The RLC Series Circuit

The resistor, inductor and capacitor connected in series in a circuit is called RLC or LRC series circuit. Mostly we call RLC but some may prefer to call LRC series circuit. We consider a simple RLC series circuit with alternating source of emf.

The voltage across the resistance it is in phase with the current. The inductor and capacitor have their corresponding reactances. The reactances are equivalent to the resistance. And those reactances give rise to the voltage across them. Of course, you learnt the phase relationship of voltage and current across both reactances in inductive reactance and capacitive reactance articles. The Figure 1 shows a RLC series circuit with an ac source.

Figure 1 RLC series circuit.

Here we take a look at the voltages across each component, add them, and find the net voltage which is equal to the source voltage. In doing so, we find the net opposing effect (impedence) due to the individual components. Since the source voltage changes sinusoidally, the voltage across each component also changes sinusoidally. The sum of the voltages across each component at an instant is equal to the source voltage at that instant.

You know that, it's easier to represent sinusoidally changing quantities in terms of their rms value. Here we consider the voltage amplitude for each component. And, each voltage amplitude is represented by a phasor in the phasor diagram as shown in the Figure 2 below whose projection is the instantaneous value of the voltage. The sum of the voltage amplitudes of each voltage is equal to the voltage amplitude of the source voltage, and again the projection of this source voltage amplitude is the instantaneous value of the source voltage. But the sum is not an algebraic sum, instead it is the vector sum of the phasors.

Figure 2 The source voltage leads the current ($X_L > X_C$).

Since, this is the series connection, the current is the same throughout the circuit. The voltage and current across the resistance are at the same phase (the maximum and minimum values of voltage and current occur at the same time). Therefore, the voltage amplitude across the resistance is

\[V_R = IR\]

The voltage across the inductor leads the current by $\pi/2\,\text{rad}$. You know that in RL circuit, due to the presence of inductor, the current can not become maximum instantaneously, and therefore, it stars from zero and when the voltage across the inductor is zero, it is maximum. The voltage amplitude across the inductor is

\[V_L = IX_L\]

The voltage across capacitor lags the current by $\pi/2\,\text{rad}$. Initially, the capacitor has zero charge i.e. zero voltage, and gradually increases as the charge starts to build up and the voltage is also maximum when it's fully charged. The voltage amplitude across the capacitor is

\[V_C = IX_L\]

We talked earlier that, the sum of these voltage amplitudes is the voltage amplitude of the source voltage. But, we do not add them algebraically, but it's the vector sum. Each amplitude is represented by the rotating vector (phasor) as shown in Figure 2 for the $X_L > X_C$ case. The diagram for the $X_L > X_C$ is shown in Figure 3.

Figure 3 The source voltage lags the current ($X_L < X_C$).

You also saw in LC circuit that the the voltage across inductor and capacitor are out of phase by $\pi\,\text{rad}$. One leads the current by $\pi / 2 \,\text{rad}$ and another lags the current by $\pi / 2 \,\text{rad}$, therefore the phasors of $V_L$ and $V_C$ are opposite to each other. The presence of resistance does not disturb the phase relationship between inductor and capacitor.

To sum up these phasors, we subtract, the phasor $V_C$ from $V_L$, and $V_L - V_C$ is at right angle to the $V_R$, and therefore

\[V = \sqrt{(V_L - V_C)^2 + V_R^2} = I \sqrt{(X_L - X_C)^2 + R^2} \]

The above relationship is similar to $V= IR $ for dc current if $Z = \sqrt{(X_L - X_C)^2 + R^2}$, that is $V = IZ$. Here $Z$ is called impedance whose role is similar to that of the resistance and has the SI unit $\Omega$ (Ohm) which is the SI unit of resistance. The impedance is defined as the ratio of voltage amplitude across the circuit to the current amplitude, that is

\[Z = \sqrt{(X_L - X_C)^2 + R^2}\]

If both inductor and capacitor are not present in the circuit, the impedance is equal to the resistance, that is $Z = R$. Although the above relationship is derived for the RLC series circuit, we define it as the ratio of voltage amplitude of any circuit of inductors, resistors and capacitors to the current amplitude in the circuit, or as the ratio of rms voltage across any the circuit to the rms current.

The impedance also depends on the frequency (or angular frequency) as reactances ($X_L = \omega\,L$ and $X_C = 1/\omega\,C$) depend on it. From the diagrams of Figure 2 or Figure 3, you can find that

\[\phi = \tan^{-1}(\frac{V_L - V_C}{V_R}) = \tan^{-1}(\frac{X_L - X_C}{R}) \]

The $\phi$ is the angle between the voltage across the whole circuit and the current. If $X_L > X_C$, the voltage is ahead of current and $\phi$ is positive, and if $X_L < X_C$, the voltage is behind the current and $\phi$ is negative. Since the reactances also depend on the frequency, the phase angle $\phi$ with respect to the current also depends on the frequency.

Electromagnetism
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