How Do Transformers and Resonance Affect AC Circuits?

[catfisherman]

A 240 RMS voltage source is connected to a step down transformer with primary windings consisting of 2000 turns or wraps. A secondary set of windings consist of 500 turns or wraps and is connected to an inductor of 0.03 Henry's, a 0.004 farad capacitor and a 120 ohm resistor. Determine the following:
a. The Maximum induced voltage in the secondary coils.
b. The frequency of the ac source required to produce maximum induced current in the secondary windings


Any help would be greatly appreciated! Thanks!

 

[dynamicsolo]

The deal here at the Homework Forum is that you have to show an attempt to solve a problem before you'll get much help.

So saying, I will say two things about the parts of this problem:

a) The voltage source uses alternating current (that's why the voltage is given as Root-Mean-Square), so you'll need to find its peak voltage. What is the relationship between the voltages in the two coils in a transformer?

b) The secondary coil is connected to an RLC circuit. If the coil and circuit are to have maximum response to being driven by the induced voltage via the transformer from the voltage source, what frequency in the circuit will make this possible?

 

[Moetasim]

I would approach this problem by first understanding the basic principles of electromagnetic induction and transformer operation. From the given information, we can calculate the turns ratio of the transformer as 2000:500, which is 4:1. This means that for every 4 turns in the primary winding, there is 1 turn in the secondary winding.

a. Using the turns ratio, we can determine the maximum induced voltage in the secondary coils by multiplying the RMS voltage of 240 by the turns ratio of 4. This gives us a maximum induced voltage of 960 volts in the secondary coils.

b. To determine the frequency required for maximum induced current, we can use the formula for reactance of an inductor and capacitor in series, X = 1/2πfC, where X is the total reactance, f is the frequency, and C is the capacitance. We can also use Ohm's law to calculate the total impedance of the circuit, Z = √(R^2 + (X_L - X_C)^2), where R is the resistance, X_L is the reactance of the inductor, and X_C is the reactance of the capacitor.

Since we are looking for the frequency that will produce the maximum induced current, we want to minimize the total impedance of the circuit. This occurs when X_L = X_C, or when the reactances of the inductor and capacitor are equal.

Substituting this into the formula for reactance, we get X = 1/2πfC = R. Solving for f, we get f = 1/(2π√(LC)) = 1/(2π√(0.03*0.004)) = 218.2 Hz.

Therefore, the frequency of the AC source required to produce maximum induced current in the secondary windings is approximately 218.2 Hz.

I hope this explanation helps. Please let me know if you have any further questions.