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1201
GATE ECE 2006 | Question: 19
The electric field of an electomagnetic wave propagating in the positive $z$-direction is given by $ \left.\text{E}=\hat{a_x} \sin (\omega t-\beta z\right)+\hat{a_y} \sin \left(\omega t-\beta z+\frac{\pi}{2}\right) $ The wave is linearly polarized in the $z$-direction elliptically polarized left-hand circularly polarized right-hand circularly polarized
The electric field of an electomagnetic wave propagating in the positive $z$-direction is given by $$ \left.\text{E}=\hat{a_x} \sin (\omega t-\beta z\right)+\hat{a_y} \si...
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1202
GATE ECE 2006 | Question: 20
A transmission line is feeding $1 \mathrm{Watt}$ of power to a horn antenna having a gain of $10 \mathrm{~dB}$. The antenna is matched to the transmission line. The total power radiated by the horn antenna into the free-space is $10$ Watts $1$ Watt $0.1$ Watt $0.01$ Watt
A transmission line is feeding $1 \mathrm{Watt}$ of power to a horn antenna having a gain of $10 \mathrm{~dB}$. The antenna is matched to the transmission line. The total...
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1203
GATE ECE 2006 | Question: 21
The eigenvalues and the corresponding eigen vectors of a $2 \times 2$ ... $\left[\begin{array}{ll}4 & 8 \\ 8 & 4\end{array}\right]$
The eigenvalues and the corresponding eigen vectors of a $2 \times 2$ matrix are given by$$\begin{array}{cc} \textbf{Eigenvalue}& \textbf{Eigenvector} \\ \lambda_1=8 & \t...
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GATE ECE 2006 | Question: 22
For the function of a complex variable $\text{W}=\ln \text{Z}\; ($where, $\mathrm{W}=u+j \mathrm{v}$ and $\mathrm{Z}=x+j y),$ the $u=$ constant lines get mapped in $\text{Z}$-plane as set of radial straight lines set of concentric circles set of confocal hyperbolas set of confocal ellipses
For the function of a complex variable $\text{W}=\ln \text{Z}\; ($where, $\mathrm{W}=u+j \mathrm{v}$ and $\mathrm{Z}=x+j y),$ the $u=$ constant lines get mapped in $\text...
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1205
GATE ECE 2006 | Question: 23
The value of the contour integral $\displaystyle{}\oint_{\mid z-j \mid =2} \;\frac{1}{z^2+4} d z$ in positive sense is $\frac{j \pi}{2}$ $-\frac{\pi}{2}$ $-\frac{j \pi}{2}$ $\frac{\pi}{2}$
The value of the contour integral $\displaystyle{}\oint_{\mid z-j \mid =2} \;\frac{1}{z^2+4} d z$ in positive sense is$\frac{j \pi}{2}$$-\frac{\pi}{2}$$-\frac{j \pi}{2}$$...
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1206
GATE ECE 2006 | Question: 24
The integral $\displaystyle{}\int_0^\pi \sin ^3 \theta\; d \theta$ is given by $\frac{1}{2}$ $\frac{2}{3}$ $\frac{4}{3}$ $\frac{8}{3}$
The integral $\displaystyle{}\int_0^\pi \sin ^3 \theta\; d \theta$ is given by$\frac{1}{2}$$\frac{2}{3}$$\frac{4}{3}$$\frac{8}{3}$
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1207
GATE ECE 2006 | Question: 25
Three companies $\text{X, Y}$ and $\text{Z}$ ... computer is defective, the probability that it was supplied by $\text{Y}$ is $0.1$ $0.2$ $0.3$ $0.4$
Three companies $\text{X, Y}$ and $\text{Z}$ supply computers to a university. The percentage of computers supplied by them and the probability of those being defective a...
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1208
GATE ECE 2006 | Question: 26
For the matrix $\left[\begin{array}{ll}4 & 2 \\ 2 & 4\end{array}\right]$, the eigen value corresponding to the eigenvector $\left[\begin{array}{l}101 \\ 101\end{array}\right]$ is $2$ $4$ $6$ $8$
For the matrix $\left[\begin{array}{ll}4 & 2 \\ 2 & 4\end{array}\right]$, the eigen value corresponding to the eigenvector $\left[\begin{array}{l}101 \\ 101\end{array}\ri...
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1209
GATE ECE 2006 | Question: 27
For the differential equation $\dfrac{d^2 y}{d x^2}+k^2 y=0$, the boundary conditions are $y=0$ for $x=0$, and $y=0$ for $x=a$ The form of non-zero solutions of $y$ (where $m$ ... $y=\displaystyle{}\sum_m\;\mathrm{~A}_{m} \;e^{-\frac{m \pi x}{a}}$
For the differential equation $\dfrac{d^2 y}{d x^2}+k^2 y=0$, the boundary conditions are$y=0$ for $x=0$, and$y=0$ for $x=a$The form of non-zero solutions of $y$ (where $...
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1210
GATE ECE 2006 | Question: 28
Consider the function $f(t)$ having Laplace transform $ \text{F}(s)=\frac{\omega_0}{s^2+\omega_0^2} \operatorname{Re}[s]>0 $ The final value of $f(t)$ would be $0$ $1$ $-1 \leq f(\infty) \leq 1$ $\infty$
Consider the function $f(t)$ having Laplace transform $$ \text{F}(s)=\frac{\omega_0}{s^2+\omega_0^2} \operatorname{Re}[s]>0 $$ The final value of $f(t)$ would be$0$$1$$-1...
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1211
GATE ECE 2006 | Question: 29
As $x$ is increased from $-\infty$ to $\infty$, the function $ f(x)=\frac{e^x}{1+e^x} $ monotonically increases monotonically decreases increases to a maximum value and then decreases decreases to a minimum value and then increases
As $x$ is increased from $-\infty$ to $\infty$, the function $$ f(x)=\frac{e^x}{1+e^x} $$monotonically increasesmonotonically decreasesincreases to a maximum value and th...
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1212
GATE ECE 2006 | Question: 30
A two-port network is represented by $\text{ABCD}$ ... $\frac{\mathrm{B}+\mathrm{AR}_{\mathrm{L}}}{\mathrm{D}+\mathrm{CR}_{\mathrm{L}}}$
A two-port network is represented by $\text{ABCD}$ parameters given by$$ \left[\begin{array}{c} \mathrm{V}_1 \\ \mathrm{I}_1 \end{array}\right]=\left[\begin{array}{ll} \m...
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1213
GATE ECE 2006 | Question: 31
In the two port network shown in the figure below, $z_{12}$ and $z_{21}$ are, respectively $r_e$ and $\beta r_o$ $0$ and $-\beta r_o$ $0,$ and $\beta r_o$ $r_e$ and $-\beta r_o$
In the two port network shown in the figure below, $z_{12}$ and $z_{21}$ are, respectively$r_e$ and $\beta r_o$$0$ and $-\beta r_o$$0,$ and $\beta r_o$$r_e$ and $-\beta r...
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1214
GATE ECE 2006 | Question: 32
The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by $\text{RL}$ network only $\text{RC}$ network only $\text{LC}$ network only $\mathrm{RC}$ as well as $\mathrm{RL}$ networks
The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero...
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1215
GATE ECE 2006 | Question: 33
A $2 \; \mathrm{mH}$ inductor with some initial current can be represented as shown below, where $s$ is the Laplace Transform variable. The value of initial current is $0.5 \mathrm{~A}$ $2.0 \mathrm{~A}$ $1.0 \mathrm{~A}$ $0.0 \mathrm{~A}$
A $2 \; \mathrm{mH}$ inductor with some initial current can be represented as shown below, where $s$ is the Laplace Transform variable. The value of initial current is$0...
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1216
GATE ECE 2006 | Question: 34
In the figure shown, assume that all the capacitors are initially uncharged. If $\text{V}_i(t)=10 u(t)$ Volts, then $\text{V}_0(t)$ is given by $8 e^{-0.004 t}$ Volts $8\left(1-e^{-0.004 t}\right)$ Volts $8 u(t)$ Volts $8$ Volts
In the figure shown, assume that all the capacitors are initially uncharged. If $\text{V}_i(t)=10 u(t)$ Volts, then $\text{V}_0(t)$ is given by$8 e^{-0.004 t}$ Volts$8\le...
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1217
GATE ECE 2006 | Question: 35
Consider two transfer functions $ \mathrm{G}_1(s)=\frac{1}{s^2+a s+b} \text { and } \mathrm{G}_2(s)=\frac{s}{s^2+a s+b} $ The $3\text{-dB}$ bandwidths of their frequency responses are, respectively $\sqrt{a^2-4 b}, \sqrt{a^2+4 b}$ $\sqrt{a^2+4 b}, \sqrt{a^2-4 b}$ $\sqrt{a^2-4 b}, \sqrt{a^2-4 b}$ $\sqrt{a^2+4 b}, \sqrt{a^2+4 b}$
Consider two transfer functions $$ \mathrm{G}_1(s)=\frac{1}{s^2+a s+b} \text { and } \mathrm{G}_2(s)=\frac{s}{s^2+a s+b} $$ The $3\text{-dB}$ bandwidths of their frequenc...
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1218
GATE ECE 2006 | Question: 36
A negative resistance $R_{\text {neg }}$ is connected to a passive network $N$ having driving point impedance $Z_{1}$ (s) as shown below. For $Z_{2}(s)$ ... $\left|\text{R}_{\text {neg }}\right| \leq \angle Z_{1}(j \omega), \forall \omega$
A negative resistance $R_{\text {neg }}$ is connected to a passive network $N$ having driving point impedance $Z_{1}$ (s) as shown below. For $Z_{2}(s)$ to be positive re...
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1219
GATE ECE 2006 | Question: 37
In the circuit shown below, the switch wa connected to position $1$ at $t<0$ and at $t=0$, it is changed to position $2$. Assume that $y$ the diode has zero voltage drop and a storage time $t_{s}$. For $0 < t \leq t_{s}, \text{V}_{\text{R}}$ is given by ... $\mathrm{V}_{\mathrm{R}}=+5$ $0 \leq \mathrm{V}_{\mathrm{R}}<5$ $-5 < \mathrm{V}_{\mathrm{R}} < 0$
In the circuit shown below, the switch wa connected to position $1$ at $t<0$ and at $t=0$, it is changed to position $2$. Assume that $y$ the diode has zero voltage drop ...
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GATE ECE 2006 | Question: 38
The majority carriers in an $n$-type semiconductor have an average drift velocity $\mathrm{V}$ in a direction perpendicular to a uniform magnetic field $\text{B}.$ The electric field $\mathrm{E}$ induced due to Hall effect acts in the direction $\mathrm{V} \times \mathrm{B}$ $\mathrm{B} \times \mathrm{V}$ along $\text{V}$ opposite to $\mathrm{V}$
The majority carriers in an $n$-type semiconductor have an average drift velocity $\mathrm{V}$ in a direction perpendicular to a uniform magnetic field $\text{B}.$ The el...
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GATE ECE 2006 | Question: 39
Find the correct match between Group $1$ and Group $2$ ... $\text{E - 3, F - 4, G - 1, H - 2}$ $\text{E - 1, F - 3, G - 2, H - 4}$
Find the correct match between Group $1$ and Group $2$$$\begin{array}{ll}\qquad \textbf{Group 1} & \qquad \textbf{Group 2} \\\text{E. Varactor diode} & \text{1. Voltage r...
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1222
GATE ECE 2006 | Question: 40
A heavily doped $n$-type semiconductor has the following data Hole-electron mobility ratio : $0.4$ Doping concentration : $4.2 \times 10^{8} \; \mathrm{atoms/m}^{3}$ Intrinsic concentration : $1.5 \times 10^{4} \; \mathrm{atoms/m}^{3}$ The ... that of the intrinsic semiconductor of same material and at the same temperature is given by $0.00005$ $2,000$ $10,000$ $20,000$
A heavily doped $n$-type semiconductor has the following dataHole-electron mobility ratio : $0.4$Doping concentration : $4.2 \times 10^{8} \; \mathrm{atoms/m}^{3}$Intrins...
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1223
GATE ECE 2006 | Question: 41
For the circuit shown in the following figure, the capacitor $C$ is initially uncharged. At $t=0$, the switch $S$ is closed. The voltage $V_{C}$ across the capacitor at $t=1$ millisecond is In the figure shown above, the $\text{OP AMP}$ ... $\nabla$. $0$ Volts $6.3$ Volts $9.45$ Volts $10$ Volts
For the circuit shown in the following figure, the capacitor $C$ is initially uncharged. At $t=0$, the switch $S$ is closed. The voltage $V_{C}$ across the capacitor at $...
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1224
GATE ECE 2006 | Question: 42
For the circuit shown below, assume that the zener diode is ideal with a breakdown voltage of $6$ Volts. The waveform observed across $R$ is
For the circuit shown below, assume that the zener diode is ideal with a breakdown voltage of $6$ Volts. The waveform observed across $R$ is
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1225
GATE ECE 2006 | Question: 44
An $\frac{\mathrm{I}}{\mathrm{O}}$ peripheral device shown in Figure $(b)$ below is to be interfaced to an $8085$ microprocessor. To select the $\frac{\mathrm{I}}{\mathrm{O}}$ device in the $\frac{\mathrm{I}}{\mathrm{O}}$ ... $(a)$ below output $7$ output $5$ output $2$ output $0$
An $\frac{\mathrm{I}}{\mathrm{O}}$ peripheral device shown in Figure $(b)$ below is to be interfaced to an $8085$ microprocessor. To select the $\frac{\mathrm{I}}{\mathrm...
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1226
GATE ECE 2006 | Question: 45
For the circuit shown in figure below, two $4$ ... $\mathrm{S}=1, \mathrm{C}_{0}=0$ $\mathrm{S}=1, \mathrm{C}_{0}=1$
For the circuit shown in figure below, two $4$-bit parallel-in serial-out shift registers loaded with the data shown are used to feed the data to a full adder. Initially,...
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1227
GATE ECE 2006 | Question: 46
A $4$-bit D/A converter is connected to a free-running $3$-bit UP counter, as shown in the following figure. Which of the following waveforms will be observed at $V_{0}$? In the figure shown above, the ground has been shown by the symbol $\nabla$
A $4$-bit D/A converter is connected to a free-running $3$-bit UP counter, as shown in the following figure. Which of the following waveforms will be observed at $V_{0}$?...
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1228
GATE ECE 2006 | Question: 47
Two $\text{D}$-flip-flops, as shown below, are to be connected as a synchronous counter that goes through the following $\mathrm{Q}_{1} \mathrm{Q}_{0}$ sequence $00 \longrightarrow 01 \longrightarrow 11 \longrightarrow 10 \longrightarrow 00 \longrightarrow \dots$ ... $\mathrm{Q}_{1} \mathrm{Q}_{0}$
Two $\text{D}$-flip-flops, as shown below, are to be connected as a synchronous counter that goes through the following $\mathrm{Q}_{1} \mathrm{Q}_{0}$ sequence $00 \long...
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1229
GATE ECE 2006 | Question: 48
Following is the segment of a $8085$ assembly language program $\text{LXISP,EFFFH}$ $\text{CALL 3000 H}$ $3000 \mathrm{H}: \mathrm{LXIH}, 3 \mathrm{CF} 4 \mathrm{H}$ $\text{PUSH PSW}$ $\text{SPIH}$ $\text{POP PSW}$ $\text{RET}$ On completion of $\text{RET}$ execution ... of $\mathrm{SP}$ is $3 \text{CFO H}$ $3 \mathrm{CF} 8 \; \mathrm{H}$ $\text{EFFD H}$ $\text{EFFF H}$
Following is the segment of a $8085$ assembly language program$\text{LXISP,EFFFH}$$\text{CALL 3000 H}$$3000 \mathrm{H}: \mathrm{LXIH}, 3 \mathrm{CF} 4 \mathrm{H}$$\text{P...
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GATE ECE 2006 | Question: 49
The point $\text{P}$ in the following figure is stuck-at-$1$. The output $f$ will be $\overline{\mathrm{AB} \overline{\mathrm{C}}}$ $\overline{\mathrm{A}}$ $\mathrm{AB} \overline{\mathrm{C}}$ $\mathrm{A}$
The point $\text{P}$ in the following figure is stuck-at-$1$. The output $f$ will be$\overline{\mathrm{AB} \overline{\mathrm{C}}}$$\overline{\mathrm{A}}$$\mathrm{AB} \ove...
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1231
GATE ECE 2006 | Question: 50
A signal $m(t)$ with bandwidth $500 \mathrm{~Hz}$ is first multiplied by a signal $g(t)$ where \[ g(t)=\sum_{\mathrm{R}=-\infty}^{\infty}(-1)^{k} \delta\left(t-0.5 \times 10^{-4} k\right) \] The resulting signal is then passed through an ideal ... with bandwidth $1 \; \mathrm{kHz}$. The output of the lowpass filter would be $\delta(t)$ $m(t)$ $0$ $m(t) \delta(t)$
A signal $m(t)$ with bandwidth $500 \mathrm{~Hz}$ is first multiplied by a signal $g(t)$ where\[ g(t)=\sum_{\mathrm{R}=-\infty}^{\infty}(-1)^{k} \delta\left(t-0.5 \times ...
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1232
GATE ECE 2006 | Question: 51
The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion \[ x(t)=5\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{3}+7\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{2} \] would be $2 \times 10^{3}$ $4 \times 10^{3}$ $6 \times 10^{3}$ $8 \times 10^{3}$
The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion\[ x(t)=5\left(\frac{\sin 2 \pi 1000 t}{\p...
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1233
GATE ECE 2006 | Question: 52
A uniformly distributed random variable $X$ with probability density function \[ f_{\gamma}(x)=\frac{1}{10}(t r(x+5)-u(x-5)) \] where $u(.)$ is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random ...
A uniformly distributed random variable $X$ with probability density function\[ f_{\gamma}(x)=\frac{1}{10}(t r(x+5)-u(x-5)) \]where $u(.)$ is the unit step function is pa...
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GATE ECE 2006 | Question: 53
A system with input $x[n]$ and output $y[n]$ is given as $y[n]=\left(\sin \frac{5}{6} \pi n\right) x(n)$. The system is linear, stable and invertible non-linear, stable and non-invertible linear, stable and non-invertible linear, unstable and invertible
A system with input $x[n]$ and output $y[n]$ is given as $y[n]=\left(\sin \frac{5}{6} \pi n\right) x(n)$.The system islinear, stable and invertiblenon-linear, stable and ...
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GATE ECE 2006 | Question: 54
The unit-step response of a system starting from rest is given by $\qquad C(t)=1-t^{-2 t} \text { for } t \geq 0$ The transfer function of the system is $\frac{1}{1+2 s}$ $\frac{2}{2+s}$ $\frac{1}{2+s}$ $\frac{2 s}{1+2 s}$
The unit-step response of a system starting from rest is given by$\qquad C(t)=1-t^{-2 t} \text { for } t \geq 0$The transfer function of the system is$\frac{1}{1+2 s}$$\f...
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1236
GATE ECE 2006 | Question: 55
The Nyquist plot of $\mathrm{G}(j \omega) \mathrm{H}(j \omega)$ for a closed loop control system, passes through $(-1, j 0)$ point in the $\mathrm{GH}$ plane. The gain margin of the system in $\mathrm{dB}$ is equal to infinite greater than zero. less than zero zero
The Nyquist plot of $\mathrm{G}(j \omega) \mathrm{H}(j \omega)$ for a closed loop control system, passes through $(-1, j 0)$ point in the $\mathrm{GH}$ plane. The gain ma...
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GATE ECE 2006 | Question: 56
The positive values of $\text{“K”}$ and “$a$” so that the system shown in the figure below oscillates at a frequency of $2 \mathrm{rad} / \mathrm{sec}$ respectively are $1,0.75$ $2, 0.75$ $1,1$ $2,2$
The positive values of $\text{“K”}$ and “$a$” so that the system shown in the figure below oscillates at a frequency of $2 \mathrm{rad} / \mathrm{sec}$ respective...
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GATE ECE 2006 | Question: 57
The unit impulse response of a system is \[h(t)=e^{-t}, t \geq 0\] For this system, the steady-state value of the output for unit step input is equal to $-1$ $0$ $1$ $\infty$
The unit impulse response of a system is\[h(t)=e^{-t}, t \geq 0\]For this system, the steady-state value of the output for unit step input is equal to$-1$$0$$1$$\infty$
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GATE ECE 2006 | Question: 58
The transfer function of a phase-lead compensator is given by \[G_{c}(s)=\frac{1+3 T s}{1+T s} \text { where } T>0\] The maximum phase-shift provided by such a compensator is $\frac{\pi}{2}$ $\frac{\pi}{3}$ $\frac{\pi}{4}$ $\frac{\pi}{6}$
The transfer function of a phase-lead compensator is given by\[G_{c}(s)=\frac{1+3 T s}{1+T s} \text { where } T>0\]The maximum phase-shift provided by such a compensator ...
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GATE ECE 2006 | Question: 59
A linear system is described by the following state equation $X(t)=\mathrm{AX}(t)+\mathrm{BU}(t), \mathrm{A}=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0\end{array}\right]$ ... $\left[\begin{array}{cc}\cos t & -\sin t \\ \cos t & \sin t\end{array}\right]$
A linear system is described by the following state equation$$X(t)=\mathrm{AX}(t)+\mathrm{BU}(t), \mathrm{A}=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]$$The...
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