Control Systems and Z-Transforms Quiz

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Questions and Answers

What is the expression for the real part of the function f(z)?

-e^y sin x

e^(-y) sin x

e^y cos x

e^(-y) cos x (correct)

According to the Cauchy-Riemann equations, what is the relationship between $\frac{\partial u}{\partial x}$ and $\frac{\partial v}{\partial y}$?

$\frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial y} = -1$

$\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ (correct)

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$

$\frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial y} = 1$

What is the value of f(x) determined from the equation derived from Cauchy-Riemann?

k (correct)

e^{-x}

e^x

What is the final expression for v when k is set to 0?

e^{-y} sin x Signup and view all the answers

What is the maximum value of the determinant for 2 × 2 real symmetric matrices with trace 14?

49 Signup and view all the answers

What is the formula to calculate the damped natural frequency of an undamped second-order system?

$\omega_d = \omega_n(\sqrt{1 - \xi^2})$ Signup and view all the answers

If the natural frequency of an undamped second-order system is 40 rad/s and the damping ratio is 0.3, what is the value of the damping ratio squared?

0.09 Signup and view all the answers

What effect does increasing the damping ratio have on the damped natural frequency?

It decreases the damped natural frequency. Signup and view all the answers

What would be the damped natural frequency if the damping ratio were increased to 1.0 with the same undamped natural frequency of 40 rad/s?

0 rad/s Signup and view all the answers

For the given transfer function examples, what form does option (3) represent?

Second-order system with a conjugate pair of poles Signup and view all the answers

What is the relationship between the zeros of x(z) indicated by the given information?

If one zero is 0.5 + j0.25, the other is 0.5 - j0.25 Signup and view all the answers

What is the defined property of x(z) in terms of z-transfer time reversal?

x(z) = x(z-1) Signup and view all the answers

How is the power of the nth harmonic component of a periodic square wave calculated?

P_n ∝ n^2 Signup and view all the answers

What is the ratio of the power in the 7th harmonic to the power in the 5th harmonic for the periodic square wave?

0.5 Signup and view all the answers

If the power in the 5th harmonic component of a periodic square wave is 25 units, what would be the power in the 7th harmonic component?

49 units Signup and view all the answers

What form does the z-transform follow according to the z-transfer time reversal property?

X(z) = x(z-1) Signup and view all the answers

Which of the following statements about the zeros of a function is true given one zero is 0.5 + j0.25?

A corresponding conjugate zero must exist. Signup and view all the answers

For a periodic square wave, what does the nth harmonic component power indicate in relation to its frequency?

It increases as the harmonic number increases. Signup and view all the answers

What is the coupling factor (k) when 56% of the total flux from one coil links the other coil?

0.56 Signup and view all the answers

What is the formula used to calculate mutual inductance (M) in magnetically coupled circuits?

M = k * L1 * L2 Signup and view all the answers

If L1 is 4H and L2 is 5H, what is the mutual inductance (M) when k is 0.56?

2.50 H Signup and view all the answers

Given electron mobility μn = 1000 cm²/V-s and concentration gradient of 1 × 10²¹/cm⁴, what type of current is being calculated?

Diffusion current Signup and view all the answers

What is the significance of kT/q being equal to 25 mV in this context?

It represents the thermal energy available to electrons. Signup and view all the answers

What is the value of electronic charge (q) used in these calculations?

1.6 × 10⁻¹⁹ C Signup and view all the answers

What is the unit of measurement for mutual inductance (M)?

Henries Signup and view all the answers

What is the magnitude of electron diffusion current density when the concentration gradient is 1 × 10²¹/cm⁴?

To be calculated using formulas Signup and view all the answers

What is the value of $xp$ calculated from the given equations?

$0.3 , \mu m$ Signup and view all the answers

What is the depletion width formula stated?

$w = \frac{2E_{si}}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right)(V_0 + V_R)$ Signup and view all the answers

Given $V_i = 5sin(wt)$, what is the maximum output voltage $V_o$?

5 Volts Signup and view all the answers

What is the voltage drop across the diode when forward-biased?

0.7 Volts Signup and view all the answers

What are the minimum and maximum output voltages $V_o$ in Volts if $V_i$ is sinusoidal?

–5 and 2.7 Signup and view all the answers

What is the relationship between $N_A$ and $N_D$ in the depletion width formula?

They add together in the denominator. Signup and view all the answers

What does the variable $V_R$ typically represent in diode circuits?

Reverse voltage Signup and view all the answers

What effect does a larger $V_0$ have on the depletion width $w$?

Increases the depletion width. Signup and view all the answers

Study Notes

Z-Transform and Signal Properties

Time reversal property: If ( x(n) = x(-n) ), then the Z-transform relation is ( x(z) = x(z^{-1}) ).
A zero of ( x(z) ) is at ( 0.5 + j0.25 ), implying ( x(0.5 + j0.25) = 0 ).
The reciprocal zero property indicates another zero at ( 0.5 + j0.25 ).

Power Ratio in Harmonics

For a periodic square wave, the power ratio of the 7th harmonic to the 5th harmonic is calculated.
The nth harmonic power of a square wave is proportional to ( \frac{1}{n^2} ).
Ratio derived: ( \frac{P_7}{P_5} = \frac{7^2}{5^2} = 0.5 ).

Natural Frequency and Damping in Control Systems

An undamped second-order system has a natural frequency ( \omega_n = 40 , \text{rad/s} ).
With a damping ratio ( \xi = 0.3 ), the damped natural frequency is calculated using ( \omega_d = \omega_n \sqrt{1 - \xi^2} ).
The result yields ( \omega_d \approx 38.157 , \text{rad/s} ).

Transfer Function in Control Systems

The transfer function for a system where ( X_1(s) = 0 ) is identified.
Possible options for the transfer function included several rational functions.
The correct transfer function option is (B).

Determinant of Symmetric Matrices

The maximum determinant value among all ( 2 \times 2 ) symmetric matrices with trace 14 is sought.
The determinant can be optimized under the constraints given, leading to a calculated value.

Magnetic Coupled Circuits

In a magnetically coupled circuit, 56% of the flux from one coil links to another.
The coupling factor ( k = 0.56 ).
Mutual inductance ( M ) is computed using ( M = k \sqrt{L_1 L_2} ), resulting in ( M = 2.50 , \text{H} ).

Electron Diffusion in Semiconductor Physics

Given parameters include electron charge ( q = 1.6 \times 10^{-19} , C ) and electron mobility ( \mu_n = 1000 , \text{cm}^2/\text{V·s} ).
A concentration gradient of ( 1 \times 10^{21}/\text{cm}^4 ) leads to calculations of electron diffusion current density.
Determination of the depletion width and its dependence on given voltage values occurs.

Diode Circuit Analysis

For a diode with ( V_{on} = 0.7 , \text{V} ) in a circuit with ( V_i = 5\sin(\omega t) , \text{V} ), analysis of min/max values of output voltage ( V_o ) is conducted.
Key answer choices include ranges for ( V_o ) based on circuit behavior.

These notes summarize concepts from signals, control systems, electromagnetism, and semiconductor physics relevant to the stated problems.

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Description

Test your knowledge on Z-Transform properties, harmonic power ratios in signals, and concepts of natural frequency in control systems. This quiz covers essential principles and calculations, including damping ratios and transfer functions. Ideal for those studying control systems and signal processing.