Sinusoidal Functions

Sinusoidal variables are of special importance in electrical and electronic systems, not only because they occur frequently in such systems, but also because any periodical signal can be represented as a linear combination of a set of sinusoidal signals of different frequencies, amplitudes, and phase angles (Fourier transform theory).

A sinusoidal variable (voltage or current) can be written as

$$x(t)=A\;\cos(\omega t + \phi),\;\;\; \;\;\; x(t)=A\;\sin(\omega t+\phi+\pi/2)$$ (3)

The three parameters $A$, $\omega$ and $\phi$ represent three important elements:

• $A$: amplitude or peak value
• $\phi$: phase angle $0 \le \phi \le 2\pi$ in radians or $0^\circ \le \phi \le 360^\circ$ in degrees.
• $\omega=2\pi f=2\pi/T$: angular frequency in radians per second, where $f=1/T$ is the frequency measured by cycles per second (in Hertz, Hz), and $T$ is cycle time or period (in second).

  $$\omega=2\pi f=2\pi/T,\;\;\;\;f=1/T=\omega/2\pi,\;\;\;T=1/f=2\pi/\omega$$ (4)

In the following discussion, all sinusoidal variables are assumed to have the same frequency $\omega=2\pi f$, for the reason that any arbitrary waveform can be represented as a linear combination of sinusoids of different amplitudes, phase angle and frequencies based on Fourier theory, Once we know how the circuit responds to a single frequency, we can find its response to any waveform by superposition based on the assumption that the circuit is a linear system.

Sinusoidal functions are closely related to complex exponentials due to Euler's formula:

$$\left\{ \begin{array}{l} e^{ j\omega t}=\cos\omega t+j\,\sin\omeg... ...omega t=(e^{j\omega t}-e^{-j\omega t})/2j=Im(e^{j\omega t}) \end{array} \right.$$ (5)

As exponential functions can be much more conveniently manipulated than sinusoidal functions, a sinusoidal function is often considered as the real or imaginary part of the corresponding complex variable:

$$A\;\cos(\omega t+\phi)=Re[A\;e^{j\omega t+\phi}],\;\;\;\; A\;\sin(\omega t+\phi)=Im[A\;e^{j\omega t+\phi}]$$ (6)

Example: (Homework)

A sinusoidal current with a frequency of 60 Hz reaches a positive maximum of 20A at $t=2 \; ms$. Give the expression of this current as a function of time $i(t)$.

Answer

Average Value

The average of a time varying current $i(t)$ is the steady or DC (direct current) value of current $I_{av}$ that in period $T$ would transfer the same charge $Q$:

$$I_{av}T=Q=\int_0^T i(t) dt,\;\;\;\; \;\;\;\; I_{av}=\frac{1}{T}\int_0^T i(t) dt$$ (7)

Similarly, the average voltage is defined as:

$$V_{av}=\frac{1}{T}\int_0^T v(t) dt$$ (8)

If the current/voltage is periodic, i.e., it repeats itself every time cycle $T$, then $i(t)=i(t+T)$ and $v(t)=v(t+T)$. The reciprocal of $T$ is called the fundamental frequency. In particular, for a sinusoidal current

$$i(t)=I_p\,\sin(\omega t)=I_p\,\sin(2\pi ft)=I_P\,\sin(2\pi t/T)$$ (9)

the average over the complete cycle $T=1/f$ is always zero (the charge transferred during the first half is the opposite to that transferred in the second). However, we can consider the half-cycle average over $T/2$:

$$I_{av} = \frac{1}{T/2}\int_0^{T/2} i(t)\; dt =\frac{2}{T}\int_0^{T/2} \;I_... ...(2\pi t/T)\;dt =-\frac{2}{T}\frac{TI_P}{2\pi} \cos(2\pi t/T)\bigg\vert _0^{T/2}$$

$$= \frac{1}{\pi}\left[\cos(0)-\cos(\pi)\right]I_P =\frac{2}{\pi}I_P\;=0.637\,I_P$$ (10)

Effective or RMS Value

The effective value of a time-varying current $i(t)$ or voltage is the constant value of current $I_{rms}$ or voltage $V_{rms}$ that in period $T$ would transfer the same amount of energy:

$$W=\int_0^T p(t) dt = R\int_0^T i^2(t) dt =R I^2_{rms}T$$

$$= \frac{1}{R}\int_0^T v^2(t) dt=\frac{1}{R}V^2_{rms}T$$ (11)

i.e.,

$$I_{rms}=\sqrt{\frac{1}{T}\int_0^T i^2(t) dt},\;\;\;\;\;\;\; \;\;\;\;\;\; V_{rms}=\sqrt{\frac{1}{T}\int_0^T v^2(t) dt}$$ (12)

As $I_{rms}$ or $V_{rms}$ is the “square root of the mean of the squared value”, it is also called the root-mean-square (rms) current or voltage.

For a sinusoidal variable $i(t)=I_p\,\cos(\omega t)$, we have ( $\cos^2\alpha=[1+\cos(2\alpha)]/2$)

$$I^2_{rms} = \frac{1}{T}\int_0^T i^2(t) dt = \frac{I^2_p}{T}\int_0... ...2\pi t/T) dt = \frac{I^2_p}{2T}\int_0^T [1+\cos(4\pi t/T)]\; dt=\frac{I^2_p}{2}$$ (13)

i.e.,

$$I_{rms}=\frac{I_p}{\sqrt{2}}=I_p\;0.707$$ (14)

Similarly, for $v(t)=V_p\,\cos(\omega t)$, we also have

$$V_{rms}=\frac{1}{\sqrt{2}}=V_p\;0.707$$ (15)