LCR Circuit (Series)

What this figure shows. A single-loop series circuit diagram showing a resistor R, an inductor L and a capacitor C connected one after another in SERIES (so the same instantaneous current i flows through all three), the whole series combination connected across an AC source, with the instantaneous emf e of the source and the common current i both labelled -- the combined circuit whose separate R-only, L-only and C-only behaviours (Figs. 13.3, 13.6, 13.9) are now superposed using phasor addition, since the three elements' individual voltage drops are not in pha …

What this figure shows. A phasor diagram built by vector (phasor) addition: the current phasor $i_0$ drawn along the positive X-axis as the common reference (since current is the same through R, L and C in series); the resistor's voltage phasor $e_R=i_0R$ drawn ALONG the same direction as $i_0$ (in phase, along OA on the X-axis); the inductor's voltage phasor $e_L=i_0X_L$ drawn perpendicular to $i_0$, rotated $90^\circ$ ANTICLOCKWISE from it (leading); the capacitor's voltage phasor $e_C=i_0X_C$ drawn perpendicular to $i_0$ but rotated $90^\circ$ CLOCKWISE (lagging), i.e. exactly opposite to $e_L$ along OY'. Since $e_L$ and $e_C$ are $180^\circ$ apart, only their DIFFERENCE $(e_L-e_C)$ survives as the net reactive voltage OB', and the diagonal OK of the rectangle formed by OA (along $e_R$) and OB' (the net reactive voltage) gives …

What this figure shows. A right-angled triangle (derived directly from dividing every side of the voltage phasor triangle in Fig. 13.13 by the common current $i_0$): the base of the triangle represents the pure Ohmic resistance R (horizontal), the vertical side (perpendicular to the base) represents the net reactance $(X_L-X_C)$, and the hypotenuse (the diagonal connecting them) represents the impedance $Z=\sqrt{R^2+(X_L-X_C)^2}$. The angle between the base (R) and the hypotenuse (Z) is marked as $\phi$, the same phase angle by which the voltage leads (or lags) the current, satisfying $\tan\phi=(X_L-X_C)/R$ -- giving a purely geometric (right-triangle) way to remember and …

Worked out. A 100 mH inductor, a 25 $\mu F$ capacitor and a 15 $\Omega$ resistor are connected in series to a 120 V (rms), 50 Hz AC source. At RESONANCE (the frequency at which $X_L=X_C$, developed fully in section 13.8), the reactive part of the impedance cancels exactly, leaving the circuit purely resistive: $Z=R=15\,\Omega$. The current at resonance is then simply $i_{rms}=e_{rms}/R=120/15=8$ A. The resonant frequency itself is found from $f_r=\dfrac{1}{2\pi\sqrt{LC}}=\dfrac{1}{2\times3.142\times\sqrt{10^{-1}\times25\times10^{-6}}}\approx100.6$ Hz -- notably this resonant frequency does NOT depend on the resistance R or on the actual 50 Hz supply frequency at all, only on L and C, and is worked out here purely to characteri …

Worked out. A coil of inductance $L=0.01$ H and resistance $R=1\,\Omega$ is connected to a 200 V, 50 Hz AC supply (no capacitor, so this is a series RL circuit, i.e. the C-terms of the general LCR formulas simply drop out). The inductive reactance is $X_L=2\pi fL=2\times3.142\times50\times0.01\approx3.142\,\Omega$, giving impedance $Z=\sqrt{R^2+X_L^2}=\sqrt{1^2+3.142^2}=\sqrt{10.872}\approx3.297\,\Omega$. The phase angle follows from $\tan\phi=X_L/R=3.142/1=3.142$, so $\phi=\tan^{-1}(3.142)\approx72.35^\circ=72.35\times\pi/180$ rad. Since this phase angle corresponds to a fraction of one full cycle ($2\pi$ rad $=T=1/f=0.02$ s), the actual TIME LAG between the peaks of voltage and current is $\Delta t=\dfrac{\phi}{\omega}=\dfrac{\phi}{2\pi f}=\dfrac{72.35\times\pi/180}{2\pi\times50}\approx4.019\times10^{-3}$ s -- this example's key device is converting a phase angle …