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By: Neil E.
Cotter
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CIRCUITS
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Kirchhoff's laws
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Voltage loops
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Def: A loop is
any continuous path (that may even cross gaps across open space) that ends
where it starts.
Note: The goal
of writing current summation and voltage loop equations is to obtain n
equations in n unknowns that we can solve to find all the currents and
voltages in a circuit.
Tool: Loops that
cross over themselves may always be treated as two smaller loops. (Smaller
loops yield simpler equations.)
Note: We may
proceed in either direction around a loop when we write a voltage loop
equation, (but we must continue in the same direction all the way around the
loop).
Tool: We set the
sum of voltage drops around a loop to zero. If we exit a circuit element from
the + sign of the voltage measurement as we proceed around the loop, then that
voltage appears with a plus sign in the loop equation. If we exit a circuit
element from the – sign of the voltage measurement as we proceed around the
loop, then that voltage appears with a minus sign in the loop equation.
Note: Using a +
sign for a voltage term if we enter a circuit element from the + sign of the
voltage measurement, and using a – sign for a voltage term if we enter a
circuit element from the – sign of the voltage measurement yields an equation
equivalent to using the opposite sign convention (as stated in the preceding
tool). Multiplying one equation by –1 on both sides yields the other equation.
Tool: We skip
voltage loops where we would be forced to define a voltage for a current
source. Writing an equation for such a loop adds a new variable and a new
equation. Thus, we merely create more equations in more unknowns rather than
moving closer to the goal of writing n equations in n unknowns.
Tool: We write
voltage loops for all inner loops, if appropriate. If we are able to write an
equation for each inner loop, we have all the voltage loop equations we need.
When we must skip an inner loop equation (because we would have to define a
voltage for a current source), we write a voltage-loop equation for the next
larger voltage loop containing some portion of that inner loop. If we must
skip that next larger loop, we proceed to the next larger loop, and so on
recursively. If we must skip even the largest voltage loop, then that voltage
loop is unnecessary. (In that case, the other voltage-loop and
current-summation equations will be sufficient to solve the circuit.)
Tool: When
necessary, we supplement voltage-loop equations with equations that equate
voltages across circuit elements that are in parallel.