In a series LRC circuit, which consists of inductors, resistors, and capacitors connected in series to an alternating current (AC) source, the current flowing through each component remains constant. However, the voltage across each element behaves differently due to the phase differences in AC circuits. Unlike direct current (DC) circuits, where the total voltage is simply the sum of the voltages across each component, in AC circuits, the maximum voltage is determined using a specific relationship. This relationship is expressed as:
\[ V_{LRC} = \sqrt{V_R^2 + (V_L - V_C)^2} \]
Here, \(V_{LRC}\) represents the maximum voltage across the entire circuit, while \(V_R\), \(V_L\), and \(V_C\) are the maximum voltages across the resistor, inductor, and capacitor, respectively. This equation arises because the peaks of the voltages across the inductor and capacitor do not occur simultaneously, necessitating the use of the square root to account for their phase differences.
Another critical concept in LRC circuits is impedance, which is the effective resistance to the flow of current. The impedance \(Z\) is calculated using the formula:
\[ Z = \sqrt{R^2 + \left(X_L - X_C\right)^2} \]
where \(R\) is the resistance, \(X_L\) is the inductive reactance given by \(X_L = \omega L\) (with \(\omega\) being the angular frequency), and \(X_C\) is the capacitive reactance given by \(X_C = \frac{1}{\omega C}\). The maximum current \(I_{max}\) in the circuit can then be determined using the equation:
\[ I_{max} = \frac{V_{max}}{Z} \]
To illustrate these concepts, consider a circuit with a 0.5 henry inductor, a 10 ohm resistor, and a 500 microfarad capacitor connected to an AC source with a root mean square (RMS) voltage of 120 volts and a frequency of 60 hertz. First, the maximum voltage is calculated as:
\[ V_{max} = \sqrt{2} \times V_{rms} = \sqrt{2} \times 120 \, \text{V} \approx 170 \, \text{V} \]
The angular frequency is calculated as:
\[ \omega = 2\pi f = 2\pi \times 60 \, \text{Hz} \approx 377 \, \text{s}^{-1} \]
Next, the impedance is calculated by substituting the values into the impedance formula. The inductive reactance is:
\[ X_L = \omega L = 377 \times 0.5 \approx 188.5 \, \Omega \]
The capacitive reactance is:
\[ X_C = \frac{1}{\omega C} = \frac{1}{377 \times 500 \times 10^{-6}} \approx 5.305 \, \Omega \]
Now, substituting these values into the impedance formula gives:
\[ Z = \sqrt{10^2 + (188.5 - 5.305)^2} \approx 183 \, \Omega \]
Finally, the maximum current in the circuit is calculated as:
\[ I_{max} = \frac{170 \, \text{V}}{183 \, \Omega} \approx 0.93 \, \text{A} \]
This example highlights the importance of understanding the relationships between voltage, current, and impedance in series LRC circuits, as well as the necessity of using angular frequency in calculations.
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