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Magazine Chapter 3: Problem 14
The input impedance \(Z\) of a particular network is related to the terminating impedance \(z\) by the equation] $$ Z=\frac{(1+j) z-2+j 4}{z+1+j} $$Find \(Z\) when \(z=0,1\) and \(j \Omega\) and sketch the variation of \(|Z|\) and \(\arg Z\) as \(z\) moves along the positive real axis from the origin.
Short Answer
Expert verified
For \( z = 0 \), \( Z = 1 + 3j \); for \( z = 1 \), \( Z = \frac{3}{5} + j \frac{11}{5} \); for \( z = j \), \( Z = \frac{7}{5} + j \frac{11}{5} \). |Z| and Arg Z vary along the real axis.
Step by step solution
01
Substitute z = 0
We begin by substituting \( z = 0 \) into the given equation for \( Z \):\[ Z = \frac{(1+j) \times 0 - 2 + j4}{0 + 1 + j} \]Simplifying the numerator, we get:\[ Z = \frac{-2 + j4}{1 + j} \].
02
Simplify for z = 0
Next, we'll simplify the expression by multiplying numerator and denominator by the conjugate of the denominator:\( 1+j \) has a conjugate \( 1-j \).\[ Z = \frac{(-2 + j4)(1 - j)}{(1 + j)(1 - j)} \]\[ Z = \frac{(-2 + 2j + j4 - j^2 4)}{1-j^2} \]Knowing that \(j^2 = -1\), the denominator becomes 2:\[ Z = \frac{-2 + 2j + j4 + 4}{2} \]\[ Z = \frac{2 + 6j}{2} = 1 + 3j \].
03
Substitute z = 1
Substitute \( z = 1 \) into the equation:\[ Z = \frac{(1+j) \times 1 - 2 + j4}{1 + 1 + j} \].Simplify:\[ Z = \frac{1 + j - 2 + j4}{2 + j} \]\[ Z = \frac{-1 + 5j}{2 + j} \].
04
Simplify for z = 1
Multiply the expression by the conjugate of the denominator:\[ Z = \frac{(-1 + 5j)(2 - j)}{(2+j)(2-j)} \]\[ Z = \frac{-2 + j + 10j - 5j^2}{4 - j^2} \]\[ Z = \frac{-2 + 11j + 5}{4+1} \]\[ Z = \frac{3 + 11j}{5} = \frac{3}{5} + j \frac{11}{5} \].
05
Substitute z = j
Next, substitute \( z = j \):\[ Z = \frac{(1+j) \times j - 2 + j4}{j + 1 + j} \]Simplify:\[ (1+j)j = j - 1 \]\[ Z = \frac{j - 1 - 2 + j4}{1 + 2j} \]\[ Z = \frac{-3 + j5}{1 + 2j} \].
06
Simplify for z = j
Simplify using the conjugate:\[ Z = \frac{(-3 + j5)(1 - 2j)}{5} \]Multiply the numerator and simplify:\[ Z = \frac{-3 + 6j + j5 - j^2 10}{5} \]\[ Z = \frac{-3 + 6j + 5j + 10}{5} \]\[ Z = \frac{7 + 11j}{5} = \frac{7}{5} + j \frac{11}{5} \].
07
Sketch |Z| and Arg Z
For \( z \) moving along the positive real axis from 0, we first plot the absolute value \(|Z|\) calculated as \( \sqrt{ \left( \text{real part} \right)^2 + \left( \text{imaginary part} \right)^2 } \).Similarly, the argument \( \arg(Z) \) can be determined using \( \tan^{-1}\left( \frac{\text{imaginary part}}{\text{real part}}\right) \).Substitute values of \( z \) varying from 0 to 10 to observe the trend in \(|Z|\) and \(\arg Z\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Impedance Transformation
Impedance transformation is a concept in electrical engineering that deals with the change in impedance from one part of an electrical circuit to another. The input impedance (
(Z)
) of a network can be different from the terminating impedance (
(z)
) due to the reactive components like capacitors and inductors that introduce phase differences and magnitude changes. This transformation becomes crucial in designing circuits to ensure maximum power transfer, reduce reflections in transmission lines, and other technical requirements.
By using the formula provided, (Z) can be calculated for different values of (z) . For example, when (z = 0, 1, ext{ and } j) , the impedance transformation can be determined through substitution into the given equation. The process involves the application of complex numbers and algebraic simplifications, often requiring steps like multiplying by the conjugate to achieve a real denominator.
By using the formula provided, (Z) can be calculated for different values of (z) . For example, when (z = 0, 1, ext{ and } j) , the impedance transformation can be determined through substitution into the given equation. The process involves the application of complex numbers and algebraic simplifications, often requiring steps like multiplying by the conjugate to achieve a real denominator.
- Impedance matching helps to ensure that maximum power is transferred from the source to the load.
- In transmission lines, impedance transformation helps minimize the reflection of signals, which is crucial for effective wiring of data, radio frequencies, etc.
Complex Numbers
Complex numbers are used extensively in electrical engineering to handle circuit quantities and analysis, including impedance transformation. Represented often as
(a + bj)
, complex numbers consist of a real part (
a
) and an imaginary part (
bj
), where
(j)
is the imaginary unit with the property that
(j^2 = -1)
.
By expressing impedances in complex form, engineers can effectively analyze AC circuits. The real part typically represents resistive elements, while the imaginary part represents reactive elements due to capacitors and inductors. In the given problem, the impedance (Z) expression involves manipulating complex numbers to find its real and imaginary parts for different values of (z) .
By expressing impedances in complex form, engineers can effectively analyze AC circuits. The real part typically represents resistive elements, while the imaginary part represents reactive elements due to capacitors and inductors. In the given problem, the impedance (Z) expression involves manipulating complex numbers to find its real and imaginary parts for different values of (z) .
- Mathematically, complex arithmetic is fundamental to simplifying expressions through operations such as addition, multiplication, and division.
- Conjugates of complex numbers are frequently used to rescale expressions and eliminate imaginary denominators.
- Understanding polar and rectangular forms of complex numbers simplifies tasks such as calculating magnitudes and phases in circuit analysis.
Electrical Engineering
Electrical engineering involves the study and application of electricity, electronics, and electromagnetism. Within this field, impedance, a measure of the opposition that a circuit presents to a current when a voltage is applied, is a key concept. Impedance is scalar in DC circuits but becomes complex in AC circuits due to the behavior of circuit components like inductors and capacitors.
Electrical engineers use complex impedance to analyze how circuits behave over different frequencies and how signals propagate through these networks. It's essential for determining resonance, bandwidth, and the stability of systems, impacting real-world applications such as telecommunications, power management, and electronic device design.
Electrical engineers use complex impedance to analyze how circuits behave over different frequencies and how signals propagate through these networks. It's essential for determining resonance, bandwidth, and the stability of systems, impacting real-world applications such as telecommunications, power management, and electronic device design.
- Impedance involves not just resistance but also consider factors like inductive and capacitive reactance.
- Knowledge in impedance and its transformation allows engineers to optimize circuit performance and predict system behaviors accurately.
- Practical tools like oscilloscopes and vector network analyzers help in visualizing impedance characteristics in laboratories and field tests.
Phasor Analysis
Phasor analysis is a technique used in electrical engineering to analyze sinusoidally varying signals by representing them as rotating vectors (phasors). This method converts differential equations into algebraic ones, simplifying the process to handle AC circuit calculations using complex numbers.
By representing voltages and currents as phasors, engineers can easily add, subtract, and manipulate them to understand circuit behavior under various sinusoidal inputs. In this exercise, phasor analysis is applied in solving for the impedance (Z) , because both the magnitude and phase of the signals are important in determining how the circuit behaves.
By representing voltages and currents as phasors, engineers can easily add, subtract, and manipulate them to understand circuit behavior under various sinusoidal inputs. In this exercise, phasor analysis is applied in solving for the impedance (Z) , because both the magnitude and phase of the signals are important in determining how the circuit behaves.
- Phasor representation links the time-varying nature of sinusoidal functions with their steady-state solution.
- When using phasors, the frequency of signals is often assumed constant, meaning calculations are carried out within the frequency domain.
- Phasor analysis is extremely useful in power systems engineering and in analyzing signals in communication systems.
