Alternating Voltages and Currents: Videos & Practice Problems

Alternating current (AC) circuits involve currents that change direction, unlike direct current (DC) circuits, which flow in a single direction. AC is produced by alternating voltages, typically sinusoidal, represented as $v=Vmax\cos (\omega t)$. The current mirrors this sinusoidal pattern, with angular frequency $\omega =2\pi f$. Understanding these principles is crucial for analyzing AC circuits effectively.

Alternating Voltages and Currents

In the study of electrical circuits, it's essential to differentiate between direct current (DC) and alternating current (AC). DC circuits involve currents that flow in a single direction, typically powered by a constant voltage source like a battery. In contrast, AC circuits feature currents that alternate directions, necessitating the use of alternating voltages. The most common form of alternating voltage is sinusoidal, represented mathematically as:

$$ V(t) = V_{\text{max}} \cos(\omega t) $$

Here, V is the instantaneous voltage, V_max is the maximum voltage (or amplitude), and ω is the angular frequency, which relates to the linear frequency f by the equation:

$$ \omega = 2\pi f $$

In AC circuits, the current also follows a sinusoidal pattern, matching the voltage's oscillation. The current can be expressed as:

$$ I(t) = I_{\text{max}} \cos(\omega t) $$

where I is the instantaneous current and I_max is the maximum current. This relationship indicates that the current and voltage oscillate between their maximum and minimum values, with negative values indicating a reversal in polarity or direction.

For example, in North America, the standard frequency of AC voltage from household outlets is 60 Hertz. If the maximum voltage is 120 volts, the voltage at a specific time can be calculated using the sinusoidal voltage equation. To find the angular frequency:

$$ \omega = 2\pi \times 60 \, \text{Hz} = 377 \, \text{s}^{-1} $$

To determine the voltage at 0.4 seconds, substitute into the voltage equation:

$$ V(0.4) = 120 \cos(377 \times 0.4) $$

This results in a voltage of approximately -97 volts, indicating a negative polarity at that moment.

Understanding these principles of AC circuits is crucial, as they form the foundation for analyzing more complex electrical systems and their behaviors.

An AC source produces an alternating current in a circuit with the function $i\left(t\right)=\left(1.5A\right)\cos\left\lbrack\left(250s^{-1}\right)t\right\rbrack$. What is the frequency of the source? What is the maximum current in the circuit?

• A. f=39.3 Hz
• B. f=250 Hz
• C. f=393 Hz
• D. f=1570 Hz

Answer: (A)

AC Circuit Graphs

In alternating current (AC) circuits, the behavior of voltage and current can be described using cosine functions. The voltage \( V(t) \) as a function of time is expressed as:

\[ V(t) = V_{\text{max}} \cdot \cos(\omega t) \]

where \( V_{\text{max}} \) is the maximum voltage produced by the source, and \( \omega \) is the angular frequency of the source. Similarly, the current \( I(t) \) can be represented as:

\[ I(t) = I_{\text{max}} \cdot \cos(\omega t) \]

Here, \( I_{\text{max}} \) is the maximum current produced by the source. To determine these functions, it is essential to identify the maximum values of voltage and current, as well as the angular frequency.

In this example, the maximum voltage \( V_{\text{max}} \) is 11 volts, and the maximum current \( I_{\text{max}} \) is 2.5 amps. It is important to note that the current can be negative, indicating it is at the negative amplitude during its oscillation.

Next, we need to calculate the angular frequency. The time taken for half a cycle is given as 0.05 seconds. Since this represents half of a period, the full period \( T \) can be calculated as:

\[ T = 2 \times 0.05 \, \text{s} = 0.1 \, \text{s} \]

The angular frequency \( \omega \) is then derived from the period using the formula:

\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{0.1 \, \text{s}} \approx 62.8 \, \text{s}^{-1} \]

With all values determined, the functions for voltage and current can be written as:

\[ V(t) = 11 \cdot \cos(62.8 t) \]

\[ I(t) = 2.5 \cdot \cos(62.8 t) \]

These equations effectively describe the oscillatory nature of voltage and current in the AC circuit, highlighting their maximum values and the angular frequency of the oscillation.

The current in an AC circuit takes 0.02 s to change direction. What is the angular frequency of the AC source?

• A. 6.4×10⁻³ s⁻¹
• B. 0.04 s⁻¹
• C. 0.25 s⁻¹
• D. 157 s⁻¹

Answer: (D)

Here’s what students ask on this topic:

What is the difference between AC and DC circuits?

AC (Alternating Current) circuits involve currents that change direction periodically, typically following a sinusoidal pattern. This is represented by the equation $v=V_{\max }\cos \left(\omega t\right)$. DC (Direct Current) circuits, on the other hand, have currents that flow in a single direction, usually produced by a constant voltage source like a battery. In DC circuits, the current remains constant over time. Understanding these differences is crucial for analyzing and designing electrical systems effectively.

How is the angular frequency related to the linear frequency in AC circuits?

The angular frequency $\omega$ in AC circuits is related to the linear frequency $f$ by the equation $\omega =2\pi f$. The linear frequency $f$ is measured in Hertz (Hz), which indicates the number of cycles per second. The angular frequency $\omega$ is measured in radians per second (s^-1), and it describes how quickly the current oscillates in terms of radians.

What is the significance of the maximum voltage in an AC circuit?

The maximum voltage, often denoted as $V_{\max }$, in an AC circuit represents the peak value of the voltage waveform. It is the highest voltage that the AC source can deliver at any point in time. This value is crucial for determining the amplitude of the sinusoidal voltage and current waveforms. For example, in a household outlet in North America, the maximum voltage is typically 120 volts, which means the voltage oscillates between +120V and -120V.

How do you calculate the voltage at a specific time in an AC circuit?

To calculate the voltage at a specific time $t$ in an AC circuit, you can use the equation $v=V_{\max }\cos \left(\omega t\right)$. Here, $V_{\max }$ is the maximum voltage, and $\omega$ is the angular frequency. For example, if the maximum voltage is 120V and the angular frequency is 377 s^-1, the voltage at $t=0.04$ seconds would be $v=120\cos (3770.04)=-97$ volts.

What is the role of angular frequency in AC circuits?

The angular frequency $\omega$ in AC circuits determines how quickly the current and voltage oscillate. It is given by the equation $\omega =2\pi f$, where $f$ is the linear frequency. The angular frequency is measured in radians per second (s^-1) and is crucial for defining the time-dependent behavior of the sinusoidal voltage and current waveforms in the circuit. It ensures that the current and voltage follow the same sinusoidal pattern, maintaining the AC circuit's functionality.