Question

Is it possible for the voltage amplitude across the inductor in a series RLC circuit to exceed the voltage amplitude of the voltage supply? Why or why not?

Short answer

Answer: The voltage amplitude across the inductor in a series RLC circuit can exceed the voltage amplitude of the voltage supply when the inductive reactance (XL) is significantly larger than the capacitive reactance (XC) and the resistance (R). In such a scenario, the vector sum of the inductor and capacitor voltages (V_L and V_C) results in a voltage that is smaller than the voltage across the inductor itself, thereby satisfying the condition V_L > V_R + V_C.

Step-by-step solution

Review the RLC circuit elements and their voltages

In a series RLC circuit, there are three main components: a resistor (R), an inductor (L), and a capacitor (C). The voltage across the resistor is in phase with the current, the voltage across the inductor leads the current by 90 degrees, and the voltage across the capacitor lags the current by 90 degrees.

Analyze the impedance and phasor relationship

The total impedance (Z) of the series RLC circuit can be determined using the following formula: Z = sqrt(R^2 + (XL - XC)^2) where XL = 2 * \pi * f * L is the inductive reactance, XC = 1 / (2 * \pi * f * C) is the capacitive reactance, and f is the frequency of the source voltage supply. Using Kirchhoff's law, the sum of the voltage drops across each component must equal the voltage supply: V_source = V_R + V_L + V_C Since each voltage has a different phase, we can represent them as vectors in a phasor diagram.

Investigate the relationship between the voltage amplitudes

To find out if the voltage amplitude across the inductor can exceed the voltage amplitude of the voltage supply, let's consider the phasor diagram. In this diagram, V_L and V_C are 180 degrees out of phase with each other, and V_R is in-phase with the supply voltage. Therefore, the phasor summation of V_L and V_C can sometimes result in a vector with a smaller magnitude than V_L itself.

Identify the condition for V_L > V_source

The condition for the voltage amplitude across the inductor to exceed the voltage amplitude of the voltage supply (V_source) is when: V_L > V_R + V_C Taking phasors into account, the condition can be written as: sqrt(V_L^2 - V_C^2) > V_R This inequality is satisfied when the reactance of the inductor (XL) is much larger than the reactance of the capacitor (XC) and the resistance (R).

Conclude the answer

It is possible for the voltage amplitude across the inductor in a series RLC circuit to exceed the voltage amplitude of the voltage supply under certain conditions. This can happen if the inductive reactance (XL) is significantly larger than the capacitive reactance (XC) and the resistance (R) in the circuit. In such a scenario, the vector sum of the inductor and capacitor voltages (V_L and V_C) can result in a voltage that is smaller than the voltage across the inductor itself, thereby satisfying the condition V_L > V_R + V_C. The phasor analysis helps in identifying the conditions where the voltage amplitude across the inductor can be greater than the source voltage.