Voltage and Current Sources

Real voltage sources can be represented as ideal voltage sources in series with a resistance r, the ideal voltage source having zero resistance. Real current sources can be represented as ideal current sources in parallel with a resistance r, the ideal current source having infinite resistance.

In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current flow through it. Real-world sources of electrical energy, such as batteries, generators, or power systems, can be modeled for analysis purposes as a combination of an ideal voltage source and additional combinations of impedance elements

Working on high voltage power lines, Pearl Harbor

Voltage, otherwise known as electrical potential difference or electric tension (denoted ∆V and measured in volts, or joules per coulomb) is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points. Voltage is equal to the work which would have to be done, per unit charge, against a static electric field to move the charge between two points. A voltage may represent either a source of energy (electromotive force), or it may represent lost or stored energy (potential drop). A voltmeter can be used to measure the voltage (or potential difference) between two points in a system; usually a common reference potential such as the ground of the system is used as one of the points. Voltage can be caused by static electric fields, by electric current through a magnetic field, by time-varying magnetic fields, or a combination of all three.

Voltage is the Cause, Current is the Effect

Voltage attempts to make a current flow, and current will flow if the circuit is complete. Voltage is sometimes described as the 'push' or 'force' of the electricity, it isn't really a force but this may help you to imagine what is happening. It is possible to have voltage without current, but current cannot flow without voltage.


Voltage and Current The switch is closed making a complete circuit so current can flow.	Voltage but No Current The switch is open so the circuit is broken and current cannot flow.	No Voltage and No Current Without the cell there is no source of voltage so current cannot flow.

Voltage, V

Connecting a voltmeter in parallel

Voltage is a measure of the energy carried by the charge.
Strictly: voltage is the "energy per unit charge".
The proper name for voltage is potential difference or p.d. for short, but this term is rarely used in electronics.
Voltage is supplied by the battery (or power supply).
Voltage is used up in components, but not in wires.
We say voltage across a component.
Voltage is measured in volts, V.
Voltage is measured with a voltmeter, connected in parallel.
The symbol V is used for voltage in equations.

Voltage at a point and 0V (zero volts)

Voltage is a difference between two points, but in electronics we often refer to voltage at a point meaning the voltage difference between that point and a reference point of 0V (zero volts). Zero volts could be any point in the circuit, but to be consistent it is normally the negative terminal of the battery or power supply. You will often see circuit diagrams labelled with 0V as a reminder.
You may find it helpful to think of voltage like height in geography. The reference point of zero height is the mean (average) sea level and all heights are measured from that point. The zero volts in an electronic circuit is like the mean sea level in geography.

Zero volts for circuits with a dual supply

Some circuits require a dual supply with three supply connections as shown in the diagram. For these circuits the zero volts reference point is the middle terminal between the two parts of the supply. On complex circuit diagrams using a dual supply the earth symbol is often used to indicate a connection to 0V, this helps to reduce the number of wires drawn on the diagram.
The diagram shows a ±9V dual supply, the positive terminal is +9V, the negative terminal is -9V and the middle terminal is 0V.

Connecting an ammeter in series

Current, I

Current is the rate of flow of charge.
Current is not used up, what flows into a component must flow out.
We say current through a component.
Current is measured in amps (amperes), A.
Current is measured with an ammeter, connected in series.
To connect in series you must break the circuit and put the ammeter acoss the gap, as shown in the diagram.
The symbol I is used for current in equations.
Why is the letter I used for current? ... please see FAQ.

1A (1 amp) is quite a large current for electronics, so mA (milliamps) are often used. m (milli) means "thousandth": 1mA = 0.001A, or 1000mA = 1A
The need to break the circuit to connect in series means that ammeters are difficult to use on soldered circuits. Most testing in electronics is done with voltmeters which can be easily connected without disturbing circuits.

Voltage and Current for components in Series

Voltages add up for components connected in series.
Currents are the same through all components connected in series. In this circuit the 4V across the resistor and the 2V across the LED add up to the battery voltage: 2V + 4V = 6V.
The current through all parts (battery, resistor and LED) is 20mA.

Voltage and Current for components in Parallel

Voltages are the same across all components connected in parallel.
Currents add up for components connected in parallel. In this circuit the battery, resistor and lamp all have 6V across them.
The 30mA current through the resistor and the 60mA current through the lamp add up to the 90mA current through the battery.

the root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. RMS is used in various fields, including electrical engineering.
It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values. It is a special case of the generalized mean with the exponent p = 2.

The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean (average) of the squares of the original values (or the square of the function that defines the continuous waveform).
In the case of a set of

n

values $\{x_1,x_2,\dots,x_n\}$ , the RMS value is given by:

$x_{\mathrm{rms}} = \sqrt {{{x_1}^2 + {x_2}^2 + \cdots + {x_n}^2} \over n}.$

The corresponding formula for a continuous function (or waveform)

f (t)

defined over the interval $T_1 \le t \le T_2$ is

$f_{\mathrm{rms}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, dt}},$

and the RMS for a function over all time is

$f_\mathrm{rms} = \lim_{T\rightarrow \infty} \sqrt {{1 \over {2T}} {\int_{-T}^{T} {[f(t)]}^2\, dt}}.$

The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.
In the case of the RMS statistic of a random process, the expected value is used instead of the mean.

RMS of common waveforms

Waveform	Equation	RMS
Sine wave	$y=a\sin(2\pi ft)\,$	$\frac{a}{\sqrt{2}}$
Square wave	$y=\begin{cases}a & \{ft\} < 0.5 \\ -a & \{ft\} > 0.5 \end{cases}$	$a\,$
Modified square wave	$y=\begin{cases}0 & \{ft\} < 0.25 \\ a & 0.25 < \{ft\} < 0.5 \\ 0 & 0.5 < \{ft\} < 0.75 \\ -a & \{ft\} > 0.75 \end{cases}$	$\frac{a}{\sqrt{2}}$
Sawtooth wave	$y=2a\{ft\}-a\,$	$a \over \sqrt 3$
Notes: t is time f is frequency a is amplitude (peak value) {r} is the fractional part of r

Uses

The RMS value of a function is often used in physics and electrical engineering.

Average electrical power

Electrical engineers often need to know the power,

P

, dissipated by an electrical resistance,

R

. It is easy to do the calculation when there is a constant current,

I

, through the resistance. For a load of

R

ohms, power is defined simply as:

$P = I^2 R.\,\!$

However, if the current is a time-varying function,

I (t)

, this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is nonetheless still meaningful to talk about the average power dissipated over time, which we calculate by taking the simple average of the power at each instant in the waveform or, equivalently, the squared current. That is,

$P_\mathrm{avg}\,\!$	$= \langle I(t)^2R \rangle \,\!$ (where $\langle \ldots \rangle$ denotes the mean of a function)
	$= R\langle I(t)^2 \rangle\,\!$ (as R does not vary over time, it can be factored out)
	$= (I_\mathrm{RMS})^2R\,\!$ (by definition of RMS)

So, the RMS value,

I RMS

, of the function

I (t)

is the constant signal that yields the same power dissipation as the time-averaged power dissipation of the current

I (t)

.
We can also show by the same method that for a time-varying voltage,

V (t)

, with RMS value

V RMS

$P_\mathrm{avg} = {(V_\mathrm{RMS})^2\over R}.\,\!$

This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, allowing us to calculate the mean power delivered into a specified load.
By taking the square root of both these equations and multiplying them together, we get the equation

$P_\mathrm{avg} = V_\mathrm{RMS}I_\mathrm{RMS}.\,\!$

Both derivations depend on voltage and current being proportional (i.e., the load, R, is purely resistive). Reactive loads (i.e., loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power.
In the common case of alternating current when

I (t)

is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If we define

I p

to be the peak current, then:

$I_{\mathrm{RMS}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {(I_\mathrm{p}\sin(\omega t)}\, })^2 dt}.\,\!$

where t is time and ω is the angular frequency (ω = 2π/T, whereT is the period of the wave).
Since

I p

is a positive constant:

$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {\sin^2(\omega t)}\, dt}}.$

Using a trigonometric identity to eliminate squaring of trig function:

$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {{1 - \cos(2\omega t) \over 2}}\, dt}}$

$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} \left [ {{t \over 2} -{ \sin(2\omega t) \over 4\omega}} \right ]_{T_1}^{T_2} }$

but since the interval is a whole number of complete cycles (per definition of RMS), the

sin

terms will cancel out, leaving:

$I_{\mathrm{RMS}} = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} \left [ {{t \over 2}} \right ]_{T_1}^{T_2} } = I_\mathrm{p}\sqrt {{1 \over {T_2-T_1}} {{{T_2-T_1} \over 2}} } = {I_\mathrm{p} \over {\sqrt 2}}.$

A similar analysis leads to the analogous equation for sinusoidal voltage:

$V_{\mathrm{RMS}} = {V_\mathrm{p} \over {\sqrt 2}}.$

Where

I P

represents the peak current and

V P

represents the peak voltage. It bears repeating that these two solutions are for a sinusoidal wave only.
Because of their usefulness in carrying out power calculations, listed voltages for power outlets, e.g. 120 V (USA) or 230 V (Europe), are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies V_p = V_RMS × √2, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × √2, or about 170 volts. The peak-to-peak voltage, being twice this, is about 340 volts. A similar calculation indicates that the peak-to-peak mains voltage in Europe is about 650 volts.
It is also possible to calculate the RMS power of a signal. By analogy with RMS voltage and RMS current, RMS power is the square root of the mean of the square of the power over some specified time period. This quantity, which would be expressed in units of watts (RMS), has no physical significance. However, the term "RMS power" is sometimes used in the audio industry as a synonym for "mean power" or "average power". For a discussion of audio power measurements and their shortcomings, see Audio power.

Amplifier power efficiency

The electrical efficiency of an electronic amplifier is the ratio of mean output power to mean input power. The efficiency of amplifiers is of interest when the energy used is significant, as in high-power amplifiers, or when the power-supply is taken from a battery, as in a transistor-radio.
Efficiency is normally measured under steady-state conditions with a sinusoidal current delivered to a resistive load. The power output is the product of the measured voltage and current (both RMS) delivered to the load. The input power is the power delivered by the DC supply, i.e. the supply voltage multiplied by the supply current. The efficiency is then the output power divided by the input power, and it is always a number less than 1, or, in percentages, less than 100. A good radio frequency power amplifier can achieve an efficiency of 60–80%.
Other definitions of efficiency are possible for time-varying signals. As discussed, if the output is resistive, the mean output power can be found using the RMS values of output current and voltage signals. However, the mean value of the current should be used to calculate the input power. That is, the power delivered by the amplifier supplied by constant voltage

V C C

$P_\mathrm{input}(t) = I_Q V_{CC} + I_\mathrm{out}(t) V_{CC}\,$

where

I Q

is the amplifier's operating current. Clearly, because

V C C

is constant, the time average of

P input

depends on the time average value of

I out

and not its RMS value. That is,

$\langle P_\mathrm{input}(t) \rangle = I_Q V_{CC} + \langle I_\mathrm{out}(t) \rangle V_{CC}.\,$

Root mean square speed

Main article: Root mean square speed

In the physics of gas molecules, the root mean square speed is defined as the square root of the average speed-squared. The RMS speed of an ideal gas is calculated using the following equation:

${v_\mathrm{RMS}} = {\sqrt{3RT \over {M}}}$

where

R

represents the Ideal Gas Constant (in this case, 8.314 J/(mol·K)),

T

is the temperature of the gas in kelvins, and

M

is the molar mass of the gas in kilograms. The generally accepted terminology for speed as compared to velocity is that the former is the scalar magnitude of the latter. Therefore, although the average speed is between zero and the RMS speed, the average velocity for a stationary gas is zero.

Root mean square error

Main article: Root mean square error

When two data sets—one set from theoretical prediction and the other from actual measurement of some physical variable, for instance—are compared, the RMS of the pairwise differences of the two data sets can serve as a measure how far on average the error is from 0.
The mean of the pairwise differences does not measure the variability of the difference, and the variability as indicated by the standard deviation is around the mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the error.

RMS in frequency domain

The RMS can be computed also in frequency domain. The Parseval's theorem is used. For sampled signal:
$\sum\limits_{n}{{{x}^{2}}(t)}=\frac{\sum\limits_{n}{{{\left| X(f) \right|}^{2}}}}{n}$ , where

X (f) = F F T {x (t)}

n

is number of

x (t)

samples.
In this case, the RMS computed in time domain is the same as in frequency domain:
$RMS=\sqrt{\frac{1}{n}\sum\limits_{n}{{{x}^{2}}(t)}}=\frac{1}{n}\sqrt{\sum\limits_{n}{{{\left| X(f) \right|}^{2}}}}=\sqrt{\sum\limits_{n}{{{\left| \frac{X(f)^{2}}{n} \right|}}}}$

Relationship to the arithmetic mean and the standard deviation

If $\bar{x}$ is the arithmetic mean and

σ x

is the standard deviation of a population or a waveform then:

$x_{\mathrm{rms}}^2 = \bar{x}^2 + \sigma_{x}^2.$

From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well.
Physical scientists often use the term "root mean square" as a synonym for standard deviation when referring to the square root of the mean squared deviation of a signal from a given baseline or fit.^{[citation needed]} This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the root mean square of a signal's variation about the mean, rather than about 0, the DC component is removed (i.e. RMS(signal) = Stdev(signal) if the mean signal is 0).

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Wednesday, October 5, 2011