Problem 8 REFLECTED IMPEDANCE How does t... [FREE SOLUTION]

Chapter 18: Problem 8

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 $R_{2}$.

Short Answer

Expert verified

The impedance seen at the primary ($ Z $ as a complex quantity) varies with $ R_2 $ according to the equation $ Z = \frac{R_{2}}{R_{2}^{2}+1}+j \frac{R_{2}^{2}}{R_{2}^{2}+1} $ and its magnitude and phase depend on the value of $ R_2 $.

Step by step solution

Understanding Transformer Impedance Reflection

When a load is connected to the secondary side of a transformer, its resistance ($R_2$) is reflected to the primary side as an impedance due to the transformer's turns ratio. In our case, since one of the simplifying assumptions is that the turns ratio is 1 ($k=1$, where k is the turns ratio), the impedance seen at the primary will be equivalent to the secondary resistance ($R_2$) reflected directly without modification by the turns ratio.

Calculating the Reflected Impedance

Using the given variables $\omega=L_1=L_2=k=1$ and $R_1=0$, we can use the formula for reflected impedance in the primary side of a transformer $Z$, which reflects the secondary side resistance $R_2$ and reactance $X_2$. The impedance is a complex number consisting of a real part (resistance) and an imaginary part (reactance). Since $L_2=1$ and $\omega=1$, the reactance $X_2$ in the secondary would be $\omega L_2 = 1$. Considering the secondary circuit as a series of $R_2$ and $X_2$, according to Ohm's Law, $ Z = R_2 + j X_2 = R_2 + j$. To find the reflected impedance, we apply the current division principle, where $Z = R_2 / (1 + R_2^2) + j (R_2^2 / (1 + R_2^2))$.

Plotting the Curves of R, X, |Z|

To visualize how the resistance and reactance (and thus the magnitude of the impedance) changes as a function of $R_2$, we need to plot the real part $R = R_2 / (R_2^2 + 1)$, the imaginary part $X = R_2^2 / (R_2^2 + 1)$, and the magnitude of the impedance $ |Z| = sqrt(R^2 + X^2) $ over a range of $ R_2 $ values from 0.1 to 10 on a log scale for $ R_2 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transformer Impedance Reflection

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.

Complex Impedance Calculation

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

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.

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Short Answer

Step by step solution

Understanding Transformer Impedance Reflection

Calculating the Reflected Impedance

Plotting the Curves of R, X, |Z|

Key Concepts

Transformer Impedance Reflection

Complex Impedance Calculation

Understanding the Calculated Expression

Impedance Magnitude Plotting

How to Approach Plotting

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