(a) How does the resistance differ from impedance? With the help of a suitable phasor diagram, obtain an expression for impedance of a series LCR…

1. Resistance is the opposition offered to both alternating current and direct current while impedance is the opposition offered to alternating current only.

2. Resistance is independent of frequency of source while impedance depends on frequency.

3. Resistance is opposition offered by material of the conductor while impedance is combined opposition offered by different electrical components such as resistor, inductor or capacitor.

\[\mathrm{V}_{\mathrm{R}}=\mathrm{i}_{\mathrm{m}} \mathrm{R}, \mathrm{~V}_{\mathrm{c}}=\mathrm{i}_{\mathrm{m}} \mathrm{X}_{\mathrm{c}}, \mathrm{~V}_{\mathrm{L}}=\mathrm{i}_{\mathrm{m}} \mathrm{X}_{\mathrm{L}} \]

\(\displaystyle \mathrm{i}_{\mathrm{m}}=\) Peak value of current in the circuit.

\[\begin{aligned} & \overrightarrow{\mathrm{V}_{\mathrm{L}}}+\overrightarrow{\mathrm{V}_{\mathrm{R}}}+\overrightarrow{\mathrm{V}_{\mathrm{C}}}=\overrightarrow{\mathrm{V}_{\mathrm{m}}} \\ &\left(\mathrm{~V}_{\mathrm{m}}\right)^{2}=\mathrm{V}_{\mathrm{R}}{ }^{2}+\left(\mathrm{V}_{\mathrm{C}}-\mathrm{V}_{\mathrm{L}}\right)^{2} \\ &=\left(\mathrm{i}_{\mathrm{m}} \mathrm{R}\right)^{2}+\left(\mathrm{i}_{\mathrm{m}} \mathrm{X}_{\mathrm{c}}-\mathrm{i}_{\mathrm{m}} \mathrm{X}_{\mathrm{L}}\right)^{2} \\ &=\mathrm{i}_{\mathrm{m}}\left[\mathrm{R}^{2}+\left(\mathrm{X}_{\mathrm{c}}-\mathrm{X}_{\mathrm{L}}\right)^{2}\right] \end{aligned} \]

\[\mathrm{i}_{\mathrm{m}}=\frac{\mathrm{V}_{\mathrm{m}}}{\sqrt{\mathrm{R}^{2}+\left(\mathrm{X}_{\mathrm{c}}-\mathrm{X}_{\mathrm{L}}\right)^{2}}} \]

\(\displaystyle \mathrm{i}_{\mathrm{m}}=\frac{\mathrm{V}_{\mathrm{m}}}{\mathrm{Z}}\) where \(\displaystyle \mathrm{Z}=\sqrt{\mathrm{R}^{2}+\left(\mathrm{X}_{\mathrm{c}}-\mathrm{X}_{\mathrm{L}}\right)^{2}}=\) impedance