Why is it called a linear system?

Thread Starter

MikeJacobs

Joined Dec 7, 2019
215
So when you look at a "linear" system something like an RC circuit. The response is exponential and is in fact not linear at all.
So why do we call them linear systems
 

LvW

Joined Jun 13, 2013
1,353
So when you look at a "linear" system something like an RC circuit. The response is exponential and is in fact not linear at all.
So why do we call them linear systems
The TIMELY response to a voltage step is exponenetial. But this kind of test has nothing to do with "linearity".
Instead, we have to apply a test signal to the device under test (for example sinusoidal) and then to modify the amplitude of the signal. When the output behaves in proportion to the input we say that the circuit is linear.
 

Ian0

Joined Aug 7, 2020
4,065
Presumably because it is determined by linear equations (i.e. those with no terms of V^2 or higher).
The RC delay is determined by a linear differential equation (of the form dV/dt=kV), the solution of which is an exponential.
 

hrs

Joined Jun 13, 2014
355
Systems defined by higher order differential equations can also be linear, in the sense that if you double the input you get double the output.
 

Ian0

Joined Aug 7, 2020
4,065
Systems defined by higher order differential equations can also be linear, in the sense that if you double the input you get double the output.
agreed:
the normal oscillatory response comes from
\(
\frac{d^2V}{dt^2}=-kV
\)
which has no terms of V^2 or higher, just the double differentiation.
 

Deleted member 115935

Joined Dec 31, 1969
0
In my experience,

linear is an old term, used to mean an output is dependent upon its inputs,
if you know the input conditions, you know the output conditions,

Now days , its tended to be used by many to cover any "analog" circuits,

But exceptions,
a noise generator is not linear, its output does not depend upon its inputs,
but it could still be analog, This would be a none linear system,
 

ZCochran98

Joined Jul 24, 2018
192
I don't know if this will help at all, but the term "linearity" comes from signal analysis and mathematics, rather than purely looking at the signal response.

From a mathematical perspective, when a circuit is "linear," it means it has a "linear mapping" between the input \(x_1(t)\) and the output \(y_1(t)\). What this means is that if we have a constant \(a_1\), then if we multiply the input by this constant, \(a_1x_1(t)\), then the output should be scaled by exactly the same amount - it should be \(a_1y_1(t)\). Furthermore, if \(x_2(t)\) is a different input with an output \(y_2(t)\), then the input \(x_1(t)+x_2(t)\) should result in an output of \(y_1(t)+y_2(t)\). Combining both thoughts, if we have \(a_1x_1(t)+a_2x_2(t)\) input to a system, the output should be \(a_1y_1(t)+a_2y_2(t)\). This is the "formal" (mostly) definition of the superposition principle, which is what is used to define linearity in circuits. You could throw in time invariance in there too (if input is \(x(t)\) and output is \(y(t)\), then if you add a time shift by T, then input \(x(t-T)\) should give output \(y(t-T)\)), but that's a specific subset of linear systems (that most linear circuits follow).

Basically, what this looks like in a circuit is if you supply some kind of signal to your circuit, and see an output, and then apply a different signal to your circuit, and see the new output, if you were to combine the two distinct input signals, you should see an output that is the combination of the two distinct outputs.

I don't know if this clears anything up, but it is, at least, the "formal" definition of linearity.
 

MrSalts

Joined Apr 2, 2020
836
An electronic circuit can be either "linear" or "digital".
Exactly.
From wikipedia
Digital electronics is a field of electronics involving the study of digital signals and the engineering of devices that use or produce them. This is in contrast to analog electronics and analog signals.

@LvW, @Ian0, @hrs, @andrewmm and @ZCochran98 can all make up stuff but, it's as simple as @Audioguru said - there is either digital or analog (also known as linear). "Linear" comes from the fact that transistors need to operate in their non-saturated regions (linear regions) to achieve analog (non-binary/non-digital) transfer functions.
 

ZCochran98

Joined Jul 24, 2018
192
Exactly.
From wikipedia
Digital electronics is a field of electronics involving the study of digital signals and the engineering of devices that use or produce them. This is in contrast to analog electronics and analog signals.

@LvW, @Ian0, @hrs, @andrewmm and @ZCochran98 can all make up stuff but, it's as simple as @Audioguru said - there is either digital or analog (also known as linear). "Linear" comes from the fact that transistors need to operate in their non-saturated regions (linear regions) to achieve analog (non-binary/non-digital) transfer functions.
The difference between analog and digital has almost nothing to do with linearity and what a linear circuit is. Analog indicates that there's infinite resolution (you can go from 0 to 1 and any real number in between, for example), whereas digital relies on your state being zero or one (with ideal state transitions being perfect jumps). Linear circuits ARE generally analog, yes, but not all analog circuits are linear (there are a great many circuits that are nonlinear, such as logarithmic gain amplifiers, the VCO, or the gilbert cell). Equating analog circuits to linearity is an inaccurate comparison.

The stuff about transistors: yes, transistors need to be in their linear regions to perform (most) analog transfer functions (which may or may not be linear!), but that doesn't explain what linearity actually is - just that it's a state that the transistors need to be in in order to be useful for most analog applications. Transistors themselves - as well as any semiconducting device - are inherently nonlinear devices; they just have small regions of behavior that approximate to linear relationships (namely, Ohm's law). That's why the linear region of a device is also called the "Ohmic region."
 

Thread Starter

MikeJacobs

Joined Dec 7, 2019
215
Exactly.
From wikipedia
Digital electronics is a field of electronics involving the study of digital signals and the engineering of devices that use or produce them. This is in contrast to analog electronics and analog signals.

@LvW, @Ian0, @hrs, @andrewmm and @ZCochran98 can all make up stuff but, it's as simple as @Audioguru said - there is either digital or analog (also known as linear). "Linear" comes from the fact that transistors need to operate in their non-saturated regions (linear regions) to achieve analog (non-binary/non-digital) transfer functions.
omg lol did you just say that? Actually the other folks who answered are completely correct when talking about the relationship to the input and output from a mathematical perspective. You are completely wrong.
 

LvW

Joined Jun 13, 2013
1,353
Exactly.
From wikipedia
Digital electronics is a field of electronics involving the study of digital signals and the engineering of devices that use or produce them. This is in contrast to analog electronics and analog signals.

@LvW, @Ian0, @hrs, @andrewmm and @ZCochran98 can all make up stuff but, it's as simple as @Audioguru said - there is either digital or analog (also known as linear). "Linear" comes from the fact that transistors need to operate in their non-saturated regions (linear regions) to achieve analog (non-binary/non-digital) transfer functions.
You are completely wrong. Of course, there are "analog" parts and circuits which are "non-linear". In fact, there are no parts or analog circuits which are really linear. All these devices show a certain degree of non-linearity.
 
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