# what does negative inductance mean?

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#### shaoyingh

##### Newbie level 5
negative inductance

Hi,

need your help and thank you!

#### LvW

what does negative inductance mean

The answer is very easy: It is an inductance with a negative sign - used for example to cancel some unwanted (parasitic) inductance like influences. In a similar manner you can cancel small capacitances (e.g. of a coaxial transmission line) with a negative capacitor.

In both cases, these artificial parts are to be realized by active circuitry like negative impedance converters (NIC) using opamps or OTA´s.

LvW

Addendum: The input impedance of a NIC is -(Z1*Z2/Z3)

(a) Z3=1/sC3 results in a negative grounded inductance
(b) Z1=1/sC1 or Z2=1/sC2 results in a negative capacitance

### shaoyingh

Points: 2

#### byteptr

##### Member level 4
can inductance be negative

Hi, in addition to the last post, here is an article about simulation of positive and negative inductors:

https://www.te.kmutnb.ac.th/~msn/w4d-2new.pdf

There are also FDNR (Frequency Dependant Negative Resistors) where R = -k*w and others based on OTAs, used for filter synthesis.
The advantage can be a solid-state filter with all capacitors and electronically tunable since gm is controlled...

Enjoy it.

### shaoyingh

Points: 2

#### shaoyingh

##### Newbie level 5
negative capacitance

Thank you very much!

but what is the physical meaning of negative inductance for a transmission line that is guiding waves?

#### byteptr

##### Member level 4
negative inductor

"but what is the physical meaning of negative inductance for a transmission line that is guiding waves?"

mmmmm... I'm not sure what are you mean. However in a passive transmission line have no physical sense.

For example, assuming lossless line we have a well know relationship for a characteristic impedance:

Zo = sqrt(L/C)

With L and C in Henrys per m and farad per m.

This implies that if L is negative and C positive,

Zo = sqrt(-L/C) = j*sqrt(L/C)

a pure imaginary characteristic impedance can not transport the energy. (In a classical filter theory is named "The pass band theorem"). However if the sign of both reactances in transmission line is the same, Zo is real.

A more complicated case is when there is a line-loss

Zo = sqrt((R+jwL)/(G+jwC)) = sqrt(Z/Y)

since a propagation constant is Gamma = sqrt(Z*Y) = sqrt((R+jwL)*(G+jwC)) -> exp(Gamma*x) = I1/I2

gives you complex current ratio, you can deduce if the wave is attenuated or ""amplificated"" in function of transmission line-length "x" for all cases of impedace signs and also if has no physical construction.

#### LvW

negative inductance?

I agree with byteptr,
and here is another simple physical interpretation:

For a positive inductance - e.g. realized by a coil - the current is 90 deg behind the voltage across the part.

For a negative inductance there will be instead a phase shift of -270 deg (resp. +90 deg) since the minus sign is identical to 180 deg phase shift.

By the way: FDNR´s are realized with opamps as well (even simpler and more exact)

LvW

#### jallem

A negative inductance can be thought
as a capacitance.

#### LvW

true negative inductance

Quote:A negative inductance can be thought
as a capacitance.

And what about the frequency dependence ? Is this a joke ?

LvW

#### shaoyingh

##### Newbie level 5
convert negative inductance to capacitance

negative inductance can only be converted to capacitance at a specific frequency.

Thank you all for the response!

LvW

May I know more details about the structure/construction of a negative impedance converters (NIC)?

Thank you!

#### Azulykit

inductor canceling capacitance

I note that the same question was posted in "Electromagnetic Design and Simulation" by the same individual.

#### shaoyingh

##### Newbie level 5
what mean by inductor

Yes I did.

The reason I do so is that I found people in the other category is more on RF/microwave circuit side. Thus I am there looking for the possible explanation from an EM point of view.

Sorry for any inconvenience caused.

#### FvM

##### Super Moderator
Staff member
what is mean by pass band transmission

As you see from the very different answers, your question isn't clear. Obviously a negative inductance can be created by an active circuit, e. g. for compensation purposes, but this isn't related to your question anyway. Can you simply give an example where you see a negative inductance value in an expression related to transmission lines? I haven't seen it yet and also don't expect it, so I would say, it has no physical menaning.

You have been talking of transmission lines that are guiding waves. If you actually mean hollow wave guides, you should tell explicitely.

#### byteptr

##### Member level 4
can you have negative inductance?

I have 2 commentaires to reply:

1.- "A negative inductance can be thought
as a capacitance."

It's not true in all cases, only for one frequency (as says one of precedent posts), since magnitude negative or positive capacitance (assuming negative capacitante in theoretical point of view) varies inversely with frequence as |1/(wC)|
and inductance varies proportionally as |wL|.

2.- "As you see from the very different answers, your question isn't clear."

It's clear I think. See my post (5 May 2008 14:14 Re: what does negative inductance mean?), please:

"a pure imaginary characteristic impedance can not transport the energy. (In a classical filter theory is named "The pass band theorem"). However if the sign of both reactances in transmission line is the same, Zo is real.

A more complicated case is when there is a line-loss

Zo = sqrt((R+jwL)/(G+jwC)) = sqrt(Z/Y)

since a propagation constant is Gamma = sqrt(Z*Y) = sqrt((R+jwL)*(G+jwC)) -> exp(Gamma*x) = I1/I2

gives you complex current ratio, you can deduce if the wave is attenuated or ""amplificated"" in function of transmission line-length "x" for all cases of impedace signs and also if has no physical construction."

In addition to this (maintaining theoretical interest), one may be attempted to imagine a real-characteristic impedance transmission line with both negative Inductance and capacitante unit per length:

Zo = sqrt((-L)/(-C)) where Zo is real

But this implies inverting direction of wave propagation or "phase advance" or changing "x" direction, this is derived by the Telegrapher's equations.

Finally, about impedance compensation there a some tecniques based on passive circuits (not all are active circuits), for example that are used in an old crystal filter synthesis for remove parallel capacitance (see the attached image).

Best regards.

#### LvW

some negative impedance convertors

Sorry, but I cannot see the relationship between the circuit diagram shown above and the topic "negative inductance". Or did I misunderstand something ?

#### byteptr

##### Member level 4
negative mutual impedance

Hello,
In my last post I say:
"Finally, about impedance compensation there a some tecniques based on passive circuits (not all are active circuits), for example that are used in an old crystal filter synthesis for remove parallel capacitance (see the attached image).
"

I'm talking about "removing" un desired parasitic impedances with passives, because some of the posts talks about the utility of negative inductances or negative capacitances for cancel parasitics impedances using active circuits.

In the case of the picture undesired parallel capacitance, you can think in "connect a negative capacitor in parallel" made of active circuits or doing this passive circuit.
The RF bridges are "magic".

It's just an example. For the inductances remains the same.

#### FvM

##### Super Moderator
Staff member
negative inductance meaning

I partcularly agree with your previous statement in a passive transmission line have no physical sense. I guess, you didn't want to raise the imaginary all negative reactances transmission line as a reservation or contradiction.

#### jallem

negative inductance

see pg. 3 at the bottom :
https://www.venable.biz/tp-19.pdf
A negative inductor is a capacitor whose impedance increases with
increasing frequency. This is potentially a useful element for modeling bizarre impedances.

https://arxiv.org/PS_cache/cond-mat/pdf/9806/9806145v1.pdf

Also see an article about a Hartley oscillator:

https://www.edn.com/index.asp?layout=ednPDF&file_name=/contents/images/6382649f1.pdf
https://www.edn.com/index.asp?layout=article&articleid=CA6382649
A traditional Hartley oscillator’s resonant circuit consists of a tapped inductor and resonating capacitor (a).
Allowing for mutual coupling between windings produces an equivalent circuit containing a negative inductance (b).
Replacing the negative inductance with a capacitor yields an easily modeled equivalent circuit (c).

#### byteptr

##### Member level 4
can you have negative inductance

"A traditional Hartley oscillator’s resonant circuit consists of a tapped inductor and resonating capacitor (a).
Allowing for mutual coupling between windings produces an equivalent circuit containing a negative inductance (b). "

Same as the circuit for compensating crystal parallel capacitance.

All of these "rare" elements are clever mathematical tools for modeling, It would be desirable keep in mind clear the concepts... if we lie in a language barrier, we can say also: "Resistors are inductors whose magnitude is constant and relative phase of V-I is 0degrees..."

Best regards.

#### LvW

inductance means

Thank you for the last sentence !! It is a good example for playing with words without sufficient engineering knowledge. Thanks.
LvW

#### jallem

does inductance have a positive or negative

It is amazing how a fool is quick to call others a fool.

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