Question

A dc voltage equal to \(V_{\mathrm{dc}} \mathrm{V}\) is applied to a resistor of \(R \Omega .\) A sinusoidal voltage equal to \(v_{s} V\) is also applied to a resistor of \(R \Omega\). Show that the de voltage will deliver the same amount of energy in \(T\) seconds (where \(T\) is the period of the sinusoidal voltage \()\) as the sinusoidal voltage provided \(V_{\mathrm{dc}}\) equals the rms value of \(v_{s}\). (Hint: Equate the two expressions for the energy delivered to the resistor.)

Short answer

Both energies equal when \(V_{\text{dc}} = V_{\text{rms}}\), the RMS value of the sinusoidal voltage.

Step-by-step solution

Define the Equations for Energy

First, understand that the energy delivered to a resistor by a voltage source is given by the formula \(E = \frac{V^2}{R} \times t\), where \(E\) is energy, \(V\) is voltage, \(R\) is resistance, and \(t\) is time. We will use this formula for both DC and sinusoidal voltages.

Calculate Energy Delivered by DC Voltage

For the DC voltage \(V_{\text{dc}}\), the energy delivered to the resistor in time \(T\) is given by: \(E_{\text{dc}} = \frac{V_{\text{dc}}^2}{R} \times T\).

Calculate Energy Delivered by Sinusoidal Voltage

The sinusoidal voltage \(v_s(t) = V_m \sin(\omega t)\) has an RMS value \(V_{\text{rms}} = \frac{V_m}{\sqrt{2}}\). The energy delivered by this sinusoidal voltage in one period \(T\) is: \(E_{\text{sin}} = \frac{(V_{\text{rms}})^2}{R} \times T\).

Equate the Two Energy Expressions

To equate the energy delivered by both the DC and sinusoidal voltage sources, set \(E_{\text{dc}}\) equal to \(E_{\text{sin}}\): \(\frac{V_{\text{dc}}^2}{R} \times T = \frac{(V_{\text{rms}})^2}{R} \times T\).

Simplify and Derive the Relationship

Cancel \(T\) and \(R\) from both sides of the equation: \(V_{\text{dc}}^2 = (V_{\text{rms}})^2\). Taking the square root on both sides, we find that \(V_{\text{dc}} = V_{\text{rms}}\). This shows that for equal energy delivery, \(V_{\text{dc}}\) must equal the RMS value of the sinusoidal voltage.

Key concepts

DC voltage

Direct current (DC) voltage is a form of electrical energy where the electric charge (flow of electrons) moves in one direction. Unlike alternating current (AC) where the flow reverses periodically, DC maintains a constant direction with a stable magnitude over time.

In practical terms, DC voltage can be visualized as a steady, constant voltage source. It is typically supplied by batteries and DC power supplies.
Working with DC voltage allows for straightforward calculations when determining how much energy is delivered to a load like a resistor.

This is because the voltage remains unchanged, making the energy calculation over a time period simple: multiplying the voltage squared by the resistance and time. For example, using the formula: \[ E = \frac{V^2}{R} \times t \] ensures accurate measurement of energy delivery over time.

sinusoidal voltage

Sinusoidal voltage is a type of alternating current (AC) voltage waveform that varies with time in a smooth, wave-like manner characterized by a sine function. In a sinusoidal voltage, the voltage level fluctuates between positive and negative peaks, creating a periodic wave pattern.

This type of voltage is common in power supply systems such as the electricity that powers homes and businesses.
The mathematical representation of sinusoidal voltage is given by: \[ v_s(t) = V_m \sin(\omega t) \] where:

• \( V_m \) is the peak voltage, the maximum voltage the waveform reaches.
• \( \omega \) is the angular frequency in radians per second, denoting how fast the waveform oscillates.

Using sinusoidal voltage introduces complexity to circuit analysis since the voltage is constantly changing, necessitating more sophisticated calculations to determine its effects over time.

energy delivered to resistor

The concept of energy delivered to a resistor is foundational in electrical engineering. A resistor is a component that dissipates electrical energy, usually as heat. Understanding the energy a resistor consumes involves the power relationship described by \[ E = \frac{V^2}{R} \times t \].

This formula allows us to calculate the total energy transferred, where:

• \( V \) denotes the voltage across the resistor.
• \( R \) is the resistance opposing the flow of current.
• \( t \) is the time for which the current flows.

Both DC and sinusoidal voltages deliver energy to the resistor, but only when the root mean square (RMS) voltage of a sinusoidal wave equals the DC voltage, does the total energy delivery match over one cycle. Thus, using this formula, the energy delivered to a resistor is a measurable way to understand power consumption and energy usage in electrical circuits.

RMS value

RMS, or root mean square value, is a key metric in analyzing AC circuits, particularly when working with sinusoidal voltages. The RMS value represents the effective voltage or current of an AC waveform that conveys the same power as a corresponding DC voltage.

For sinusoidal voltages, the RMS value is derived from the peak voltage using the relationship: \[ V_{\text{rms}} = \frac{V_m}{\sqrt{2}} \].

Here,

• \( V_m \) is the peak voltage.

This key value is crucial because it allows engineers and students alike to compare AC and DC circuits directly.
The RMS value is especially important in the context of energy delivery because it provides a means of equating sinusoidal and DC voltages, as shown in energy equivalence problems. To equate energy delivery, the DC voltage applied must equal the RMS value of the sinusoidal input. This equivalence illustrates how RMS is a practical and necessary approach when working with alternating voltages in electrical engineering.