Circuit Theory/Sinusoidal Sources

Sinusoidal Sources

With the advent of AC power, analysis of circuits has become a much more complicated task that requires a whole set of new analysis tools. Many of the mathematical concepts and tools used to work with sinusoids are very different from the kinds of tools that people are used to working with.

Sinusoidal Forcing Functions

Let us consider a general AC forcing function:

$v(t)=M\sin (\omega t+\varphi )$

In this equation, the term M is called the "Magnitude", and it acts like a scaling factor that allows the peaks of the sinusoid to be higher or lower then +/- 1. The term ω is what is known as the "Radial Frequency". The term φ is an offset parameter known as the "Phase".

Sinusoidal sources can be current sources, but most often they are voltage sources.

Other Terms

There are a few other terms that are going to be used in many of the following sections, so we will introduce them here:

Period

The period of a sinusoidal function is the amount of time, in seconds, that the sinusoid takes to make a complete wave. The period of a sinusoid is always denoted with a capital T. This is not to be confused with a lower-case t, which is used as the independant variable for time. A sinusoid moving through an entire

Frequency

Frequency is the reciprocal of the period, and is the number of times, per second, that the sinusoid completes an entire cycle. Frequency is measured in Hertz (Hz). The relationship between frequency and the Period is as follows:

$f=\frac{1}{T}$

Where f is the variable most commonly used to express the frequency.

Radian Frequency

Radian frequency is the value of the frequency expressed in terms of Radians Per Second, instead of Hertz. Radian Frequency is denoted with the variable $\omega$. The relationship between the Frequency, and the Radian Frequency is as follows:

$\omega =2\pi f$

Phase

The phase is a quantity, expressed in radians, of the time shift of a sinusoid. A sinusoid phase-shifted $\varphi =+2\pi$ is moved forward by 1 whole period, and looks exactly the same. An important fact to remember is this:

$\sin (t+\frac{\pi }{2})=\cos (t)$

Phase is often expressed with many different variables, including $\varphi ,\psi ,\theta ,\gamma$ etc... This wikibook will try to stick with the symbol $\varphi$, to prevent confusion.

Lead and Lag

A circuit element may have both a voltage across it's terminals, and a current flowing through it. If one of the two (current or voltage) is a sinusoid, then the other must be a sinusoid. (remember, voltage is the derivative of the current, and the derivative of a sinusoid is always a sinusoid). However, the sinusoids of the voltage and the current may differ by quantities of magnitude and phase.

If the current has a lower phase angle then the voltage, the current is said to lag the voltage. If the current has a higher phase angle then the voltage, it is said to lead the voltage. Many circuits can be classified and examined using lag and lead ideas.

Fundamental Principal

There is a fundamental principal in engineering that states: you get out what you put in. While this might apply metaphorically to scholastic effort, it definately applies concretely to electric circuits. For instance, if we have a polynomial forcing function, our circuit response will be a polynomial output. If our input is exponential, our output will similarly be exponential. And (most important for our present uses) Sinusoidal input will produce sinusoidal output. This principal only holds for passive circuit elements, because active circuit elements (particularly a rectifier) can convert a sinusoid into a constant value, but we don't worry about that for this book.

Sinusoidal Response

In general, we say that given an input sinusoid, we will produce an output sinusoid:

$A_{in}\cos ({\omega }_{in}t+{\varphi }_{in})\to A_{out}\cos ({\omega }_{out}t+{\varphi }_{out})$

Of course, as we can see from all the terms in the above equations, there is plenty of room for the input signal to be altered by the circuit. The fundamental tenet holds throughout this book that you get out what you put in. For that reason, if the input is a polynomial, the output will be a polynomial. If the input is a sinusoid, the output will be a sinusoid. If the input is an exponential, the output will be an exponential, et cetera.

For ease of use, we simply say that if the input sinusoid is a sine function, then the output will also be a sine function. If the input sinusoid is a cosine function, the output will also be a sinusoid. Sometimes it might be easier to write one or the other, but we like to stick to this convention.

Sinusoidal Conventions

For the purposes of this book we will generally use cosine functions, as opposed to sine functions. If we absolutely need to use a sine, we can remember the following trigonometric identity:

$\cos (\omega t)=\sin (\omega t+\pi /2)$

We can express all sine functions as cosine functions. This way, we dont have to compare apples to oranges per se. This is simply a convention that this wikibook chooses to use to keep things simple. We could easily choose to use all sin( ) functions, but further down the road it is often more convenient to use cosine functions instead by default.

RMS

Sinusoidal functions for current and voltage can be best considered as single values, through the root mean square (RMS) calculation. RMS is essentially an averaging operation of all the magnitude values of the sinusoid. The RMS formula for a continuous function $f(t)$ defined over the interval $T_1\le t\le T_2$ (for a periodic function the interval should be a whole number of complete cycles) is:

$f_{\mathrm{r}\mathrm{m}\mathrm{s}}=\sqrt{\frac{1}{T_2-T_1}\underset{T_1}{\overset{T_2}{\int }}{[f(t)]}^{2}dt}$

Since we are only considering sinusoids, we can reduce this calculation to:

$f_{\mathrm{r}\mathrm{m}\mathrm{s}}=\frac{f_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\sqrt{2}}$

Where $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the peak value of the sinusoid. RMS values for current and voltage, $I_{\mathrm{r}\mathrm{m}\mathrm{s}}$ and $V_{\mathrm{r}\mathrm{m}\mathrm{s}}$ can both be used in the standard power equation like normal:

$P_{\mathrm{a}\mathrm{v}\mathrm{g}}=I_{\mathrm{r}\mathrm{m}\mathrm{s}}{V}_{\mathrm{r}\mathrm{m}\mathrm{s}}$