An introduction to electrical circuits, and their use in physiology
Equivalent resistance in parallel circuits
Returning to our simple
parallel circuit containing two resistors, we could in principle
replace the R1 and R2 branches in the diagram
with a single branch containing the resistor Rt (the subscript t
standing for "total"). This new circuit
is considered equivalent to the old, because the current I being
driven by the battery is the same. What then is the value of Rt,
the equivalent total resistance against which the battery is
driving current?
By the arguments given previously, the voltages recorded across either R1, R2 or Rt will all be equal to the "total" voltage V produced by the battery. Using Ohm's law:
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V = I.Rt = I1R1 |
(12) |
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Substituting in our expression for I1 from Equation (11), and solving for Rt, we find: |
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This can also be expressed as: |
Rt = R1R2 / (R1 + R2) 1/Rt = 1/R1 + 1/R2 |
(13) (14) |
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In fact, if we have any number, n, of resistors in parallel, the equivalent total resistance Rt can be calculated as: |
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1/Rt = 1/R1 + 1/R2 + ... + 1/Rn |
(15) |
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As you may be able to see, or could easily calculate if you put in some example values, the more resistors you have in parallel the lower the equivalent total resistance, Rt. This is in stark contrast to putting more resistors in series, where the total resistance is simply the sum of each individual resistance, and would be increased by adding another.
As well as allowing us to change the pattern of flow in the circulation, as discussed earlier, the body's arrangement of blood vessels in parallel considerably reduces the overall resistance of the circulation, sometimes referred to as the total peripheral resistance (TPR). This greatly reduces the pressure that has to be developed by the heart in order to drive flow around the circulation, in comparison with a hypothetical animal in which one very long blood vessel supplies each of the organs in turn!
We could use Equation (15) to calculate a single, equivalent resistance Rt, representing all of our arteries (or arterioles, capillaries etc.) in parallel: this is how we would generate the model of the circulation with resistors in series, considered earlier. Although an individual capillary has a higher resistance than an individual arteriole, there are many more capillaries in parallel and, as a result, their collective resistance is actually lower.
If there are other physics-related areas that you would like to see introduced in a similar way on-line, please contact Dr Matt Mason