The overall properties of series ac circuits are the same as
those for dc circuits. For instance, the total impedance of a system is the
sum of the individual impedances:
$$ \bbox[10px,border:1px solid grey]{Z_T = Z_1 + Z_2+Z_3+...+Z_N} \tag{1}$$
Fig. 1: Series impedances.
Example 1: Determine the input impedance to the series network
of Fig. 2. Draw the impedance diagram.
Fig. 2: For Example 1.
Solution:
$$\begin{split}
Z_T &= Z_1 + Z_2 + Z_3 \\
&= R \angle 0^\circ+X_L \angle 90^\circ + X_C \angle -90^\circ\\
&= R +j X_L -jX_C \\
&= R +j( X_L - X_C) \\
&= 6 Ω+j( 10 - 12)Ω = 6-j2Ω \\
Z_T &= 6.325 \angle -18.43^\circ\\
\end{split}
$$
Fig. 3: For Example 1.
The impedance diagram appears in Fig. 3. Note that in this
example, series inductive and capacitive reactances are in direct opposition. For the circuit of Fig. 2, if the inductive reactance were equal to the capacitive reactance, the input impedance would be purely resistive.
For the representative series ac configuration of Fig. 4 having
two impedances, the current is the same through each element (as it
was for the series dc circuits) and is determined by Ohm's law:
$$\bbox[10px,border:1px solid grey]{Z_T = Z_1 + Z_2}$$
Fig. 4: Series ac circuit.
and
$$\bbox[10px,border:1px solid grey]{I = { E \over Z_T}}$$
The voltage across each element can then be found by another application of Ohm's law:
$$\bbox[10px,border:1px solid grey]{V_1 = IZ_1}$$
$$\bbox[10px,border:1px solid grey]{V_2 = IZ_2}$$
Kirchhoff's voltage law can then be applied in the same manner as it
is employed for dc circuits. However, keep in mind that we are now
dealing with the algebraic manipulation of quantities that have both
magnitude and direction.
$$E - V_1 + V_2 = 0$$
or
$$\bbox[10px,border:1px solid grey]{E = V_1 + V_2}$$
The power to the circuit can be determined by
$$P = EI \cos \theta_T$$
where $\theta_T$ is the phase angle between E and I.
Now that a general approach has been introduced, the simplest of
series configurations will be investigated in detail to further emphasize
the similarities in the analysis of dc circuits. In many of the circuits to
be considered, $3 + j4 =5 \angle 53.13^\circ$ and $4 + j3 = 5 \angle36.87^\circ$ will be
used quite frequently to ensure that the approach is as clear as possible
and not lost in mathematical complexity.
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