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Issue No 24, 24 April 2023
By: Anthony O. Ives
For simplicity in most cases it is assumed that batteries discharge and charge at a linear rate [1], that is the battery charges and discharges proportional to time. A simple example is if a battery takes 5 minutes to lose half of its charge then in 10 minutes it will be fully discharged, or in a charging scenario if it takes 1 hour to regain half of its charge then in 2 hours the battery will be fully charged. The above example assumes the charging and discharging current is constant which means it does not change with time. In reality, batteries charge and discharge in a way similar to the way a capacitor charges and discharges which is exponential relationship [2]. A capacitor is an electronic component that is used in electrical circuits to hold charge, see [3] and [4].
Battery capacity is measured by the amount of electrical charge they can hold however, the standard units that are used for batteries are not Coulombs but Milliampere Hours or Ampere Hours. Electric current is the flow of electric charge so if you multiply electric current by the time it has charging or discharging then in theory that will give you the amount of electric charge that has been gained or lost therefore electrical charge can be calculated using the following simple equation:
Q = I t
Where Q is electric charge, I is electric current and t is time. From the above equation it should be obivous how Milliampere Hours or Ampere Hours could be used as units of charge, as Milliampere and Ampere are a units for current and Hours is unit for time. For electronic components like capacitors the unit for charge used is usually Coulombs.
However, capacitor size is defined by a property call capacitance which is defined as the amount charge held for a unit voltage. The unit for capacitance is Farad, the equation for capacitance is given below:
\[C = \frac{Q}{V}\]
Where C is capacitance and V is voltage. Capacitors are introduced in this article because the charging and discharging equation could be used to calculate how a battery charges and discharges. Capacitors are also an important electrical component and a concept that has many different applications.
Capacitors typically consist of two metal plates with a dielectric material sandwiched between them. The type of dielectric material can vary greatly it can even be something as diverse as jet fuel. The capacitor concept can be used to measure the mass fuel in a fuel tank whereas a conventional float type fuel gauge can only measure volume and it does not take account of aircraft attitude change, capacitor type fuel gauge is a lot more accurate. However, in most electronic capacitors the dielectric material is usually a semi-conductor type of material such as an advanced polymer material. The equation relating the capacitor dimensions to the value of the capacitance.
\[C = \frac{A \epsilon}{d}\]
Where A is the surface area of the metal plates, d is the distance between the metal plates occupied by the dielectric material and ε is the dielectric constant. Equations showing how the various different electrical properties vary as the capacitor charges is given below:
\[I(t) = \frac{V_{0}}{R} e^{-\frac{t}{\tau_{0}}}\]
\[V(t) = V_{0} \left( 1 - e^{-\frac{t}{\tau_{0}}} \right) \]
\[Q(t) = C V_{0} \left( 1 - e^{-\frac{t}{\tau_{0}}} \right) \]
\[\tau_{0} = R C\]
Where V0 is the intial voltage, R is the electrical circuit resistance, τ is the time constant which is given by multiplying the capacitor capacitance by the circuit resistance. The equations for discharge of a capacitor are similar except V0 is replaced with VCi. VCi is the voltage due to intial charge of the capacitor. The equations for voltage and charge are also slightly different, the equations are given below:
\[I(t) = \frac{V_{Ci}}{R} e^{-\frac{t}{\tau_{0}}}\]
\[V(t) = V_{Ci} e^{-\frac{t}{\tau_{0}}} \]
\[Q(t) = C V_{Ci} e^{-\frac{t}{\tau_{0}}} \]
The equations for how capacitor charge varies with time are plotted in a graph below for both the charge and discharge senarios:
A typically charging and discharging circuit for capacitor is given below:
Similar equations could be used to calculate how a battery charges and discharges defining equivalent properties of capacitance, time constant, etc for the battery. Generally speaking however most of time the battery is charging and discharging the process is linear and only deviates from the linear region at the end of the charging and discharging process. So the charge of a battery can be calculated for most cases using the simple linear equation given below:
QBattery = I t
Where QBattery is the battery charge given in milliampere hours (mAhr), I the charging or discharging current of the battery and t is the time battery is being charged or drained. In most cases this type calculation will give a fairly accurate prediction of the time the battery can provide power. However, for certain applications you may need to use the exponetial equation similar to those used for capacitors to give a more accurate prediction.
For typical electronic circuits used to measure battery power and charge batteries see [5] however, please be aware these circuits are only suitable for NiCad or Nickel Metal Hydride batteries, do not use them for other batteries such as Lithium Ion or Lithium Ploymer. It is also highly recommended to always follow the battery and charger manufacturers instructions.
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References:
[1] http://www.eiteog.com/EiteogBLOG/No15EiteogBlogLinear.html
[2] http://www.eiteog.com/EiteogBLOG/No9EiteogBlogLogs.html
[3] Hughes Electric Technology, Edward Hughes, 7th Edition, 1995, Prentice Hall
[4] Electronic Circuits Fundamentals and Applications, Mike Tooley, 3rd Edition, 2006, Newnes-Elsevier
[5] Electronic Projects for Model Aircraft, Ken Ginn, 1998, Nexus Special Interests
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