Constant power implies steady rate of accumulation

A larger constant power in any pathway accumulates energy in a store at a larger steady rate. This accumulation can be either positive or negative, thereby augmenting or depleting the energy in the store.

For the electrical pathway, you choose how to make the power larger: set the character of the pathway by using either a larger current or a larger potential difference, or both.

Accumulating over more time, at the same constant rate, also leads to larger quantities of energy being shifted to or from stores.

• ## 02 Using a motor to liftEe04PNnugget02 Exposition

### A simple view of a motor

One view of a motor is as a complex assembly of wires, magnets, axles, brushes and commutators. There are many different designs of electrical motor (e.g. brushed DC, stepper DC, synchronous AC) and considerable skill is deployed in adapting a design to a particular task. This might be for a particularly high-torque application, perhaps to generate large accelerations for an electric sports car, or an application where the working power is very small, but very little power can be wasted, such as in a solar-powered car.

A much simpler view is provided by developing a description in terms of energy stores or the power in pathways.

### Working power and maximum power

There are many working powers for motors. Even in the domestic sphere you can find motors designed to agitate clothes in a washing machine, blend soup, grind coffee or to spin the hard disk in your portable music player. Precision movement is required for the last example – but perhaps not for much longer. The reproduction of music has depended on precise rotation from 78 rpm (revolutions per minute) singles, through the steady high speed for audio CD-ROM, to the steady high speed for hard disk drives (7200 rpm is commonly available at the time of writing). Perhaps flash memory players will supersede both hard disks and portable audio cassette players, with their need for very precise motors to draw the tape over the head at 1 78 inches per second. Nevertheless, if robots are to play any part in the future, there will be a great need for engineers to design motors with a range of precision and rated power.

But there is a simpler view (and physicists like simpler views) that provides guidance for all of this engineering. A motor is simply a device designed to switch from an electrical pathway to a mechanical pathway, so it's a kind of a transducer. A perfect motor will switch all of the power – there is no waste.

Use a battery to drive a motor, which is lifting a pallet (perhaps on a fork-lift truck). Then the device/pathway description provides a useful level of detail for one strategic view of the engineering demands. An even more abstract view is provided by a description in terms of stores – one that is even less concerned with the how?, and so even more focused on the how much?.

### Perfect and less than perfect switching from one pathway to another

Engineers are very much concerned with real motors, which are not perfect. Perfection is only an unattainable target: engineering is all about the right compromise.

Any motor has a significant length of wire, in which there is a current, and across which there is a potential difference, the value constrained by this relationship: V = R × I. There will therefore be some warming in these motor wires. So a more realistic view of the motor is a device that switches from the electrical pathway to the mechanical working pathway and to the heating by particles pathway.

A less wasteful, more efficient motor is one where more of the power ends up in the mechanical working pathway.

As motors don't glow very much we can neglect the heating by radiation pathway; in other words, there is not much power in this pathway for most motors, until they are run beyond their design power, often resulting in a burnt out motor. This happens when the power switched from the input (electrical) pathway exceeds the three output pathways (mechanical working, heating by particles, heating by radiation). The energy shifted in exceeds the energy shifted out, and the thermal store is augmented, so the wire gets hotter and hotter until it eventually melts.

### How to calculate accumulations in stores as a result of lifting

Back to the motor of the fork-lift truck lifting a pallet. As the motor runs, so the battery is flattened and the load is lifted. The faster the load is lifted, the greater the rate at which the chemicals react in the battery and so it is depleted at a greater rate.

The motor switches from the electrical pathway – the energy shifted from the chemical store of the battery, and to the gravitational store, accumulates over time.

The power in the electrical pathway (set by the current and the potential difference, as before) and the time both set the accumulation.

### Power in two pathways

The energy shifted to the gravitational store depends both on the force (mass  × gravitational field strength) and the distance – here the height. You can check these connections in the SPT: Forces and the SPT: Energy topics. So you can calculate the energy shifted to the gravitational store accumulating as the height is varied. The force is fixed by the contents of the pallet that the fork lift picks up.

The energy accumulated in the gravitational store, as a result of the change in height, can be calculated by mass  × gravitational field strength  × height change. Check this makes sense using the units: kilogram  × Newton kilogram -1  × metre, which simplifies to:Newton  × metre. Moving from units back to quantities, this is force  × distance.

This calculates energy (see the SPT: Energy topic for more detail).

These two accumulations, due to the electrical and mechanical pathways, will be equal if the motor is perfect. We often use this simplified model because it's often a good guide to action. But for real motors, some energy will inevitably be shifted to thermal stores, so the accumulation calculated from the electrical pathway will be equal to the energy shifted to the gravitational store and to these thermal stores.

• ## 03 Warming an ironEe04PNnugget03 Exposition

### A simple view of an iron

Many domestic appliances switch from an electrical pathway to warm things up (and even cool them down as well). The heating by particles pathway and the heating by radiation pathway are both used, depending on what needs to be warmed.

Consider which of these pathways might be used by the following:

• An electric bar heater
• An electric hob
• A toaster

Imagine using an iron.

Again, there are two theoretical perspectives: the engineer's detailed view and the physicist's energy view.

These are less distinct than in the case of the motor but they are still separate. The engineer will concentrate on the resistive element that provides the heating: on its dimensions (both cross-sectional area, and length), on how effectively the plate next to it is warmed, and on how to insulate the operator from this warming. The pathways view is much simpler, at least for the perfect iron: all of the power in the electrical pathway is switched to the heating by particles pathway.

### Perfect switching – and less than perfect switching

Real-world irons don't only warm clothes: the surroundings are also warmed, so you might model the process by including two heating by particles pathways, one filling the intended thermal store of energy (perhaps associated with a blouse warming up), the other filling thermal stores associated with the surroundings.

The better the design of the iron, the more power in the pathway directed to the energy store associated with the blouse.

### How to calculate accumulations in stores as a result of warming

The power in the electrical pathway (set by the current and the potential difference, as ever) and the time both set the accumulation. As the iron is a mains appliance, the current will in practice be set by choosing the resistance of the element in the iron.

The energy accumulating in the thermal store is set by the power in the heating by particles pathway and the time. But there is no simple way of finding the power in this pathway from measurements made with commonly available laboratory instruments, so watch the energy accumulating in the store directly, as the temperature rises (more on this in the SPT: Energy topic).

The number of joule kelvin -1  × temperature rise in kelvin gives this accumulation. The more mass there is, the greater the number of joule kelvin -1, so we often factor this out, writing

mass  × a constant for the material.

So the energy accumulated in this store, as a result of the temperature rise can be calulated by mass  × constant  × temperature rise.

• ## 04 Battery-accelerated carEe04PNnugget04 Exposition

### A simple view of accelerating masses

Using batteries to accelerate a model car will involve different complex technologies. The energy picture is both simple and clear: a chemical store empties and and a kinetic store fills.

The electrical link is missing altogether: the energy picture is that abstract.

The description using pathways fills in a little detail, without providing the precise mechanisms and detailed thinking necessary to make a device that works. Rather, this level of description provides a way of understanding what can be achieved and what cannot.

The motor switches from the electrical pathway – the energy shifted from the chemical store of the battery, and to the kinetic store, accumulates over time. The power in the electrical pathway (set by the current and the potential difference, as before) and the time both set the accumulation. This power in the electrical pathway sets the maximum rate at which the kinetic store accumulates.

Why the maximum? Simple: the motor is not perfect. Some of the power in the electrical pathway will accumulate energy in a thermal store.

### Energy from store to store through switched pathways

That not all of the energy is shifted from the chemical store to the kinetic store is certain, at least in the real world. That still leaves you with a decision about how to model this as a useful description using pathways. There are at least two possibilities.

In the first, we suggest that you focus on the motor as the switching device. This isn't completely effective because the wires of the motor present some electrical resistance and so cause some warming. Some of the power in the electrical pathway is switched to the heating by particles pathway, leaving most still switched to the mechanical working pathway.

In the second, the car as a whole is the switching device. The input is now still the electrical pathway, but the output paths are two mechanical working pathways: one serves to accelerate the car, the other works on the air, warming it, and so accumulating energy in the thermal store.

The choice of which model is more appropriate depends on the task at hand: improving the motor, or improving the ease with which the car acquires speed as it slips through the air and propels itself along the road.

### How to calculate accumulations in stores as a result of accelerating

Here is a common model of the battery-driven car: a good-enough guide to action.

The power in the electrical pathway (set by the current and the potential difference, as ever) and the time both set the accumulation. You choose the character of the pathway to suit the devices at either end: the battery and the motor.

The energy accumulating in the kinetic store is set by the power in the mechanical working pathway and the time.

### A non-obvious, but illuminating, graph

Another way is to plot a seemingly uninteresting graph – of mass  ×  velocity against velocity. This will always be a straight line through the origin, with a gradient of the mass. However, the area of the graph is interesting: it will be measured in metre second -1 ×  kilogram ×  metre second -1.

Why is this interesting? Only because it is exactly metre × kilogram × metre second -2.

This can be shown with simple algebra: get out the pencil and paper and convince yourself. I hope you'll recognise these collections of units as distance × mass × acceleration, and then see that you can collect them as distance × force, and then (really the last step) remember that this calculates energy shifted (covered in the SPT: Energy topic).

Looking at the triangle on the graph that represents the energy one last time, you should be able to see that its area is 12 × mass × velocity × velocity (12 × height × base is the area of a right-angled triangle).

Tidy this expression up to get 12 × mass × velocity2. More simple algebra: again it's best to convince yourself with a pencil and paper.

So the energy accumulated in the kinetic store as a result of the acceleration can be calculated by 12  × mass  × velocity2.

• ## 05 Differing accumulationEe04PNnugget05 Exposition

Power in a pathway accumulates over time to fill or empty a store. A constant power leads to a steady change in the energy of stores. How much energy is shifted can be found from the area under a power/time graph.

### Accumulations that are simple to calculate

The energy in these stores can be calculated, and therefore dealt with during studies for those aged 14–16, mostly in physics.

The areas under the graphs again show the energy accumulated in the store as the independent variable is altered:

• For the kinetic store, the speed.
• For the elastic store, the stretch.
• For the thermal store, the temperature.
• For the gravity store, the change in separation.
• For the chemical store, clues about the number of particles that have reacted.

Changes in these independent variables were exactly the changes you were asked to look out for in the SPT: Energy topic when identifying likely variation in the energy in each store. This is what justified identifying these as stores in pre-quantitative work – that you would be able to calculate the energy stored due to particular changes not too far in the (pupil's) future.

### Accumulations that are less simple to calculate

These stores are mostly dealt with in post-16 physics, largely pre-19 (magnetic stores are mathematically tricky, and only the electric bit of the electromagnetic store can easily be calculated by A-level students).

Again, the areas under the graphs show the energy accumulated in the store as the independent variable is altered:

• For the vibration store, the amplitude.
• For the nuclear store, the change in mass.
• For the electric and magnetic store, the change in separation.

Again, these changes map onto those used to identify the stores, although looking for the minuscule changes in mass associated with energy shifted to or from the nuclear store is hard to imagine, which is perhaps why that one seemed a bit mysterious in the SPT: Energy topic. Perhaps surprisingly, the magnetic aspect of the electromagnetic store was easiest to relate to the stretched elastic band teaching tip in the SPT: Energy topic, and turns out to be hardest to calculate.

• ## 06 Calculating energy efficienciesEe04PNnugget06 Exposition

### How to find efficiencies

Efficiency is simply a measure of success – it tells you how much energy is squandered and how much ends up in the store that you designed the system for. Often, but not always, the squandered energy will be shifted to many stores – dissipation has occurred.

Calculate the percentage of the input energy ending up in the desired output and you have the efficiency: efficiency = energyoutenergyin × 100 %.

Here's a support sheet showing how to calculate efficiency:

Support sheet

• ## 07 Quantifying pathwaysEe04PNnugget07 Exposition

### Calculating power in all four pathways

Here is how to calculate the power in each pathway directly – at least in principle. In practice, some of the required measurements might be hard to do.

Electrical working: joulecoulomb ×  coulombsecond

Mechanical working: joulemetre ×  metresecond

Heating by particles: jouleparticle ×  particlesecond

Heating by radiation: joulephoton ×  photonsecond

You might be more familiar with the first two as:

Electrical working: p.d. × current

Mechanical working: force × velocity

### Checking the switching from pathway to pathway

If you can calculate the power in each pathway in a particular case, then this can give another route to finding energy efficiency. Remember that power is just energy per second.

Because the pathways description is often closer to the mechanism, this may be a better practical route to improving the efficiency. Find the leaky pathways, decide which can be fixed and then plug the leaks. More of the input power will be directed to the desired output pathway.

Imagine running the system for exactly 1 second. Calculate the percentage of the input energy ending up in the desired output and you have the efficiency: efficiency = energyoutenergyin × 100 %.

• ## 08 Cells revisitedEe04PNnugget08 Expansion – tell me more

### A model of internal resistance in terms of energy

An electric cell is a part of the circuit in which there is current. Therefore there must be a current in the cell, as well as in the wires in the circuit. Remember that putting a cell into a circuit starts the charge moving everywhere – including charge in the cell.

### Real cells are made of materials with a resistance

Just like a wire, a cell isn't perfect. Both are special arrangements of atoms through which the charged particles move while there is a current. Both have a resistance, which is usually ignored in our models of the circuit. However, cells, like wires, can get very hot as a result of large currents in the cell. Even a small resistance, and therefore a small potential difference, can result in a large power in the pathway if the current is large. The energy dissipated in the cell as a result of this action is then not available to the rest of the circuit. And a cell that provides less energy for each charge is a cell with a smaller potential difference. So drawing large currents through cells significantly drops the potential difference across the cell.

This resistance of the cell is called the internal resistance, and it can usually be safely ignored in models that describe circuits with moderate currents. A car starter motor draws 30 ampere – not a moderate current. You will notice the headlights dimming when you start the car as a result of this reduced potential difference.

The output coil of a transformer acts like a small cell when the magnetic effect through the coil changes. This cell may contain a lot of wire, so it too has an internal resistance. This could be put to use, designing sources with high internal resistance, so that the current that can be drawn is limited.

In fact, all sources of potential difference have some internal resistance, and this is a feature that is often considered in choosing a particular source. One reason why the domestic mains supply is to be treated with respect is that its effective internal resistance is rather small – it has to be, or the sockets and supply wiring would get rather warm.

• ## 09 Some efficienciesEe04PNnugget09 Exposition

### Real-world efficiencies – but compare like with like

There are both practical and theoretical limits to efficiency. Both were of concern to the engineers, physicists and artisans who developed the idea of the conservation of energy and the science of thermodynamics. They were concerned about how much work could be extracted from burning coal, and how many horses this would replace. (This is the historical origin of the horsepower as a unit of power.)

Efficiencies are likely to be of more and more interest as energy resources are depleted. That all domestic appliances are rated (A++ to G) is a start, but it seems likely that only changing our habits will make a real difference. These are rated by the energy (KWh) dissipated to perform a particular standard task, so they aren't strictly efficiencies.

Here are some efficiencies – for the mechanical parts of bicycles:chain and derailleur 92.1–98.5 percent ; three-speed hub gear 87.3–97.9 percent ; single speed 96–99 percent . A human can only manage a continuous power output of about 75 watt, so this needs not to be squandered. (Interestingly, you can feel these differences.)

Here are some for motor vehicles, starting with the fuel, so they're not directly comparable. The best practical results are 18–20 percent. The theoretical limit for metal engine blocks is 37 percent. Rocket motors run hotter and can run at up to 70 percent (but you might not want to tailgate in dense traffic).

But take care in drawing conclusions: these are not intended as direct comparisons.

Here are some interesting domestic values, for heat pumps: Groundâ€“air heat exchanger, 400 percent ; Airâ€“air heat exchanger, 300 percent.

One of the most significant things you can do to reduce your domestic heating bills is to fit a device that shifts energy from one thermal store (air or ground) to the thermal store that is your home. For each kilojoule of energy that is dissipated, several kilojoules of energy are shifted to the thermal store. That's why the efficiencies quoted above exceed 100 percent.

Here's another view of efficiencies:

Support sheet

• ## 10 Using efficienciesEe04PNnugget10 Summary

### Devices and pathways

Use either power or energy to calculate efficiencies. Your choice is determined by a focus on either the power in pathways for processes that are still happening, or on the energy that has accumulated in the stores during the process.

### Practical guides to action

Efficiencies are useful as guides to action because they show how far short one has fallen of perfection (100 percent efficiency in almost all cases). The difference between what you have achieved and what could be achieved may help you to see whether it's possible to make significant improvements. You might seek to develop what you are working on, or to adopt a completely new approach. In either case you'll be comparing your calculated efficiency with other approaches, or perhaps even with a theoretical maximum, calculated from first principles. (We don't show you how to try to do that here, but the principles of thermodynamics are often useful. These take the story of energy developed here even further.)

Energy efficiencies, using calculations based on stores, will be more general than power efficiencies, based on pathways. The latter incorporate more information about the devices. That's why they're more specific and often better suited to development work by engineers. However, the most general approach, using energy descriptions, often has no connection with the mechanism at all (a distinction made much of in the SPT: Energy topic). This makes comparisons possible: energy is a universal non-arbitrary currency, which allows the outcomes of completely different physical options to be compared reliably and objectively.

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