GATE EC question library

Two identical copper wires $W1$ and $W2$ placed in parallel as shown in the figure, carry currents $I$ and $2I$, respectively, in opposite directions. If the two wires are separated by a distance of $4r$, then the magnitude of the magnetic field $\overrightarrow{B}$ between the wires at a distance $r$ from $W1$ is

The dispersion equation of a waveguide, which relates the wavenumber $k$ to the frequency $\omega$ is $$k(\omega)= (1/c) \sqrt{\omega^{2}-\omega_{\circ}^{2}}$$ where the speed of light $c= 2 \times 10^{8}\: m/s$ and $\omega_{\circ}$ is a constant . If the group velocity is $2 \times 10^{8}\: m/s$, then the phase velocity is

In the circuit shown, the breakdown voltage and the maximum current of the Zener diode are $20\:V$ and $60\:mA$, respectively. The values of $R_{1}$ and $R_{L}$ are $200\: \Omega$ and $1\:k\Omega,$ respectively. What is the range of $V_{i}$ that will maintain the Zener diode in the ‘on’ state?

The state transition diagram for the circuit shown is

In the circuits shown the threshold voltage of each $\text{nMOS}$ transistor is $0.6\:V.$ Ignoring the effect of channel length modulation and body bias. the values of $\text{Vout}1$ and $\text{Vout} 2,$ respectively, in volts, are

The $\text{RC}$ circuit shown below has a variable resistance $R(t)$ given by the following expression: $$R(t)=R_{0}\left(1-\frac{t}{T}\right) \text{for} \:\: 0 \leq t < T$$ where $R_{0}=1\: \Omega,$ and $C=1\:F.$ We are also given that $T=3\: R_{0}C$ and the source voltage is $V_{s}=1\: V.$ If the current at time $t=0$ is $1\: A,$ then …

Consider a unity feedback system, as in the figure shown, with an integral compensator $\dfrac{K}{s}$ and open-loop transfer function $$G(s)=\dfrac{1}{s^{2}+3s+2}$$ where $K>0.$ The positive value of $K$ for which there are exactly two poles of the unity feedback system on the $j\omega$ axis is equal to ________ (rounded off to two decimal …

Consider the homogenous ordinary differential equation $$x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+3y=0, \quad x>0$$ with $y(x)$ as a general solution. Given that $$y(1)=1 \quad \text{and} \quad y(2)=14$$ the value of $y(1.5),$ rounded off to two decimal places, is________.

Let $h[n]$ be a length – $7$ discrete-time finite impulse response filter, given by $$h[0]=4, \quad h[1]=3,\quad h[2]=2,\quad h[3]=1,$$ $$\quad h[-1]=-3, \quad h[-2]=-2, \quad h[-3]=-1,$$ and $h[n]$ is zero for $|n|\geq4.$ A length – $3$ finite impulse response approximation $g[n]$ of $h[n]$ has to be obtained such that …

Let a random process $Y(t)$ be described as $Y(t)=h(t) \ast X(t)+Z(t),$ where $X(t)$ is a white noise process with power spectral density $S_{x}(f)=5$W/Hz. The filter $h(t)$ has a magnitude response given by $\mid H(f) \mid =0.5$ for $-5\leq f\leq5,$ and zero elsewhere. $Z(t)$ is a stationary random process, uncorrelated with $X(t)$, with …

A voice signal $m(t)$ is in the frequency range $5\:kHz$ to $15\:kHz$. The signal is amplitude-modulated to generated an AM signal $f(t)=A\left(1+m(t)\right)\cos 2\pi f_{c}t,$ where $f_{c}=600\: kHz.$ The AM signal $f(t)$ is to be digitized and archived. This is done by first sampling $f(t)$ at $1.2$ times the Nyquist frequency, and then …

A random variable $X$ takes values $-1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $N,$ so that the random variable at the channel output is $Y=X+N$. The noise $N$ is independent of $X,$ and is uniformly distributed over the interval $[-2,2].$ The receiver makes a …

A Germanium sample of dimensions $1\: cm \times 1\: cm$ is illuminated with a $20\:mW,$ $600\: nm$ laser light source as shown in the figure. The illuminated sample surface has a $100\: nm$ of loss-less Silicon dioxide layer that reflects one-fourth of the incident light. From the remaining light, one-third of the power is reflected from the Silicon …

In an ideal $pn$ junction with an ideality factor of $1$ at $T=300\:K,$ the magnitude of the reverse-bias voltage required to reach $75\%$ of its reverse saturation current, rounded off to $2$ decimal places, is ______ $mV.$ $$[k=1.38 \times 10^{-23} JK^{-1}, h=6.625 \times 10^{-34} J-s, q=1.602 \times 10^{-19}C]$$

Consider a long-channel MOSFET with a channel length $1\:\mu m$ and width $10\: \mu m.$ The device parameters are acceptor concentration $N_{A}=5 \times 10^{16}\: cm^{-3},$ electron mobility $\mu_{n}=800\: cm^{2}/V-s,$ oxide capacitance/area $C_{ox}=3.45 \times 10^{-7}\:F/cm^{2},$ threshold voltage $V_{T}=0.7\:V.$ The drain saturation current …

A rectangular waveguide of width $w$ and height $h$ has cut-off frequencies for $TE_{10}$ and $TE_{11}$ modes in the ration $1:2$ . The aspect ratio $w/h$, rounded off to two decimal places , is _______.

In the circuit shown. $V_{s}$ is a $10\:V$ square wave of period, $T=4\: ms$ with $R=500\: \Omega$ and $C= 10\:\mu F.$ The capacitor is initially uncharged at $t=0,$ and the diode is assumed to be ideal. The voltage across the capacitor $(V_{c})$ at $3\:ms$ is equal to _____ volts (rounded off to one decimal place)

A CMOS inverter, designed to have a mid-point voltage $V_{1}$ equal to half of $V_{dd}.$ as shown in the figure, has the following parameters: $V_{dd}=3V$ $\mu_{n} C_{ox}=100\: \mu A/V^{2}; V_{tn}=0.7\:V$ for $\text{nMOS}$ $\mu_{p} C_{ox}=40\: \mu A/V^{2}; \mid V_{tp} \mid=0.9\:V$ for $\text{pMOS}$ The ration of $\left(\frac{W}{L}\right)_{n}$ …

In the circuit shown, the threshold voltages of the $pMOS\:\: (|V_{tp}|)$ and $nMOS\:\: (V_{tn})$ transistors are both equal to $1\:V.$ All the transistors have the same output resistance $r_{ds}$ of $6\:M\Omega.$ The other parameters are listed below: $\mu_{n}=C_{ox}=60\: \mu A/V^{2}; \left(\frac{W}{L}\right)_{nMOS}=5$ …

In the circuit shown, $V_{1}=0$ and $V_{2}=V_{dd}.$ The other relevant parameters are mentioned in the figure. Ignoring the effect of channel length modulation and the body effect, the value of $I_{out}$ is _________ $mA$ (rounded off to $1$ decimal place).