Integral Of A Square Wave

Aug 10, 2018  You can use the square function to create a square wave with the time period of 2.pi and amplitude between-1 and 1. And then use trapz to evaluate the integral numerically. Fourier Series. Sine and cosine waves can make other functions! Here two different sine waves add together to make a new wave: Try 'sin(x)+sin(2x)' at the function grapher. (You can also hear it at Sound Beats.).

Triangular wave generator using opamp

This is an interesting project for newbies. This article is about creating a triangular wave generator using opamp IC. There are many methods for generating triangular waves but here we focus on the method using opamps. This circuit is based on the fact that a square wave on integration gives a triangular wave.

What are Triangular waves?

Triangular waves are a periodic, non-sinusoidal waveform with a triangular shape. People often get confused between the triangle and sawtooth waves. The most important feature of a triangular wave is that it has equal rise and fall times while a sawtooth wave has un-equal rise and fall times.

To generate triangular waves we need an input wave. In this project, we are using square waves for input. Like triangular waves, square waves have equal rise and fall times so they are more convenient to be converted to a triangular waveform.

The main parts of this project are 1. A square wave generator 2. An integrator which converts square waves to triangular waves.

The circuit uses an opamp based square wave generator for producing the square wave and an opamp based integrator for integrating the square wave. The circuit diagram is shown in the figure below.

Circuit diagram

The square wave generator section and the integrator section of the circuit are explained in detail below.

Square wave generator

Rocksmith 2014 no sound. The square wave generator is based on a uA741 opamp (IC1). Resistor R1 and capacitor C1 determines the frequency of the square wave. Resistor R2 and R3 forms a voltage divider setup which feedbacks a fixed fraction of the output to the non-inverting input of the IC.

Initially, when power is not applied the voltage across the capacitor C1 is 0. When the power supply is switched ON, the C1 starts charging through the resistor R1 and the output of the opamp will be high (+Vcc). A fraction of this high voltage is fed back to the non- inverting pin by the resistor network R2, R3. When the voltage across the charging capacitor is increased to a point the voltage at the inverting pin is higher than the non-inverting pin, the output of the opamp swings to negative saturation (-Vcc). The capacitor quickly discharges through R1 and starts charging in the negative direction again through R1. Now a fraction of the negative high output (-Vcc) is fed back to the non-inverting pin by the feedback network R2, R3. When the voltage across the capacitor has become so negative that the voltage at the inverting pin is less than the voltage at the non-inverting pin, the output of the opamp swings back to the positive saturation. Now the capacitor discharges trough R1 and starts charging in positive direction. This cycle is repeated over time and the result is a square wave swinging between +Vcc and -Vcc at the output of the opamp.

If the values of R2 and R3 are made equal, then the frequency of the square wave can be expressed using the following equation:

F=1 / (2.1976 R1C1)

Integrator

Next part of the triangular wave generator is the opamp integrator. Instead of using a simple passive RC integrator, an active integrator based on opamp is used here. The opamp IC used in this stage is also uA741 (IC2). Resistor R5 in conjunction with R4 sets the gain of the integrator and resistor R5 in conjunction with C2 sets the bandwidth. The square wave signal is applied to the inverting input of the opamp through the input resistor R4. The opamp integrator part of the circuit is shown in the figure below.

Let’s assume the positive side of the square wave is first applied to the integrator. By virtue capacitor C2 offers very low resistance to this sudden shoot in the input and C2 behaves something like a short circuit. The feedback resistor R5 connected in parallel to C2 can be put aside because R5 has almost zero resistance at the moment. A serious amount of current flows through the input resistor R4 and the capacitor C2 bypasses all these current. As a result the inverting input terminal (tagged A) of the opamp behaves like a virtual ground because all the current flowing into it is drained by the capacitor C2. The gain of the entire circuit (Xc2/R4) will be very low and the entire voltage gain of the circuit will be close to the zero.

After this initial “kick” the capacitor starts charging and it creates an opposition to the input current flowing through the input resistor R4. The negative feedback compels the opamp to produce a voltage at its out so that it maintains the virtual ground at the inverting input. Since the capacitor is charging its impedance Xc keeps increasing and the gain Xc2/R4 also keeps increasing. This results in a ramp at the output of the opamp that increases in a rate proportional to the RC time constant (T=R4C2) and this ramp increases in amplitude until the capacitor is fully charged.

When the input to the integrator (square wave) falls to the negative peak the capacitor quickly discharges through the input resistor R4 and starts charging in the opposite polarity. Now the conditions are reversed and the output of the opamp will be a ramp that is going to the negative side at a rate proportional to the R4R2 time constant. This cycle is repeated and the result will be a triangular waveform at the output of the opamp integrator.

Testing the triangular wave generator.

Testing videos, practical circuit diagram, and photographs will be added soon.

Applications:

The applications of triangular wave include sampling circuits, thyristor firing circuits, frequency generator circuits, tone generator circuits etc.

In mathematics, a square-integrable function, also called a quadratically integrable function or ${\displaystyle L^2}$ function^[1], is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line ${\displaystyle (-\mathrm{\infty },+\mathrm{\infty })}$ is defined as follows.

One may also speak of quadratic integrability over bounded intervals such as ${\displaystyle [a,b]}$ for ${\displaystyle a\le b}$.^[2]

An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part.

The vector space of square integrable functions (with respect to Lebesgue measure) form the L^p space with ${\displaystyle p=2}$. Among the L^p spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the L^p spaces are complete under their respective p-norms.

Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.

Properties[edit]

The square integrable functions (in the sense mentioned in which a 'function' actually means an equivalence class of functions that are equal almost everywhere) form an inner product space with inner product given by

${\displaystyle ⟨f,g⟩={\int }_A\overline{f(x)}g(x)\mathrm{d}x}$

where

• ${\displaystyle f}$ and ${\displaystyle g}$ are square integrable functions,
• ${\displaystyle \overline{f(x)}}$ is the complex conjugate of ${\displaystyle f(x)}$,
• ${\displaystyle A}$ is the set over which one integrates—in the first definition (given in the introduction above), ${\displaystyle A}$ is ${\displaystyle (-\mathrm{\infty },+\mathrm{\infty })}$; in the second, ${\displaystyle A}$ is ${\displaystyle [a,b]}$.

Since ${\displaystyle a{}^2=a\cdot \overline{a}}$, square integrability is the same as saying

It can be shown that square integrable functions form a complete metric space under the metric induced by the inner product defined above.A complete metric space is also called a Cauchy space, because sequences in such metric spaces converge if and only if they are Cauchy.A space which is complete under the metric induced by a norm is a Banach space.Therefore, the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product.As we have the additional property of the inner product, this is specifically a Hilbert space, because the space is complete under the metric induced by the inner product.

This inner product space is conventionally denoted by ${\displaystyle \left(L_2,⟨\cdot ,\cdot {⟩}_2\right)}$ and many times abbreviated as ${\displaystyle L_2}$.Note that ${\displaystyle L_2}$ denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation.The set, together with the specific inner product ${\displaystyle ⟨\cdot ,\cdot {⟩}_2}$ specify the inner product space.

The space of square integrable functions is the L^p space in which ${\displaystyle p=2}$.

Examples[edit]

• ${\displaystyle \frac{1}{x^n}}$ , defined on (0,1), is in L² for but not for ${\displaystyle n=\frac{1}{2}}$.^[1]

• Bounded functions, defined on [0,1]. These functions are also in L^p, for any value of p.^[3]
• ${\displaystyle \frac{1}{x}}$, defined on ${\displaystyle [1,\mathrm{\infty })}$.^[3]

Counterexamples[edit]

• ${\displaystyle \frac{1}{x}}$, defined on [0,1], where the value of f(0) is arbitrary. Furthermore, this function is not in L^p for any value of p in ${\displaystyle [1,\mathrm{\infty })}$.^[3]

See also[edit]

References[edit]

1. ^ ^abTodd, Rowland. 'L^2-Function'. MathWorld--A Wolfram Web Resource.
2. ^G.Sansone (1991). Orthogonal Functions. Dover Publications. pp. 1–2. ISBN978-0-486-66730-0.
3. ^ ^abc'Lp Functions'(PDF).