Sinusoidal Variables

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

$$x(t)=A\;\cos(\omega t + \phi),\;\;\;\mbox{or}\;\;\; x(t)=A\;\sin(\omega t+\phi+\pi/2)$$

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.
• $\omega$: angular frequency in radians per second.

Frequency can also be measured by cycles per second. i.e., $f=1/T$ where $T$ is cycle time or period (in seconds). Since one cycle is $2\pi$ radians, we have

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

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 varying current $i(t)$ is the steady value of current $I_{av}$ that in period $T$ would transfer the same charge $Q$:

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

Similarly, the average voltage is defined as:

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

In general, when $i(t)$ and $v(t)$ are periodic, the time period $T$ is one complete cycle. For a sinusoidal variable $i(t)=I_m \cos(\omega t)$, the average over the complete cycle is always zero (the charge transferred during the first half is the opposite to that transferred in the second). We can consider the half-cycle average:

$$I_{av} = \frac{1}{T/2}\int_{-T/4}^{T/4} i(t) dt =\frac{2}{T}\int_{-T/4}^{T... ...cos(2\pi t/T)dt =\frac{2I_m}{T}\frac{T}{2\pi} \sin(2\pi t/T)\vert _{-T/4}^{T/4}$$

$$= \frac{I_m}{\pi}[\sin(\pi/2)-\sin(-\pi/2)] =2I_m/\pi=0.637\;I_m$$

Effective Value

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

$$W=R I^2_{eff}T=R\int_0^T i^2(t) dt,\;\;\;\;\mbox{i.e.}\;\;\;\; I_{eff}=\sqrt{\frac{1}{T}\int_0^T i^2(t) dt}$$

As $I_{eff}$ is the ``square root of the mean squared value'', it is also called the root-mean-square (rms) current $I_{rms}$. Similarly, the effective voltage is defined as:

$$V_{eff}=\sqrt{\frac{1}{T}\int_0^T v^2(t) dt}$$

For a sinusoidal variable $i(t)=I_m \cos(\omega t)$, we have

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

i.e.,

$$I_{eff}=\frac{I_m}{\sqrt{2}}=0.707 I_m$$

(trigonometric identity: $\cos^2\alpha=[1+\cos(2\alpha)]/2$)