# Working Range of Transformer

• AIR&SPACE
In summary, the goal was to find a transformer with satisfying relationships between L_{S}, L_{L}, and a while a=10 and a coupling coefficient of 1. The transformer should also work properly at 1kHz and above, as well as at 100kHz and above. By using a graph and equations for the coupling coefficient and input impedance, it was determined that the low frequency resonance for the transformer can be described by the equation \frac{1}{\sqrt{2}}\approx\frac{1}{\sqrt{1+\left(\frac{R_{pa}}{2*\pi*f*L_{oc}}\right)^{2}}}. Using this equation, the values for L1 and L2 were
AIR&SPACE

## Homework Statement

By trying a variety of values for $$L_{S}$$ and $$L_{L}$$ find a transformer for which the three relationships are satisfied while, a=10. Use smallest possible inductances you need to make the transformer work properly @ 1kHz and above. Print plot with the three ratios that demonstrate the transformer works at range specified. Repeat this process so that the transformer works at 100kHz and up.
It is given that the coupling coefficient is 1

## Homework Equations

$$Z_{in}$$=$$\frac{R_{L}}{a^{2}}$$

a=$$\frac{V_{L}}{V_{S}}$$

a=$$\frac{I_{S}}{I_{L}}$$

Problem statement goes on to tell me to produce a graph using these equations as traces, note that they want me to graph 1/a instead of a.

$$\frac{1}{a}$$=$$\frac{V_{L}}{V_{S}}$$

$$\frac{1}{a}$$=$$\frac{I_{L}}{I_{S}}$$

$$\frac{1}{a}$$=$$\sqrt{\frac{1}{50}\frac{V_{S}}{I_{S}}}$$

## The Attempt at a Solution

So really, my question boils down to this $$\Rightarrow$$ How is the working frequency of this circuit defined? Here is a graph I made with 8mH and 800mH respectively.

First eqn is straight line @ 100m, second eqn is bottom curved line, 3rd is the middle line.

Now I know (or at least I think I do) that by "work" they mean where it acts as a high-pass filter, since that is what this is. My "real world" insight tells me that the two curved lines will never hit 100m. But, in this circuit we assumed a perfect coupling coefficient. Does that mean these lines actually do intersect? If they do, I would assume that is what is being looked for. Otherwise, I would assume there is one of two things at work.
1) A ratio of $$\frac{Z_{out}}{Z_{in}}$$ that defines when this circuit works. If this is the case, what is this ratio?
2) It has something to do with the resonant (corner) frequency of the filter, for an RL Circuit $$f_{c}$$=$$\frac{R}{2\pi L}$$ But this, to me, wouldn't account for the mutual inductance.

I would prefer insight or direction as opposed to formula's or a road map. But I won't complain. Thanks in advance.

P.S. For those curious, circuit designed in Orcad Capture, simulated in Orcad PSpice, class is Electronic Instrumentation @ RPI

Well, I think I have it figured out.

The low frequency resonance can be described as satisfying this equation:

$$\frac{1}{\sqrt{2}}\approx\frac{1}{\sqrt{1+\left(\frac{R_{pa}}{2*\pi*f*L_{oc}}\right)^{2}}}$$

Where $$R_{pa}$$ is the resistance there would be if the resistors were in parallel ; $$L_{oc}$$ is the Inductor's open circuit inductance and $$f$$ is frequency in Hz.

Using this, I found in the first case for L1: .000156 Henries ; L2: .0156 Henries
For the second case, L1: .00156 Henries ; L2: .156 Henries

## 1. What is the working range of a transformer?

The working range of a transformer refers to the range of input voltage that the transformer can handle while still providing a stable output voltage.

## 2. Why is the working range important?

The working range is important because it determines the wide range of input voltages that the transformer can handle without causing damage or fluctuations in the output voltage.

## 3. How is the working range of a transformer determined?

The working range of a transformer is determined by the transformer's design and its rated voltage. The transformer's design and construction materials determine its maximum voltage rating, while the transformer's rated voltage determines its minimum voltage rating.

## 4. What happens if the input voltage exceeds the working range of a transformer?

If the input voltage exceeds the working range of a transformer, it can cause damage to the transformer and may result in fluctuations or complete failure of the output voltage. This can lead to equipment malfunction or even damage.

## 5. Can a transformer be used outside of its working range?

No, a transformer should only be used within its specified working range. Using a transformer outside of its working range can lead to damage, equipment malfunction, and potential safety hazards.

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