Power Practice Potpourri

The questions below are due on Monday March 03, 2025; 05:00:00 PM.
 
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Back to Exercise 04

The capacity of a battery is usually quantified using the term milliAmp Hours, which means that the battery can source that much current at the battery's nominal voltage for one hour. With lithium polymer batteries, the nominal voltage is actually a rather complex value to estimate since it is based off of temperature, discharge rates, how much/little it has been abused, etc... but we can usually say the nominal voltage is about 3.7 Volts. Therefore if we had a lithium polymer battery that was rated for 1000 milliAmp Hours (or mAH) that means it could (roughly):

  • Provide 1000 milliAmps at 3.7V for one hour
  • Provide 500 milliAmps at 3.7V for two hours
  • etc...

Note that there are a lot of caveats with this calculation above! Batteries also have upper limits on the amount of current they can instantanteously source! These details are found in the battery's data sheet and should be paid close attention to!

How much energy is contained within a battery having a nominal voltage of 3.7V and a capacity of 350 milliAmp Hours? Answer in Joules below:

Some important notes that will help you out for the remainder of this exercise:

  • For all of the checkers on this page, you can enter arithmetic expressions like (2+4)*(3+9)/5 if you'd like. This may come in handy.
  • Recall that power (P), energy (E), and time (t) are related by the equation E = P \cdot t when the power is constant with time. When the power is varying over time, one must integrate to get energy.
  • In all of the circuits on this page, the currents and voltages are constant (as long as the batteries still have power), so the power consumption is constant as well.
  • In this exercise, we'll give you circuits with a battery (and/or solar cell) and a bunch of power-consuming devices. When we ask you to calculate the fraction of energy consumed by a device, you may find it easier to think in terms of power. For example, it will probably be helpful to calculate the total amount of power sourced by the battery (and/or solar cell), and then calculate the power consumed by each device. The total power sourced ought to be equal to the total power consumed.

For all questions on this page, please answer to within 0.1% precision. The checker is set to be looser than that, but keep as many significant figures as possible while going through chained math implications of using just barely right rounded answers in follow-on answers. If it helps, think of the provided numbers as having a lot more units of precision (3.3V is 3.3000000V). Try to keep those significant figures throughout your calculations and don't use interim values even if they themselves pass the checker with fewer sig figs.

Scenario 1

Consider the circuit below which starts with a fully charged battery and is run until the battery is depleted.

What will happend if this system starts fully charged, and is run until the battery is depleted?

How long will the circuit above last given the the battery starts fully charged and the components have the reported current consumptions. (assume the battery voltage stays constant during span of battery's "ON" time). Provide your answer in hours:

What fraction of the battery's total stored energy is consumed by component D_a?

What fraction of the battery's total stored energy is consumed by component D_b?

What fraction of the battery's total stored energy is consumed by component D_c?

Scenario 2

Now consider this system

How long will the circuit above last given that the battery starts fully charged and the components have the reported current consumptions. (assume the battery voltage stays constant during span of battery's "ON" time). Provide your answer in hours:

What fraction of the battery's total stored energy is consumed by component D_a?

What fraction of the battery's total stored energy is consumed by component D_b?

What fraction of the battery's total stored energy is consumed by component D_c?

What fraction of the battery's total stored energy is consumed by component D_d?

Scenario 3

Now consider the situation where in addition to our original battery we've now also added an energy harvesting element: a solar cell. In full brightness, the solar cell can provide 20 mA. When incorporated into our circuit from above with an additional element we get the following, where V_{sc}=5.10V:

A system with an energy-harvesting element

How long will the circuit above last given that the battery starts fully charged and the components have the reported current consumptions (assume the battery voltage stays constant during span of battery's "ON" time). Provide your answer in hours:

What fraction of the total provided energy (from both sources) is consumed by component D_a?

What fraction of the total provided energy (from both sources) is consumed by component D_b?

What fraction of the total provided energy (from both sources) is consumed by component D_c?

What fraction of the total provided energy (from both sources) is consumed by component D_d?

What fraction of the total provided energy (from both sources) is consumed by component D_e?

During the total span of the system being on, what fraction of the energy came from the battery?

What about the fraction from the solar cell?

Real World: Ohm's Law

In order to calculate power, we need to know the current through a device and the voltage across its terminals. In the real-world, it is much more difficult to measure current than it is to measure voltage. There are many types of analog-to-digital voltage sensors and circuits, but not many current sensors (though they do exist). Usually, the way to measure a current is to turn it into a voltage, which can do by inserting a low-valued resistor in series with the circuit we're analyzing and then using Ohm's Law to figure out the current.

Ohm's Law from 8.02 tells us a relationship between the voltage (v) across a resistor, the current (i) through a resistor, and its resistance R:

Ohm's Law

v=iR

This is expressed in terms of a device voltage, but we've also been doing most of our later analysis using node voltages. In that sense we can rethink of our resistor like that shown below:

Ohm's Law using node voltages rather than component voltages as variables.

And in this case Ohm's Law takes the form:

v_a - v_b = iR

Final Circuit

So using what you know, consider the circuit below with current-sensing resistor and measured node voltages included throughout at particular locations where V_{bat}=5.60 V.

Ohm's Law using node voltages rather than component voltages as variables

Answer the questions below. It'll probably help to draw out the circuit on paper.

How many hours will the circuit run before the battery runs out?

What fraction of the total provided energy is consumed by component D_a?

What fraction of the total provided energy is consumed by component D_b?

What fraction of the total provided energy is consumed by component D_c?

What fraction of the total provided energy is consumed by component D_d?

What fraction of the total provided energy is lost to the resistors?

During the total span of the system being on, what fraction of energy came from the battery?

What about the fraction from the wireless power source?

Back to Exercise 04