Question

The instantaneous voltages at three terminals marked \(X, Y\) and \(Z\) are given by $$ \begin{aligned} &V_{x}=V_{0} \sin \omega t \\ &V_{Y}=V_{0} \sin \left(\omega t+\frac{2 \pi}{3}\right) \text { and } \\ &V_{z}=V_{0} \sin \left(\omega t+\frac{4 \pi}{3}\right) . \end{aligned} $$ An ideal voltmeter is configured to read \(r m s\) value of the potential difference between its terminals. It is connected between points \(X\) and \(Y\) and then between \(Y\) and \(Z\). The reading(s) of the voltmeter will be [A] \(\quad V_{X Y}^{r m s}=V_{0} \sqrt{\frac{3}{2}}\) [B] \(\quad V_{Y Z}^{r m s}=V_{0} \sqrt{\frac{1}{2}}\) [C] \(V_{X Y}^{r m s}=V_{0}\) [D] independent of the choice of the two terminals

Short answer

The voltmeter reading would be the same for both connections and is \( V_0 \sqrt{\frac{3}{2}} \), so the correct option is [A] \( V_{XY}^{rms} = V_{YZ}^{rms} = V_0 \sqrt{\frac{3}{2}} \).

Step-by-step solution

Express the voltage between terminals X and Y

To find the voltage between terminals X and Y, calculate the difference between the voltages at X and Y: \[ V_{XY} = V_X - V_Y = V_{0} \sin (\omega t) - V_{0} \sin \left(\omega t + \frac{2 \pi}{3}\right) \]

Apply Trigonometric Identity

Use the sine difference identity to simplify the expression for \( V_{XY} \):\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right) \]Where \( A = \omega t \) and \( B = \omega t + \frac{2\pi}{3} \), this gives:\[ V_{XY} = 2V_0 \cos\left(\omega t + \frac{\pi}{3}\right)\sin\left(-\frac{\pi}{3}\right) \]

Calculate the RMS value for \( V_{XY} \)

The RMS (root mean square) value for a sinusoidal function is \( \frac{V_0}{\sqrt{2}} \), where \( V_0 \) is the peak voltage. However, due to the factor of 2 and the sine term \( \sin\left(-\frac{\pi}{3}\right) \), we adjust the RMS calculation: \[ V_{XY}^{rms} = \frac{2V_0}{\sqrt{2}} \left| \sin\left(-\frac{\pi}{3}\right) \right| = V_0 \sqrt{2} \left| \sin\left(-\frac{\pi}{3}\right) \right| = V_0 \sqrt{2} \left| \sin\left(\frac{\pi}{3}\right) \right| \]Since \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \), we find:\[ V_{XY}^{rms} = V_0 \sqrt{2} \cdot \frac{\sqrt{3}}{2} = V_0 \sqrt{\frac{3}{2}} \]

Express the voltage between terminals Y and Z

Calculate the difference between the voltages at Y and Z in the same way:\[ V_{YZ} = V_Y - V_Z = V_{0} \sin \left(\omega t + \frac{2\pi}{3}\right) - V_{0} \sin \left(\omega t + \frac{4\pi}{3}\right) \]

Apply Trigonometric Identity for \( V_{YZ} \)

Apply the sine difference identity to \( V_{YZ} \) just as we did for \( V_{XY} \). With \( A = \omega t + \frac{2\pi}{3} \) and \( B = \omega t + \frac{4\pi}{3} \):\[ V_{YZ} = 2V_0 \cos\left(\omega t + \pi\right)\sin\left(-\frac{\pi}{3}\right) \]

Calculate the RMS value for \( V_{YZ} \)

Using the RMS formula for sinusoidal function and considering the cosine value at \( \pi \), which is -1, we get:\[ V_{YZ}^{rms} = \frac{2V_0}{\sqrt{2}} \left| \sin\left(\frac{\pi}{3}\right) \right| = V_0 \sqrt{2} \left| \sin\left(\frac{\pi}{3}\right) \right| \]But since \( \cos(\omega t + \pi) = -1 \), this introduces a negative sign which doesn’t affect the RMS value because it's square-rooted. We end up with the same RMS value as for \( V_{XY} \):\[ V_{YZ}^{rms} = V_0 \sqrt{2} \cdot \frac{\sqrt{3}}{2} = V_0 \sqrt{\frac{3}{2}} \]

Key concepts

Sinusoidal Voltage

When we deal with electrical systems, particularly in the context of AC (alternating current) circuits, we often encounter sinusoidal voltages. These are voltages that vary over time in a sinusoidal pattern, typically described by the function
\( V(t) = V_0 \times \text{sin}(ωt + φ) \),
where \( V_0 \) is the peak voltage, \( ω \) is the angular frequency, and \( φ \) is the phase angle. This sinusoidal behavior is prevalent due to the way AC generators produce electricity, with the rotation of coils within magnetic fields resulting in an output voltage that rises and falls in a consistent, wave-like manner.
Understanding sinusoidal voltage is pivotal because this form of waveform is used to deliver power to businesses and residences. Moreover, the characteristics of sinusoidal voltages, such as peak values and phase differences, play a crucial role in various applications, from power distribution to signal processing.

Trigonometric Identity

In mathematics and engineering, trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. One particularly useful identity in the context of AC circuit analysis is the sine difference formula:
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right) \].
This identity allows us to simplify the expression when calculating the potential difference between two points in an AC circuit with sinusoidally varying voltages. When we apply this identity, we can transform a difficult-to-manage subtraction of two sine functions into a product of sine and cosine functions, which are easier to work with, especially when calculating RMS values or analyzing circuit behavior.

AC Circuit Analysis

Analyzing AC circuits involves understanding how alternating currents operate within various electrical components. Unlike DC circuits, where the current flows in a single direction, AC circuits carry current that changes direction periodically. Components in these circuits, such as resistors, capacitors, and inductors, respond differently to alternating current than to direct current.
A key aspect of analyzing AC circuits is the concept of phase difference, which occurs due to the sinusoidal nature of AC. This phase difference is vital when calculating the net voltage across multiple components, as seen in the exercise where we calculate the difference in voltage between terminals (X and Y, and Y and Z) with phase shifts. AC circuit analysis also relies heavily on the RMS value to characterize the effective voltage or current that produces the same power as a DC voltage or current.

Root Mean Square (RMS)

The Root Mean Square (RMS) value is a statistical measure used in AC circuit analysis to determine the equivalent DC value—meaning the value of direct current that would deliver the same power to a load as the AC current does over one cycle. For a sinusoidal voltage or current, the RMS value can be calculated using the formula:
\[ V_{RMS} = \frac{V_0}{\sqrt{2}} \],
where \( V_0 \) is the peak voltage of the sinusoidal waveform. RMS is critical in electrical power calculations because it represents the effective value of a varying voltage or current. For example, the RMS value of the voltage tells us how much power the voltage will deliver to a resistive load. It is, therefore, a more practical quantity than the peak voltage when designing and analyzing electrical systems and components.