REFLECTED IMPEDANCE How does the impedance at the terminals of the primary of a transformer vary with the resistance \(R_{2}\) in the secondary circuit? Consider a simple case where \(\omega=L_{1}=L_{2}=k=1, R_{1}=0\). a) Show that $$ Z=\frac{R_{2}}{R_{2}^{2}+1}+j \frac{R_{2}^{2}}{R_{2}^{2}+1} $$ (Note that, dimensionally, this equation is absurd. The reason is that we have assigned numerical values to some of the parameters. For example, the 1 in the denominators. is really \(\omega^{2} L_{2}^{2}\), b) Draw curves of \(R, X,|Z|\) as functions of \(R_{2}\), for values of \(R_{2}\) ranging from \(0.1\) to 10 . Use a log scale for \(R_{2}\).

One of the fundamental concepts in electrical engineering is the principle of transformer impedance reflection. When you have a transformer and you're trying to figure out how it will behave under varying conditions, understanding how impedance behaves is crucial.

Imagine you have connected a load with a certain resistance, denoted as \( R_2 \), to the secondary winding of a transformer. Due to the nature of transformers and their windings, this resistance doesn't stay confined to the secondary side. It gets reflected back to the primary side as an impedance. This is vital when designing and analyzing circuits because it changes how the primary side sees the electrical load.

In the simple case where the turns ratio is 1, the reflection is direct and unmodified. This means that whatever resistance is on the secondary side, the primary side will see the same value but in the form of impedance. This becomes even more complex when we consider non-unity turns ratios, requiring a multiplication or division by the square of the turns ratio to find the reflected impedance. Recognizing this relationship helps prevent overloading equipment and allows one to precisely tailor circuits for specific responses.

Let's dive deeper into the realm of complex impedance calculation. Impedance is not just a simple resistance value; it is a complex number that encompasses both resistance and reactance. Complex impedance is expressed as \( Z = R + jX \), where \( R \) is the real part, representing the resistance, and \( jX \) is the imaginary part, representing the reactance, with \( j \) being the imaginary unit.

In the exercise, we consider a special case where \( \(omega\)\), \( L_1 \), and \( L_2 \) are all equal to 1, and the primary resistance \( R_1 \) is set to 0. Under these conditions, the complex impedance in the primary circuit is directly determined by the resistance in the secondary circuit \( R_2 \) as shown in the given formula.

Understanding the Calculated Expression

The real and imaginary parts of the impedance are computed based on the resistance \( R_2 \). The fact that the reactance of the secondary coil is equal to the resistance \( R_2 \) squared over the sum of \( R_2 \) squared plus 1 is grounded in the physics of inductive reactance and is crucial for understanding the behavior under AC current. In a real-world scenario, you would replace the '1' in the denominators with \( \(omega\)^2 L_2^2 \) to account for different frequencies and inductance values.

By mastering the art of complex impedance calculation, students can accurately analyze AC circuits, predict their behavior, and ensure their designs will function correctly under any given set of conditions.

Impedance magnitude plotting is a graphical representation that offers a visual understanding of how impedance changes with another variable, in our case \( R_2 \), the resistance in the secondary winding. By plotting these values, one can quickly identify the relationship between resistance and the total impedance seen by the primary winding of the transformer.

To illustrate this, let's refer back to the exercise. We're tasked with plotting the magnitude of impedance \( |Z| \), resistance \( R \), and reactance \( X \) as functions of \( R_2 \). This can be quite informative as it shows us, at a glance, how the impedance behavior varies with resistance in a logarithmic scale, which is particularly useful for a wide range of values.

How to Approach Plotting

To plot these curves, one would typically use software capable of handling complex numbers and log scales. However, the process remains the same: for each value of \( R_2 \), calculate \( R \), \( X \), and \( |Z| = \sqrt{R^2 + X^2} \). Then, plot these calculated points on a graph with a log scale for the horizontal axis representing \( R_2 \) and a linear scale for the vertical axis representing the impedance, resistance, and reactance.

This plotting task illuminates the nonlinear relationship between the impedance components and the resistance in the secondary circuit, providing an invaluable tool for predicting circuit behavior. Understanding this graphing process is essential for future engineers and technicians who may need to quickly assess the impact of changes in circuit components on overall performance.