Hi everyone,

I am wondering how to calculate the impedance of an inductor when a square wave is applied. I have an idea but I don't know if it's right.

100mH inductor, 10V, 50uS pulse

I would think V=L*di/dt could be rearranged and used to calculate the impedance.

di/dt=V/L

10V/.100H= 100 amp-seconds x .00005 seconds = 5mA
Then could we just use ohms law V/I=Z

So, 10/.005 = 2kΩ

The math works, but is this the correct calculate the impedance of an inductor when a square wave is applied to it?

Thanks in advance.

 

What you suggest doesn't work as a means to finding an equivalent impedance.

Consider the apparent "impedance" at 25usec for instance - you'll obtain a different value.

Impedance generally has meaning at one particular sinusoidal frequency of interest, which is normally the source frequency - whereas a pulse has an infinite bandwidth. Hence, in the latter case impedance has no specific meaning.

 

Well, I would not call it resistance either, that would make it even more confusing.

I know what you mean though, because if what I said above was correct then that would also mean the impedance (if we can call it that) was also related to the applied voltage (V/L=di/dt).

The formula is correct in determining the rate of current rise for an inductor.
And if that is related to voltage, then why are the calculations not correct in determining "impedance"?

What would be the correct way? Is there a correct way? Or am I just looking at this incorrectly and asking a question with no significance?

 

Pulse frequency: 1/50 MHz
Inductance Impedance: X(l)=w*l

w=2*π *f
= 2*3.14*(1/50)*10^6
So, X(l)=w*100*10^-3
X(l)=12.56 kΩ

 

With respect to [linear] impedance there is proportional relationship between current and voltage when one is considering steady-state behavior. So for a given ideal inductance driven by a single frequency AC source this proves to be the case - albeit with the added complexity of phase displacement between terminal voltage and current.

Under transient conditions one applies differential equations or other methods such as Laplace transforms to develop the relationship between stimulus and response. Your context is transient behavior. I guess one can for instance think of a Laplace transform representation of passive energy storage circuit elements as being akin to representing their impedances.

Are you familiar with Laplace Transforms and their use in circuit analysis?

 

Hi everyone,

I am wondering how to calculate the impedance of an inductor when a square wave is applied. I have an idea but I don't know if it's right.

100mH inductor, 10V, 50uS pulse

I would think V=L*di/dt could be rearranged and used to calculate the impedance.

di/dt=V/L

10V/.100H= 100 amp-seconds x .00005 seconds = 5mA
Then could we just use ohms law V/I=Z

So, 10/.005 = 2kΩ

The math works, but is this the correct calculate the impedance of an inductor when a square wave is applied to it?

Thanks in advance.

You made an assumption that may not be correct. Does the voltage across the inductor really change by 10 V in 50 μS? or is it faster?

 

You made an assumption that may not be correct. Does the voltage across the inductor really change by 10 V in 50 μS? or is it faster?

Presumably the OP is considering current response in an inductor driven by a DC pulse of magnitude 10V and duration 50usec.

 

It gets weirder....

Suppose this is an ideal inductor driven by an ideal voltage pulse source. At the completion of 50usec the current would indeed be 5mA. When the source pulse voltage drops to zero at 50usec the current of 5mA flows on forever - assuming this was a single pulse stimulus. What is the apparent "impedance" at t=100usec with a current of 5mA and terminal voltage of zero?

 

Presumably the OP is considering current response in an inductor driven by a DC pulse of magnitude 10V and duration 50usec.

I would expect the driving circuit to be able to produce the desired waveform.
What I was trying to point out was whether the current was still increasing, or had it been stable for some time (say 20 μs).

 

Hi everyone,

I am wondering how to calculate the impedance of an inductor when a square wave is applied. I have an idea but I don't know if it's right.

100mH inductor, 10V, 50uS pulse

I would think V=L*di/dt could be rearranged and used to calculate the impedance.

di/dt=V/L

10V/.100H= 100 amp-seconds x .00005 seconds = 5mA
Then could we just use ohms law V/I=Z

So, 10/.005 = 2kΩ

The math works, but is this the correct calculate the impedance of an inductor when a square wave is applied to it?

Thanks in advance.

That di/dt is the instantaneous current change in the circuit.

I don't follow where you are getting your numbers from.

10V/100mH = 100 amps PER second, not 100 amp-seconds.

The 50μs time you are grabbing is the width of the pulse, which is NOT the differential time associated with the ramp. You are making the mistake of grabbing anything that happens to have the units you want and just throwing it as an equation. Doesn't work that way.

Your pulse can be written as the sum of a series of sinusoidal signals. Each signal has it's own frequency, amplitude and phase. You can find the response of the circuit to each of those signals, with the impedance of the inductor being calculated for each signal frequency in turn. You can then add the individual responses up to get your total response.

But even then you can't use steady-state techniques to find the response to a single, isolated pulse. You need to use either Laplace transforms or Fourier transforms (or just solve the differential equations directly) to do that.

 

It gets weirder....

Suppose this is an ideal inductor driven by an ideal voltage pulse source. At the completion of 50usec the current would indeed be 5mA. When the source pulse voltage drops to zero at 50usec the current of 5mA flows on forever - assuming this was a single pulse stimulus. What is the apparent "impedance" at t=100usec with a current of 5mA and terminal voltage of zero?

Not all that weird at all. This is exactly what you do with a superconducting magnet! You apply a charging voltage to it long enough to get the desired current. You then short the supply leads with a superconducting "persistent-mode" switch and then remove the supply. The current then runs forever (well, okay, it generally drops at the rate of a percent per century or so due to parasitics).

Since it is in DC steady state, the impedance calculation is trivial. X=jωL with ω=0r/s. So the impedance is 0Ω.

 

Your pulse can be written as the sum of a series of sinusoidal signals. Each signal has it's own frequency, amplitude and phase. You can find the response of the circuit to each of those signals, with the impedance of the inductor being calculated for each signal frequency in turn. You can then add the individual responses up to get your total response.But even then you can't use steady-state techniques to find the response to a single, isolated pulse. You need to use either Laplace transforms or Fourier transforms (or just solve the differential equations directly) to do that.

Ok, Well I have to admit I know nothing about Laplace & Fourier transforms. I only got as far as algebra in school. But I can learn anything.

So, anyone care to go into detail?

I'll start by reading the book section 'Square wave signals'.

Thanks everyone for all the help so far.

 

The frequency-domain concept of impedance value in ohms is used for sinusoidal excitation and doesn't readily apply in the time-domain for a square-wave or other non-sinusoidal signal unless you use Laplace transforms which can get rather messy. In the frequency-domain the inductive impedance goes from 0 ohms at DC, to infinite ohms for the infinite frequency of the zero rise and fall time of an ideal square-wave. So the easiest way to calculate the resultant current is in the time-domain using V = L*di/dt or di = V/L dt. Thus the application of one 10V, 50μs pulse from an ideal voltage source into a grounded 100mH inductor will cause an increase in the inductor current of 10/100mH * 50μs = 5mA, as previously calculated. If there is no other impedances in the circuit then each additional pulse will increase the current by another 5mA. So you can't really assign a simple impedance value to that type of behavior.

 

Get a cap whose value is know accurately, put in series with a 10 K resistor and the inductor ( resistor on high side )
Scope the voltage across the LC
Excite with a square wave. If you are any where near resonance you will see a ripple that makes it easy to adjust the square wave to the resonate frequency.
Calculate the inductance from the cap and frequency.

I am thinking of doing an article on this technique, but the above should serve.

 

If the wave is square with a high (the pulse) of 50 us, then I'd think it would also have a low of 50 us and would be described as a 10 kHz square wave.

A 10 kHz square wave will have a 10 kHz fundamental sinusoidal component plus odd harmonics at lower strengths, 1/3 the strength of the fundamental at 30 kHz, 1/5 the strength at 50 kHz, and so on.

So your inductor will react most strongly to the strongest 10 kHz component.

Calculating the impedance at 10 kHz gives:

Xl = 2 pi fl = 6.28 * 10000 hz * 0.1 H = 6280 Ohms

That value is probably pretty close to how the part will behave in a circuit. It will present 3 times that impedance to the 30 kHz component, around 19k ohms of impedance, and it will present proportionally higher impedances to the higher frequency components of the square wave, so you'll mostly see a 10 kHz sinewave at the output.

If the inductor is resonated with a capacitor, it will go even farther at filtering the sinusoidal components of the square wave, showing peaks at each sinusoidal component.

 

What if it is a unipolar square wave?

Would this change the Fourier transform?

 

What if it is a unipolar square wave?

Would this change the Fourier transform?

As described, it IS a unipolar square wave.

But, yes, if you change the signal you change the Fourier transform of that signal.

Think of the Fourier transform for what, at the end of the day, it is -- nothing more than a different way of writing a function. There is a one-to-one correspondence between the time-domain representation for a function and its representation in the complex frequency-domain (assuming a representation actually exists in both domains). Change one and you change the function. Change the function and you need to change both representations.

 

If it is an ideal Inductor,it doesn't really have an Impedance-----the correct term is Reactance!

 

If it is an ideal Inductor,it doesn't really have an Impedance-----the correct term is Reactance!

But inductors do have impedance. Impedance is a term the includes all the resistance and reactance at the node of interest. So if there is only an ideal inductor then the impedance of the inductor is its inductive reactance. A real inductor will also have some stray capacitance and resistance so its impedance would include those terms also.

 

If it is an ideal Inductor,it doesn't really have an Impedance-----the correct term is Reactance!

Sure it does. Resistance and reactance are proper subsets of impedance. Just as a pure real number and a pure imaginary number are both members of the set of complex numbers.