# Frequency Response of the Parallel RL Network

In the earlier Section 8.14.2, the frequency response of a series R-C circuit was analyzed. Let us now note the impact of frequency on the total impedance and inductive current for the parallel R-L network of Fig. 1 for a frequency range of zero through 40 kHz.
Fig. 1: Determining the frequency response of a parallel R-L network.
$Z_T$Before getting into specifics, let us first develop a "sense" for the impact of frequency on the network of Fig. 1 by noting the impedance-versus-frequency curves of the individual elements, as shown in Fig. 2. The fact that the elements are now in parallel requires that we consider their characteristics in a different manner than occurred for the series R-C circuit of Section 12.4. Recall that for parallel elements, the element with the smallest impedance will have the greatest impact on the total impedance at that frequency.
Fig. 2: The frequency response of the individual elements of a parallel R-L network
In Fig. 2, for example, $X_L$ is very small at low frequencies compared to $R$, establishing $X_L$ as the predominant factor in this frequency range. In other words, at low frequencies the network will be primarily inductive, and the angle associated with the total impedance will be close to $90^\circ$, as with a pure inductor. As the frequency increases, $X_L$ will increase until it equals the impedance of the resistor (220 Ω). The frequency at which this situation occurs can be determined in the following manner: $$X_L = 2 \pi f_2 L = R$$ and $$f_2 = {R \over 2 \pi L} = { 220 Ω \over 2 \pi (4 \times 10^{-3} H)}\\ =8.75 kHz$$ which falls within the frequency range of interest.
For frequencies less than $f_2, X_L < R$, and for frequencies greater than $f_2, X_L > R$, as shown in Fig. 2. A general equation for the total impedance in vector form can be developed in the following manner: $$\begin{split} Z_T &= {Z_R Z_L \over Z_R + Z_L}\\ &={(R \angle 0^\circ)(X_L \angle 90^\circ) \over R+jX_L}\\ &={R X_L \angle 90^\circ \over \sqrt{R^2 + X_L^2} \angle \tan^{-1} X_L/R} \\ &={R X_L \over \sqrt{R^2 + X_L^2}} \angle 90^\circ - \tan^{-1} X_L/R\\ \end{split}$$ so that $$\bbox[10px,border:1px solid grey]{Z_T = {RX_L \over \sqrt{R^2 + X_L^2}}} \tag{1}$$ and $$\bbox[10px,border:1px solid grey]{\theta_T = 90^\circ -\tan^{-1} {X_L \over R} =\tan^{-1} {R \over X_L}} \tag{2}$$ The magnitude and angle of the total impedance can now be found at any frequency of interest simply by substituting Eqs. (1) and (2).
f = 1 kHz $$X_L = 2 \pi f L\\ = 2 \pi (1kHz)(4 \times 10^{-3} H)=25.12Ω$$ and $$Z_T = {RX_L \over \sqrt{R^2 + X_L^2}}\\ ={(220 Ω)(25.12 Ω) \over \sqrt{(220Ω)^2 + (25.12Ω)^2}} = 24.96 Ω$$ with $$\theta_T = \tan^{-1} {R \over X_L} = \tan^{-1} {220 \over 25.12}\\ =\tan^{-1} 8.76 = 83.49 ^\circ$$ and $$Z_T = 24.96 \angle 83.49 ^\circ$$ This value compares very closely with $X_L = 25.12Ω \angle 90^\circ$, which it would be if the network were purely inductive ($R = \infty Ω$). Our assumption that the network is primarily inductive at low frequencies is therefore confirmed.
Continuing: $$f = 5 kHz: \,\, Z_T = 109.1 Ω \angle 60.23^\circ$$ $$f = 10 kHz: \,\, Z_T = 165.5 Ω \angle 41.21^\circ$$ $$f = 15 kHz: \,\, Z_T = 189.99 Ω \angle 30.28^\circ$$ $$f = 20 kHz: \,\, Z_T = 201.19 Ω \angle 23.65^\circ$$ $$f = 30 kHz: \,\, Z_T = 211.19 Ω \angle 16.27^\circ$$ $$f = 40 kHz: \,\, Z_T = 214.91 Ω \angle 12.35^\circ$$ At $f = 40 kHz$, note how closely the magnitude of $Z_T$ has approached the resistance level of 220Ω and how the associated angle with the total impedance is approaching zero degrees. The result is a network with terminal characteristics that are becoming more and more resistive as the frequency increases, which further confirms the earlier conclusions developed by the curves of Fig. 2.
Plots of $Z_T$ versus frequency in Fig. 3 and $\theta_T$ in Fig. 4 clearly reveal the transition from an inductive network to one that has resistive characteristics. Note that the transition frequency of 8.75 kHz occurs right in the middle of the knee of the curves for both $Z_T$ and $\theta_T$.
Fig. 3: The magnitude of the input impedance versus frequency for the network of Fig. 1.
A review of Figs. 3 of Section 8.14.2, the frequency response of a series R-C circuit and Fig. 3 will reveal that a series R-C and a parallel R-L network will have an impedance level that approaches the resistance of the network at high frequencies. The capacitive circuit approaches the level from above, whereas the inductive network does the same from below. For the series R-L circuit and the parallel R-C network, the total impedance will begin at the resistance level and then display the characteristics of the reactive elements at high frequencies.
Fig. 4: The phase angle of the input impedance versus frequency for the network of Fig. 1.