Question

Express each current or voltage as a complex number in polar form. $$v=57 \sin \left(\omega t-90^{\circ}\right)$$

Short answer

The voltage in complex polar form is \(57\angle-90^\circ\) or \(57\angle-\frac{\pi}{2}\) radians.

Step-by-step solution

Identify the Amplitude

The amplitude of the sinusoidal function, represented by the coefficient in front of the sine function, is the maximum value the function reaches. In this case, the amplitude is 57.

Determine the Phase Shift

The phase shift of the function is given by the angle inside the sine function. It is represented as a shift from the standard position. For the function \(v = 57 \sin(\omega t - 90^\circ)\), the phase shift is \(-90^\circ\), which corresponds to \(-\frac{\pi}{2}\) radians.

Convert to Cosine Function

By the sine to cosine conversion, we know that \(\sin(x - 90^\circ) = \cos(x)\). So we can rewrite the function as \(v = 57 \cos(\omega t)\) since the cosine function naturally leads the sine function by 90 degrees.

Express as a Complex Exponential

Using Euler's formula, which states that \(\cos(\theta) + i \sin(\theta) = e^{i \theta}\), we replace the cosine function with the real part of the complex exponential to express the voltage as a complex number: \(v(t) = 57e^{i(\omega t)}\).

Write in Polar Form

Given that \(e^{i(\omega t)}\) is already in polar form, with the magnitude of 57 and angle \(\omega t\), the voltage in the complex polar form is simply \(57\angle\omega t\) radians or \(57\angle\omega t \times \frac{180}{\pi}\) degrees.

Key concepts

Amplitude of Sinusoidal Function

The amplitude of a sinusoidal function is a key concept in understanding oscillations in various physical contexts including electrical engineering. In simple terms, amplitude refers to the maximum displacement from the function's average or equilibrium value.

For a function described by \(v = A \sin(\omega t + \phi)\), the coefficient \(A\) represents the amplitude. This numerical value denotes the peak value that the sinusoidal wave reaches, either above or below the centerline during its cycle.

For the given exercise, the voltage waveform \(v=57 \sin(\omega t-90^\circ)\) specifies an amplitude of 57. In practical terms, this means that the maximum voltage deviation from the equilibrium (zero voltage in this case) is 57 units. This value is critical for understanding the strength of the signal, as it directly affects the energy carried by the wave.

Phase Shift

Phase shift in sinusoidal functions represents the horizontal displacement of the waveform from its standard or original position. It's an important concept to grasp when analyzing waves as it indicates how much a wave is shifted in time.

In the expression \(v = A \sin(\omega t + \phi)\), the term \(\phi\) is the phase shift. If \(\phi\) is positive, the function is shifted to the left, meaning the wave starts earlier; if negative, the function is shifted to the right, indicating a delay in the start of the wave.

In the exercise provided, we have a phase shift of \( -90^\circ \) or equivalently represented in radians as \( -\frac{\pi}{2} \) since there are \( \frac{\pi}{180} \) radians in a degree. The negative sign denotes that the sinusoidal function is delayed by a quarter of its cycle.

Euler's Formula

Euler's formula is a fundamental bridge between complex numbers and trigonometry, and is expressed as \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\). This formula demonstrates the relationship between the exponential function of a complex number and the trigonometric functions sine and cosine.

In electrical engineering, Euler's formula enables representing sinusoidal functions, like current and voltage, using complex exponential functions. This simplifies calculations, especially when dealing with alternating currents (AC) and voltages.

The provided exercise employs Euler's formula during Step 4 by expressing the voltage as a complex exponential \(v(t) = 57e^{i(\omega t)}\). This step simplifies the representation of the sinusoidal voltage function and prepares it for conversion to polar form, which is essentially a form that combines magnitude and phase into a single complex number, easy for manipulation in various engineering applications.