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# Electronics/RC

Electronics | Foreword | Basic Electronics | Complex Electronics | Electricity | Machines | History of Electronics | Appendix |

## Introduction

An RC circuit is short for 'Resistor-Capacitor' circuit. A capacitor takes an infinite amount of time to discharge through a resistor, which varies with the values of the resistor and capacitor. A capacitor acts interestingly in an electronic circuit, practically speaking as a combination of a voltage source and a variable resistor.

## Basics

Below is a simple RC circuit:
There is a capacitor in parallel with the resistor and current probe. The way the capacitor functions is by acting as a very low resistance load when the circuit is initially turned on. This is illustrated below:
Initially, the capacitor has a very low resistance, almost 0. Since electricity takes the path of least resistance, almost all the electricity flows through the capacitor, not the resistor, as it has considerably higher resistance. As a capacitor charges, its resistance increases as it gains more and more charge. As the resistance of the capacitor climbs, electricity begins to flow not only to the capacitor, but through the resistor as well:
Once the capacitor's voltage equals that of the battery, meaning it is fully charged, it will not allow any current to pass through it. All the electricity eventually flows through the resistor.
A capacitor's resistance becomes higher and higher as it becomes more charged. Once it is fully charged, for all practical reasons, it has infinite resistance (an open connection). Once the voltage source is disconnected, however, the capacitor acts as a voltage source itself:
As time goes on, the capacitor's charge begins to drop, and so does its voltage. This means less current through the resistor:
Once the capacitor is fully discharged, you are back to square one:
If one were to do this with a light and a capacitor connected to a battery, what you would see is the following:
* Switch is hit. Light does not light up.
* Light gradually becomes brighter and brighter...
* Light is at full luminosity.
* Switch is released. Light continues to shine.
* Light begins to fade...
* Light is off.
This is how a capacitor acts. However, what if you changed the values of R1? C1? The voltage of the battery? We will examine the mathematical relationship between the resistor, capacitor, and charging rate below.

## The Time Constant

In order to find out how long it takes for a capacitor to fully charge or discharge, or how long it takes for the capacitor to reach a certain voltage, you must know a few things. First, you must know the starting and finishing voltages. Secondly, you must know the time constant of the circuit you have. Time constant is denoted by the Greek letter 'tau' or τ. The general formula for percentage change is:
# Electronics/RL transient

For a series RL of one resistor connected with one inductor in a closed loop

## Circuit Impedance

In Polar Form Z/_θ
    $Z=Z_{R}+Z_{L}$ = R/_0 + ω L/_90
    Z = |Z|/_θ = ${\sqrt {R^{2}+(\omega L)^{2}}}$/_Tan$^{-1}$$\omega {\frac {L}{R}}$
In Complex Form Z(jω)
    $Z=Z_{R}+Z_{L}=R+j\omega L$
    $Z=R+j\omega L=R(1+j\omega T)$
    $T={\frac {L}{R}}$

## Differential Equation of circuit at equilibrium

    $L{\frac {dI}{dt}}+IR=0$
    ${\frac {dI}{dt}}=-I{\frac {R}{L}}$
    $\int {\frac {1}{I}}dI=-\int {\frac {R}{L}}dt$
    $lnI=(-{\frac {R}{L}}t+c)$
    $I=e^{-{\frac {R}{L}}t+c}=e^{c}\times e^{-{\frac {R}{L}}t}$
    $I=Ae^{-{\frac {t}{T}}}$

## Time Constant

    $T={\frac {L}{R}}$
    tI(t) % I$_o$ 0A = e$^C$ = I$_o$100% R/L.63 I$_o$60% I$_o$ 2 R/LI$_o$ 3 R/LI$_o$ 4 R/LI$_o$ 5 R/L.01 I$_o$10% I$_o$

## Angle Difference between Voltage and Current

Voltage leads Current at an angle ? When a determining process is necessary many problems arise in a diagram. We need to expend on one process for the determing factor in this type of formulae
    Tan? = ${\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}=t{\frac {1}{2\pi RC}}$
Change the value of R and L will change the value Angle Difference, Angular Frquency, Frequency, Time
    $\omega ={\frac {1}{Tan\theta RC}}$
    $f={\frac {1}{2\pi Tan\theta RC}}$
    $t=2\pi Tan\theta RC$
# Electronics/RCL frequency domain

Figure 1: RCL circuit
Define the pole frequency $\omega _{n}$ and the dampening factor $\alpha$ as:
${\frac {R}{L}}=2\alpha$
${\frac {1}{LC}}=\omega _{n}^{2}$
To analyze the circuit first calculate the transfer function in the s-domain H(s). For the RCL circuit in figure 1 this gives:
$H(s)={\frac {s{\bigs+2\alpha {\big )}}{s^{2}+2\alpha s+\omega _{n}^{2}}}$
$H(s)={\frac {s{\bigs+2\alpha {\big )}}{{\bigs+\alpha +j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}{\bigs+\alpha -j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}}}$
When the switch is closed, this applies a step waveform to the RCL circuit. The step is given by $Vu(t)$. Where V is the voltage of the step and u(t) the unit step function. The response of the circuit is given by the convolution of the impulse response h(t) and the step function $Vu(t)$. Therefore the output is given by multiplication in the s-domain H(s)U(s), where $U(s)=V{\frac {1}{s}}$ is given by the Laplace Transform available in the appendix.
The convolution of u(t) and h(t) is given by:
$H(s)U(s)={\frac {V{\bigs+2\alpha {\big )}}{{\bigs+\alpha +j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}{\bigs+\alpha -j{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}{\big )}}}$
Depending on the values of $\alpha$ and $\omega _{n}$ the system can be characterized as:
3. If $\alpha <\omega _{n}$ the system is said to be underdamped The solution for h(t)*u(t) is given by:
$h(t)*u(t)=Ve^{-\alpha t}{\big\cos({\sqrt {\omega _{n}^{2}-\alpha ^{2}}}t)+{\frac {\alpha }{\sqrt {\omega _{n}^{2}-\alpha ^{2}}}}\sin({\sqrt {\omega _{n}^{2}-\alpha ^{2}}}t){\big )}$

## Example

Given the following values what is the response of the system when the switch is closed?
| R | L | C | V |
| 0.5H | 1kΩ | 100nF | 1V |
First calculate the values of $\alpha$ and $\omega _{n}$:
$\alpha ={\frac {R}{2L}}=1000$
$\omega _{n}={\frac {1}{\sqrt {LC}}}\approx 4472$
From these values note that $\alpha <\omega _{n}$. The system is therefore underdamped. The equation for the voltage across the capacitor is then:
$h(t)*u(t)=e^{-1000t}{\big\cos(4359t)+0.229\sin(4359t){\big )}$
Figure 2: Underdamped Resonse
# Electronics/RCL time domain simple

Figure 1: RCL circuit
When the switch is closed, a voltage step is applied to the RCL circuit. Take the time the switch was closed to be 0s such that the voltage before the switch was closed was 0 volts and the voltage after the switch was closed is a voltage V. The voltage across the capacitor consists of a forced response $v_{f}$ and a natural response $v_{n}$ such that:
$v_{c}=v_{f}+v_{n}$
The forced response is due to the switch being closed, which is the voltage V for $t\geq 0$. The natural response depends on the circuit values and is given below:
Define the pole frequency $\omega _{n}$ and the dampening factor $\alpha$ as:
$\alpha ={\frac {R}{2L}}$
$\omega _{n}={\frac {1}{\sqrt {LC}}}$
Depending on the values of $\alpha$ and $\omega _{n}$ the system can be characterized as:
1. If $\alpha >\omega _{n}$ the system is said to be overdamped. The solution for the system has the form:
$v_{n}(t)=A_{1}e^{{\big-\alpha +{\sqrt {\alpha ^{2}-\omega _{n}^{2}}}{\big )}t}+A_{2}e^{{\big-\alpha -{\sqrt {\alpha ^{2}-\omega _{n}^{2}}}{\big )}t}$
2. If $\alpha =\omega _{n}$ the system is said to be critically damped The solution for the system has the form:
$v_{n}(t)=Be^{-\alpha t}$
3. If $\alpha <\omega _{n}$ the system is said to be underdamped The solution for the system has the form:
$v_{n}(t)=e^{-\alpha t}{\bigB_{1}\cos({\sqrt {\omega _{n}^{2}-\alpha ^{2}}}t)+B_{2}\sin({\sqrt {\omega _{n}^{2}-\alpha ^{2}}}t){\big ]}$
How do you calculate these equations?

## Example

Given the following values what is the response of the system when the switch is closed?
| R | L | C | V |
| 1kΩ | 0.5H | 100nF | 1V |
First calculate the values of $\alpha$ and $\omega _{n}$:
$\alpha ={\frac {R}{2L}}=1000$
$\omega _{n}={\frac {1}{\sqrt {LC}}}\approx 4472$
From these values note that $\alpha <\omega _{n}$. The system is therefore underdamped. The equation for the voltage across the capacitor is then:
$v_{c}(t)=1+e^{-1000t}{\bigB_{1}\cos(4359t)+B_{2}\sin(4359t){\big ]}$
Before the switch was closed assume that the capacitor was fully discharged. This implies that v(t)=0 at the instant the switch was closed (t=0). Substituting t=0 into the previous equation gives:
$0=1+B_{1}$
Therefore $B_{1}=-1$. Similarly at the instant the switch is closed, the current in the inductor must be zero as the current can not instantly change. Substituting the equation for $v_{c}(t)$ into the equation for the inductor and solving at the instant the switch was closed (t=0) gives:
$i(t)={\frac {dv_{c}(t)}{dt}}C$ $0=100\cdot 10^{-9}{\big4359B_{2}-1000B_{1}{\big ]}$
Therefore $B_{2}\approx -0.229$. Once $v_{c}(t)$ is known, the voltage across the inductor and resistor ($V_{out}$) is given by:
$v_{out}=e^{-1000t}{\big\cos(4359t)+0.229\sin(4359t){\big ]}$
You have missed a lot of steps, where are they?
Figure 2: Underdamped Resonse
File:Example1 underdamped.png
# Electronics/RF Basics

Electronics | Foreword | Basic Electronics | Complex Electronics | Electricity | Machines | History of Electronics | Appendix |
Originally people thought that radio waves propagate through the ether.
The ether was disproved by showing that it did not flow.
# Economic Sophisms/45

<pagequality level="4" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"service which they rendered a remuneration regulated by the general rate of earnings. Among them there arose one who discovered the means of multiplying copies of the same work rapidly. He invented printing.
In the first instance, one man was enriched, and many others were impoverished. At first sight, marvellous as the invention proves itself to be, we hesitate to decide whether it is hurtful or useful. It seems to introduce into the world, as I have said, an indefinite element of inequality. Guttemberg profits by his invention, and extends his invention with its profits indefinitely, until he has ruined all the copyists. As regards the public, in the capacity of consumer, it gains little; for Guttemberg takes care not to lower the price of his books, but just enough to undersell his rivals.
But the intelligence which has introduced harmony into the movements of the heavenly bodies, has implanted it also in the internal mechanism of society. We shall see the economic advantages of the invention when it has ceased to be individual property, and has become for ever the common patrimony of the masses.
At length the invention comes to be known. Guttemberg is no longer the only printer; others imitate him. Their profits at first are large. They are thus rewarded for having been the first to imitate the invention; and it is right that it should be so, for this higher remuneration was necessary to induce them to concur in the grand definite result which is approaching. They gain a great deal, but they gain less than the inventor, for competition now begins its work. The price of books goes on falling. The profit of imitators goes on diminishing in proportion as the invention becomes of older date; that is to say, in proportion as the imitation becomes less meritorious.… The new branch of industry at length reaches its normal state; in other words, the remuneration of printers ceases to be exceptionally high, and comes, like that of the copyist, to be regulated by the ordinary rate of earnings. Here we have production, as
# Economic Sophisms/48

<pagequality level="4" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"may have embodied it in another product. Whence we see that pains, efforts, and exertions are the real subjects of exchange. It is not, indeed, the oxygen gas that I pay for, since it is at my disposal everywhere, but the labour necessary to disengage it, labour which has been saved me, and which must be recompensed. Will it be said that there is something else to be paid for, materials, apparatus, etc.? Still, in paying for these, I pay for labour. The price of the coal employed, for example, represents the labour necessary to extract it from the mine and to transport it to the place where it is to be used.
We do not pay for the light of the sun, because it is a gift of nature. But we pay for gas, tallow, oil, wax, because there is here human labour to be remunerated; and it will be remarked that, in this case, the remuneration is proportioned, not to the utility produced, but to the labour employed, so much so that it may happen that one of these kinds of artificial light, though more intense, costs us less, and for this reason, that the same amount of human labour affords us more of it.
Were the porter who carries water to my house to be paid in proportion to the absolute utility of water, my whole fortune would be insufficient to remunerate him. But I pay him in proportion to the exertion he makes. If he charges more, others will do the work, or, if necessary, I will do it myself. Water, in truth, is not the subject of our bargain, but the labour of carrying it. This view of the matter is so important, and the conclusions which I am about to deduce from it throw so much light on the question of the freedom of international exchanges, that I deem it necessary to elucidate it by other examples.
The alimentary substance contained in potatoes is not very costly, because we can obtain a large amount of it with comparatively little labour. We pay more for wheat, because the production of it costs a greater amount of human labour. It is evident that if nature did for the one what it does for the other, the price of both would tend to equality. It is impossible that the producer of wheat should permanently gain much more than the producer of potatoes. The law of competition would prevent it.
If by a happy miracle the fertility of all arable lands should come to be augmented, it would not be the agriculturist, but the
# Economic Sophisms/49

<pagequality level="4" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"consumer, who would reap advantage from that phenomenon for it would resolve itself into abundance and cheapness. There would be less labour incorporated in each quarter of corn, and the cultivator could exchange it only for a smaller amount of labour worked up in some other product. If, on the other hand, the fertility of the soil came all at once to be diminished, nature's part in the process of production would be less, that of human labour would be greater, and the product dearer. I am, then, warranted in saying that it is in consumption, in the human element, that all the economic phenomena come ultimately to resolve themselves. The man who has failed to regard them in this light, to follow them out to their ultimate effects, without stopping short at immediate results, and viewing them from the producer's standpoint, can no more be regarded as an economist than the man who should prescribe a draught, and, instead of watching its effect on the entire system of the patient, should inquire only how it affected the mouth and throat, could be regarded as a physician.
Tropical regions are very favourably situated for the production of sugar and of coffee. This means that nature does a great part of the work, and leaves little for human labour to do. But who reaps the advantage of this liberality of nature? Not the producing countries, for competition causes the price barely to remunerate the labour. It is the human race that reaps the benefit, for the result of nature's liberality is cheapness, and cheapness benefits everybody.
Suppose a temperate region where coal and iron-ore are found on the surface of the ground, where one has only to stoop down to get them. That, in the first instance, the inhabitants would profit by this happy circumstance, I allow. But competition would soon intervene, and the price of coal and iron-ore would go on falling, till the gift of nature became free to all, and then the human labour employed would be alone remunerated according to the general rate of earnings.
Thus the liberality of nature, like improvements in the processes of production, is, or continually tends to become, under the law of competition, the common and gratuitous patrimony of consumers, of the masses, of mankind in general. Then, the countries which do not possess these advantages have everything
# Economic Sophisms/41

<pagequality level="4" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"of life are most easily procured; and at the present day, in Prussia, in Austria, in Saxony, in Switzerland, in Italy, we see manufactures on an immense scale founded and supported by English capital, worked by English operatives, and directed by English engineers."
You see very clearly, then, that nature, or rather that Providence, more wise, more far-seeing than your narrow and rigid theory supposes, has not ordered this concentration of industry, this monopoly of all advantages upon which you found your reasoning as upon a fact which is unalterable and without remedy. Nature has provided, by means as simple as they are infallible, that there should be dispersion, diffusion, solidarity, simultaneous progress; all constituting a state of things which your restrictive laws paralyze as much as they can; for the tendency of such laws is, by isolating communities, to render the diversity of condition much more marked, to prevent equalization, hinder fusion, neutralize countervailing circumstances, and segregate nations, whether in their superiority or in their inferiority of condition.
III. In the third place, to contend that by a protective duty you equalize the conditions of production, is to give currency to an error by a deceptive form of speech. It is not true that an import duty equalizes the conditions of production. These remain, after the imposition of the duty, the same as they were before. At most, all that such a duty equalizes are the conditions of sale. It may be said, perhaps, that I am playing upon words, but I throw back the accusation. It is for my opponents to show that production and sale are synonymous terms; and if they cannot do this, I am warranted in fastening upon them the reproach, if not of playing on words, at least of mixing them up and confusing them.
To illustrate what I mean by an example: I suppose some Parisian speculators to devote themselves to the production of oranges. They know that the oranges of Portugal can be sold in Paris for a penny apiece, whilst they, on account of the frames and hot-houses which the colder climate would render necessary, could not sell them for less than a shilling as a remunerative price. They demand that Portuguese oranges
# Economic Sophisms/44

<pagequality level="4" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"Consumption is the end and final cause of all the economic phenomena, and it is in consumption consequently that we must expect to find their ultimate and definitive solution.
Nothing, whether favourable or unfavourable, can abide permanently with the producer. The advantages which nature and society bestow upon him, the inconveniences he may experience, glide past him, so to speak, and are absorbed and mixed up with the community in as far as the community represents consumers. This is an admirable law both in its cause and in its effects, and he who shall succeed in clearly describing it is entitled, in my opinion, to say, "I have not passed through life without paying my tribute to society."
Everything which favours the work of production is welcomed with joy by the producer, for the immediate effect of it is to put him in a situation to render greater service to the community, and to exact from it a greater remuneration. Every circumstance which retards or interrupts production gives pain to the producer, for the immediate effect of it is to circumscribe his services, and consequently his remuneration. Immediate good or ill circumstances—fortunate or unfortunate—necessarily fall upon the producer, and leave him no choice but to accept the one and eschew the other.
In the same way, when a workman succeeds in discovering an improved process in manufactures, the immediate profit from the improvement results to him. This was necessary, in order to give his labour an intelligent direction; and it is just, because it is fair that an effort crowned with success should carry its recompense along with it.
But I maintain that these good or bad effects, though in their own nature permanent, are not permanent as regards the producer. If it had been so, a principle of progressive, and, therefore, of indefinite, inequality would have been introduced among men, and this is the reason why these good or evil effects become very soon absorbed in the general destinies of the human race.
How is this brought about? I shall show how it takes place by some examples.
Let us go back to the thirteenth century. The men who then devoted themselves to the art of copying received for the
# Economic Sophisms/47

<pagequality level="4" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"would probably not have made out to furnish me with one, and I should have had to pay him the same wages for his day's work. The utility produced by the saw is then, as far as I am concerned, a gratuitous gift of nature, or rather it is a part of that inheritance which, in common with all my brethren, I have received from my ancestors. I have two workmen in my field. The one handles the plough, the other the spade. The result of their labour is very different, but the day's wages are the same, because the remuneration is not proportioned to the utility produced, but to the effort, the labour, which is exacted.
I entreat the reader's patience, and beg him to believe that I have not lost sight of free trade. Let him only have the goodness to remember the conclusion at which I have arrived: Remuneration is not in proportion to the utilities which the producer brings to market, but to his labour.
I have drawn my illustrations as yet from human inventions. Let us now turn our attention to natural advantages.
In every branch of production, nature and man concur. But the portion of utility which nature contributes is always gratuitous. It is only the portion of utility which human labour contributes which forms the subject of exchange and, consequently, of remuneration. The latter varies, no doubt, very much in proportion to the intensity of the labour, its skill, its promptitude, its suitableness, the need there is of it, the temporary absence of rivalry, etc. But it is not the less true, in principle, that the concurrence of natural laws, which are common to all, counts for nothing in the price of the product.
We do not pay for the air we breathe, although it is so useful to us, that, without it, we could not live two minutes. We do not pay for it, nevertheless; because nature furnishes it to us without the aid of human labour. But if, for example, we should desire to separate one of the gases of which it is composed, to make an experiment, we must make an exertion; or if we wish another to make that exertion for us, we must sacrifice for that other an equivalent amount of exertion, although we