# Measuring the L/R ratio of an LR circuit

I have the circuit below (fig.1) that I found in a board of an old machine. The input signal is a square wave with 10 Hz frequency and 10 V amplitude. I measured the waveform on the primary side of the transformer with an oscilloscope (fig.2) and found that the time 5*L/R is equal to 7.8 ms. I calculated this time value by measuring the inductance: it was 1.6 H with an internal resistance of 330 Ω.

With these measured values I found that the time is equal to L/(R1+r) = 1.6/(5600+330) = 1.3 ms, where L is the inductance of the primary, R1 is the series resistor in the circuit and r is the internal resistance of the coil.

And to be sure that my LCR meter is good and there's no problem, I put a coil instead of a transformer and I found that the answers are very close.

Then I thought: can we consider the transformer and coil with the same logic or should we consider the transformer differently?

• Repeat the experiment with a short-circuit on the secondary and again with, maybe, 10 Ω on it and see what you find. – Transistor Aug 2 '16 at 10:52
• What is your question? What are you trying to determine about this LR circuit and/or transformer? What is the purpose of this measurement exercise? – FiddyOhm Aug 2 '16 at 11:01
• @FiddyOhm The question is : what is the difference between coil and transformer when measuring inductance ? The goal is to analyse this circuit and to know what's the problem. – Zara Zara Aug 2 '16 at 13:18
• @transistor : I repeated it , and I found that the time L/R is decreasing with 10 ohm load resistance and decreases more and more when shorted circuit so the time is greater than period of signal – Zara Zara Aug 2 '16 at 13:21
• @ZaraZara: Well done. The point is that the secondary load affects the primary and in the transformer that's what we want. Primary impedance should drop as secondary load impedance drops otherwise the transformer would not be an efficient electrical device. Hopefully this will help you understand the transformer mathematical models better than I do! – Transistor Aug 2 '16 at 14:57

As long as the secondary is left open, a transformer primary behaves exactly as an inductor, since the secondary has substantially no effect on the circuit.

The reason behind this behavior is that the primary and secondary are coupled by the magnetic flux that flows trough the transformer core. The magnetic flux depends on the current flowing in a transformer winding. Therefore, if in the secondary doesn't flow any current, no flux is generated by the secondary that can interact with the flux generated by current in the primary, hence the secondary doesn't interact with the primary whatsoever.

Keep in mind, though, that even with the secondary open, a transformer core is usually made with ferromagnetic materials (or ferrimagnetic materials, in case of ferrite cores), which is highly non-linear. This means that the inductance of the primary is not constant (as that of an air-wound coil), but depends on the amplitude of the current in the winding.

Therefore you cannot simply substitute the transformer primary with an air-wound coil with the same inductance and internal resistance and expect identical behavior (always assuming the secondary is open). To have identical behavior you should replace the transformer primary with a coil wound on a core made of the same material with which the transformer core is made (and even the same shape).

EDIT (prompted by a comment - integrating information I provided in comments)

So you are puzzled by a theoretical primary inductance value that is about 5 times higher than what you measure. In reality this is not surprising, for a variety of factors I'll try to explain below.

You should understand that non-linear inductors (NLI in the following) are nasty beasts, and a cored transformer is just a NLI when you leave the secondary open.

The main problem for a NLI is that the very concept of inductance is not well-defined, if for inductance you intend a constant parameter that expresses the relationship between current and magnetic flux. Even if you treat L as function of the current, instead, you are not getting the right picture, it does account for the non-linearity (saturation), but it doesn't account for the memory effect due to residual magnetization in the core, i.e. the effect that causes hysteresis in the core.

The fact that a NLI is so complicated also implies that your calculations are somewhat wrong in theory: they assume the classical LR circuit where L is a constant and the step response is an exponential decay. In reality, with a NLI, the curve you see on the scope is not a true exponential, but only something that resembles it. Therefore you shouldn't expect that the rule you apply to infer L (i.e. that the decay brings the curve under 1% level after 5 time constants) is perfectly valid.

This is a first source of error. So your computed value is affected by an error just before you measure the real thing (call it a model approximation error, if you want).

Then there are the differences in measurement techniques: the "inductance" of a NLI is influenced by many parameters of the signal used to probe the inductor, especially amplitude and frequency. Hence the shape of the signal may also alter the results. As you have seen even two different LCR meters give results that differ by a factor of 3 or 4.

I won't enter in further details, but I'll point you to this application note of Keysight (former Agilent, former HP): Impedance Measurement Handbook . The following are relevant excerpts:

5.2 Inductor measurement

5.2.1 Paracitics of an inductor

An inductor consists of wire wound around a core and is characterized by the core material used. Air is the simplest core material for making inductors, but for volumetric efficiency of the inductor, magnetic materials such as iron, permalloy, and ferrites are commonly used. A typical equivalent circuit for an inductor is shown in Figure 5-9 (a). In this figure, Rp represents the magnetic loss (which is called iron loss) of the inductor core, and Rs represents the copper loss (resistance) of the wire. C is the distributed capacitance between the turns of wire. For small inductors the equivalent circuit shown in Figure 5-9 (b) can be used. This is because the value of L is small and the stray capacitance between the lead wires (or between the electrodes) becomes a significant factor.

[...]

5.2.2 Causes of measurement discrepancies for inductors

Inductance measurement sometimes gives different results when a DUT is measured using different instruments. There are some factors of measurement discrepancies as described below: Test signal current Inductors with a magnetic core exhibit a test signal current dependency due to the nonlinear magnetization characteristics of the core material as shown in Figure 5-13 (a). The level of test signal current depends on the impedance measurement instrument because many of the instruments output a voltage-driven test signal. Even when two different instruments are set to output the same test signal (OSC) voltage, their output currents are different if their source resistance, Rs, is not the same as shown in Figure 5-13 (b). To avoid the measurement discrepancies, the OSC level should be adjusted for a defined test current by using the auto level control (ALC) function or by determining the appropriate test voltage setting from the equation shown in Figure 5-13 (b).

[...]

5.3 Transformer measurement

A transformer is one end-product of an inductor so, the measurement techniques are the same as those used for inductor measurement. Figure 5-18 shows a schematic with the key measurement parameters of a transformer. This section describes how to measure these parameters, including L, C, R, and M.

[...]

So, in the end, you cannot expect to get a result which is extremely precise, especially with a non-controlled test setup. If you get a value which is of the same order of magnitude of the real value (i.e. within an 1:10 ratio) you should be happy. Then yes, the values you are getting are perfectly in line given the test setup you have described.

• The inductance that I measure on primary is 1.5 H , and what I expect is 9.5 H, Is this logic to have a difference of 8 H between the first and the second because the magnetization curve of iron? Furthermore , the coil that I use is ferromagnetic, so I think my comparison is not bad. what do you think, what I measure is the real inductance or with mutual ? – Zara Zara Aug 4 '16 at 12:21
• The comparison should be done not by difference but by ratio. 9.5H is about 5 times higher, which is in the same order of magnitude of what you measure. So you're in the same ballpark. The difference between measured an theoretical values may be due to the non-linearity of the core and the different measurement techniques, especially amplitude of the excitation signal. You have already noticed that using a different LCR meter you got a different value (4.2H vs. 1.5H), so your measurement are substantially in agreement with the theory. – Lorenzo Donati supports Monica Aug 4 '16 at 13:57
• Moreover, keep in mind that for calculating the L/R ratio you assumed the circuit behaves like a regular linear LR circuit, with exponential decay. But since the circuit is non-linear, that behavior is only approximate: from a mathematical standpoint, what you see on your scope is not a true exponential decay, but something only resembling it. So you cannot expect that the 5 time constant rule = less than 1% will hold precisely. – Lorenzo Donati supports Monica Aug 4 '16 at 14:00
• Beware that making measurement on non-linear components is always a nasty can of worms. From a theoretical standpoint a non-linear inductor doesn't have an inductance as intended usually, i.e. a constant parameter that characterize the item. The true characterization comes from curves expressing the dependance of a "small signal inductance" vs. the current level in the coil. – Lorenzo Donati supports Monica Aug 4 '16 at 14:05
• Moreover, while for linear inductors the value of inductance is independent from the signal shape, for non-linear components the harmonics of the signal may change the behavior. I.e. superimposition doesn't hold. So all this adds up as an error in the measurement intended to measure the inductance of linear inductors. – Lorenzo Donati supports Monica Aug 4 '16 at 14:05