AC Series Configuration

The general characteristics of AC series circuits are the same as those of DC circuits. For example, the total impedance of a system is the sum of all the individual impedances:
$$ \bbox[10px,border:1px solid grey]{Z_T = Z_1 + Z_2+Z_3+...+Z_N} \tag{1}$$
Series impedances
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.
$$\begin{split} Z_T &= Z_1 + Z_2 + Z_3 \\ &= R \angle 0^\circ+X_L \angle 90^\circ + X_C \angle -90^\circ\\ &= R +j X_L -jX_C \\ &= R +j( X_L - X_C) \\ &= 6 Ω+j( 10 - 12)Ω = 6-j2Ω \\ Z_T &= 6.325 \angle -18.43^\circ\\ \end{split} $$
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:
$$\bbox[10px,border:1px solid grey]{Z_T = Z_1 + Z_2}$$
Series ac circuit.
Fig. 4: Series ac circuit.
$$\bbox[10px,border:1px solid grey]{I = { E \over Z_T}}$$
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.
$$E - V_1 + V_2 = 0$$
$$\bbox[10px,border:1px solid grey]{E = V_1 + V_2}$$
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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