Ohm's low
When
an applied voltage E causes a current I to flow through an impedance Z, the
value of the impedance Z is equal to the voltage E divided by the current I.
Impedance
= Voltage / Current |
Z
= E / I |
Similarly,
when a voltage E is applied across an impedance Z, the resulting current I
through the impedance is equal to the voltage E divided by the impedance Z.
Current
= Voltage / Impedance |
I
= E / Z |
Similarly,
when a current I is passed through an impedance Z, the resulting voltage drop
V across the impedance is equal to the current I multiplied by the impedance
Z.
Voltage
= Current * Impedance |
V
= IZ |
Alternatively,
using admittance Y which is the reciprocal of impedance Z:
Voltage
= Current / Admittance |
V
= I / Y |
Resistance
The
resistance R of a circuit is equal to the applied direct voltage E divided
by the resulting steady current I:
R = E / I
Resistances
in Series
When
resistances R_{1}, R_{2}, R_{3}, ... are connected
in series, the total resistance R_{S} is:
R_{S} = R_{1} + R_{2} + R_{3} + ...
Voltage
Division by Series Resistances
When
a total voltage E_{S} is applied across series connected resistances
R_{1} and R_{2}, the current I_{S} which flows
through the series circuit is:
I_{S} = E_{S} / R_{S} = E_{S} / (R_{1}
+ R_{2})
The
voltages V_{1} and V_{2} which appear across the respective
resistances R_{1} and R_{2} are:
V_{1} = I_{S}R_{1} = E_{S}R_{1} / R_{S}
= E_{S}R_{1} / (R_{1} + R_{2})
V_{2} = I_{S}R_{2} = E_{S}R_{2} / R_{S}
= E_{S}R_{2} / (R_{1} + R_{2})
In
general terms, for resistances R_{1}, R_{2}, R_{3},
... connected in series:
I_{S} = E_{S} / R_{S} = E_{S} / (R_{1}
+ R_{2} + R_{3} + ...)
V_{n}
= I_{S}R_{n} = E_{S}R_{n} / R_{S} =
E_{S}R_{n} / (R_{1} + R_{2} + R_{3}
+ ...)
Note
that the highest voltage drop appears across the highest resistance.
Resistances
in Parallel
When
resistances R_{1}, R_{2}, R_{3}, ... are connected
in parallel, the total resistance R_{P} is:
1 / R_{P} = 1 / R_{1} + 1 / R_{2} + 1 / R_{3}
+ ...
Alternatively,
when conductances G_{1}, G_{2}, G_{3}, ... are
connected in parallel, the total conductance G_{P} is:
G_{P} = G_{1} + G_{2} + G_{3} + ...
where G_{n} = 1 / R_{n}
For
two resistances R_{1} and R_{2} connected in parallel, the
total resistance R_{P} is:
R_{P} = R_{1}R_{2} / (R_{1} + R_{2})
R_{P} = product / sum
The
resistance R_{2} to be connected in parallel with resistance R_{1}
to give a total resistance R_{P} is:
R_{2} = R_{1}R_{P} / (R_{1} - R_{P})
R_{2} = product / difference
Current
Division by Parallel Resistances
When
a total current I_{P} is passed through parallel connected
resistances R_{1} and R_{2}, the voltage V_{P} which
appears across the parallel circuit is:
V_{P} = I_{P}R_{P} = I_{P}R_{1}R_{2}
/ (R_{1} + R_{2})
The
currents I_{1} and I_{2} which pass through the respective
resistances R_{1} and R_{2} are:
I_{1} = V_{P} / R_{1} = I_{P}R_{P} /
R_{1} = I_{P}R_{2} / (R_{1} + R_{2})
I_{2} = V_{P} / R_{2} = I_{P}R_{P} /
R_{2} = I_{P}R_{1} / (R_{1} + R_{2})
In
general terms, for resistances R_{1}, R_{2}, R_{3},
... (with conductances G_{1}, G_{2}, G_{3}, ...)
connected in parallel:
V_{P} = I_{P}R_{P} = I_{P} / G_{P} =
I_{P} / (G_{1} + G_{2} + G_{3} + ...)
I_{n} = V_{P} / R_{n} = V_{P}G_{n} =
I_{P}G_{n} / G_{P} = I_{P}G_{n} / (G_{1}
+ G_{2} + G_{3} + ...)
where G_{n} = 1 / R_{n}
Note that the highest current passes through the highest conductance (with
the lowest resistance).