Calculating Transformer Input Voltage

[MechEngJordan]

Homework Statement

Calculate the primary current, and hence the voltage at the transformer input winding, V₁.

Transformation ratio, a = N1/N2 = 0.1
R₁ = 0.12 Ω; R₂ = 12 Ω
X₁ = 0.4 Ω ; X₂ = 40 Ω
R_C = 560 Ω
V₂ = 2300 V
R_L = 1 kΩ
'X_m = 800 Ω'

Homework Equations

R_1eq = R₁ + a²R₂
X_1eq = j(X₁ + a²X₂)
V₂' = aV₂

Current divider equation.

The Attempt at a Solution

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Approximate equivalent circuit:

R_1eq = 0.12 + (0.1)²(12)
R_1eq = 0.24 ΩX_1eq = j(0.4 + (0.1)²(40))
X_1eq =j0.8 Ω

V₂' = 0.1(2300)
V₂' = 230 V

a²Z_L = (0.1)²(1000)
a²Z_L = 10 Ω

I_p = V₂'/a²Z_L
I_p = 23 AFrom here is where I believed that there is likely a more efficient way to solve the problem -- particularly because the value of X_m was not actually given in the paper, but told to us during the tutorial, more-or-less made up on the spot.

Here is the outline of the given solution:

I_p= I₁*Z₂/(Z₁+Z₂)

⇒I₁ = I_p*(Z₁+Z₂)/Z₂

⇒I₁ = 23*(R_c // X_m + X_1eq + R_1eq + a²Z_L)/ (R_c // X_m)I₀ = I₁ - I_p

∴ V₁ = I₀(R_c // X_m)

I'd be thankful for any input.

 

[Hesch]

From here is where I believed that there is likely a more efficient way to solve the problem

It seems that in the given solution another equivalent is used (T-equivalent) than the one in #1. It's hard to explain what happens in that other equivalent because I cannot see what is meant by Z₁ and Z₂.

Don't you like to calculate with complex numbers? I don't know if you find complex calculations "more efficient", but they will be more structured. Example:

Z₀ = (R_c || jX_m) = (560 || j800) = (375.8 + j263). That's it.

In the same way you calculate other impedances, that can be used in Kirchhoffs laws, etc.

It becomes simple as in the given solution: I_p= I₁*Z₂/(Z₁+Z₂). As for the numerical calculation I assume a calculator can do the job for you.

 

[MechEngJordan]

It seems that in the given solution another equivalent is used (T-equivalent) than the one in #1. It's hard to explain what happens in that other equivalent because I cannot see what is meant by Z₁ and Z₂.

Don't you like to calculate with complex numbers? I don't know if you find complex calculations "more efficient", but they will be more structured. Example:

Z₀ = (R_c || jX_m) = (560 || j800) = (375.8 + j263). That's it.

In the same way you calculate other impedances, that can be used in Kirchhoffs laws, etc.

It becomes simple as in the given solution: I_p= I₁*Z₂/(Z₁+Z₂). As for the numerical calculation I assume a calculator can do the job for you.

Hi,

Thanks for the reply.

I should have probably clarified that in this case,

Z₁ = R_c // X_m

and

Z₂ = X_1eq + R_1eq + a²Z_L

I have no issues with complex numbers or the numerical computations; I suppose what I was referring to by 'efficient' was that in the original question, there was no value of X_m given (this was only given during the tutorial and seemed to be done off the cuff). This gave me the impression that the solution to this problem should have been different than the one given. Also, this was a 5-mark question and relative to other 5 mark questions, the given approach seemed like a lot of work.