# Series Combinations of R, C & L in A.C. Circuits

#### R and L in series:

The voltages dropped across the inductor and the
resistor must be added together using phasor summation, i.e. adding the two voltages as
they would be shown on a phasor diagram, if the second phasor is drawn starting from the
end of the first phasor. Since V_{L }is at right angles to V_{R }then
the components are added using pythagoras's theorem, a^{2} = b^{2} + c^{2}.

The triangle this makes shows the voltages V_{R}
and V_{L}, and the total voltage V on the hypotenuse (see below).

Since V = IZ, dividing all the sides by I gives R, X_{L}
and Z, the impedances.

Similarly, multiplying all sides by I gives I^{2}R,
I^{2}X_{L }and I^{2}Z , the power
dissipated in the components.

The term impedance (Z) is used for the combination
of component resistances and reactances, e.g. the 'circuit impedance' refers to the total
impedance of a circuit containing a number of resistors, inductors and capacitors whose
resistances and reactances have been combined using phasor summation.

#### Phase Angle:

The angle between the adjacent and the hypotenuse of
the triangles is the phase angle of the combined components. I lags V.

f = cos^{-1} (R/Z)

#### R and C in series:

As for R and L in series, except that for the phase
angle of the two combined components, I leads V.

#### R, L and C in series:

This can be visualised from drawing all 3 components
on a phasor diagram.
The resistance is in phase, therefore it's phasor will be horizontal. The inductance will
then have it's phasor going upwards, but the capacitance phasor will come down again from
this point. The hypotenuse of the triangle is from the end of the capacitance phasor back
to the start. This hypotenuse is the resultant voltage drop.

Again, the values can be divided and multiplied by I
to give impedance and power respectively.

V^{2} = (V_{R}^{2} + (V_{L} - V_{C})^{2})

Take the square root of these to get the voltage dropped, V.

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