Given below is the KCL equation of a circuit. Draw the circuit.
Given below is the KCL equation of a circuit. Draw the circuit. \(\left[\begin{array}{ccc}1+\frac{1}{4}+\frac{1}{3} & -\frac{1}{4} & -\frac{1}{3} \\ -\frac{1}{4} & 1+\frac{1}{4}+\frac{1}{3} & -1 \\ -\frac{1}{3} & -1 & 1+\frac{1}{3}+\frac{1}{5}\end{array}\right]\left[\begin{array}{c}V_{1} \\ V_{2} \\ V_{3}\end{array}\right]=\left[\begin{array}{c}10 \\ -20 \\ 0\end{array}\right]\)
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Consider the 3 × 3 matrices
3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).4. (3pts) Let \(T(x)=A x\) be the linear transformation given by$$...
The positive, negative and zero sequence bus impedance and admittance matrices of a system are given as follows:
The positive, negative and zero sequence bus impedance and admittance matrices of a system are given as follows:\(Z^{+}=Z^{-}=j\left[\begin{array}{ccc}0.14 & 0.11 & 0.125 \\ 0.11 & 0.14 & 0.125 \\ 0.125 & 0.125 & 0.175\end{array}\right] \quad Y^{+}=Y^{-}=j\left[\begin{array}{ccc}-24 & 10 & 10 \\ 10 & -24 & 10 \\ 10 & 10 & -20\end{array}\right]\)\(Z^{0}=j\left[\begin{array}{ccc}0.10 & 0.10 & 0.10 \\ 0.10 & 0.30 & 0.20 \\ 0.10 & 0.20 & 0.30\end{array}\right] \quad Y^{0}=j\left[\begin{array}{ccc}-16.66 & 3.33 & 3.33 \\ 3.33 & -6.66 & 3.33...
Defining the cross product The cross product of two nonzero vectors
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
Consider the heat conduction problem
5. If \(f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.\)is periodio of period 4 , and whose Fourier series is given by \(\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad\) find \(a_{n}\)A. \(\frac{2}{n^{2} \pi^{2}}\)B. \(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\)C. \(\frac{4}{n^{2} \pi^{2}}\)D. \(\frac{2}{n \pi}\)\(\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}\)F. \(\frac{4}{n \pi}\)6. Let \(f(x)-2 x-l\) on \([0,2]\). The Fourier sine series for \(f(x)\) is \(\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)\), What is \(b, ?\)A. \(\frac{4}{3 \pi}\)B. \(\frac{2}{\pi}\)C. \(\frac{4}{\pi}\)D. \(\frac{-4}{3 \pi}\)E. \(\frac{-2}{\pi}\)F. \(\frac{-4}{\pi}\)7. Let \(f(x)\) be periodic...
Consider the linear system A.r = b where A = 1
Consider the linear system \(A x=b\) where \(A=\left[\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right], b=\left[\begin{array}{l}1 \\ 1\end{array}\right], x=\left[\begin{array}{l}1 \\ 1\end{array}\right]\).We showed in class, using the eigenvlaues and eigenvectors of the iteration matrix \(M_{G S}\), that for \(x^{(0)}=\left[\begin{array}{ll}0 & 0\end{array}\right]^{T}\) the error at the \(k^{t h}\) step of the Gauss-Seidel iteration is given by$$ e^{(k)}=\left(\frac{1}{4}\right)^{k}\left[\begin{array}{l} 2 \\ 1 \end{array}\right] $$for \(k \geq 1\). Following the same procedure, derive an analogous expression for the error in Jacobi's method for the same system.
Given an LTI system with
Given an LTI system with$$ \begin{aligned} &A=\left(\begin{array}{cc} 1 / 2 & 0 \\ 0 & -1 / 4 \end{array}\right), B=\left(\begin{array}{l} 0 \\ 1 \end{array}\right), C=(1-1), \\ &D=0 \quad X(0)=\left(\begin{array}{l} -1 \\ -1 \end{array}\right), U(n)=(-1)^{n} u[n] \end{aligned} $$Calculate \(y[n], y[4]\) and \(y[\) Steady State \(]\)
op-amp & capacitorplease solve this problem6.76 Given the network in Fig. P6.76. (a) Determine...
op-amp & capacitorplease solve this problem6. 76 Given the network in Fig. \(\mathrm{P} 6.76 .\)(a) Determine the equation for the closed-loop gain \(|\mathrm{G}|=\left|\frac{v_{0}}{v_{i}}\right|\)(b) Sketch the magnitude of the closed-loop gain as a function of frequency if \(R_{1}=1 \mathrm{k} \Omega, R_{2}=10 \mathrm{k} \Omega\), and \(C=2 \mu \mathrm{F}\).
Find the eigenvalues and eigenvectors of the matrix.
Find the eigenvalues and eigenvectors of the matrix. $$ A=\left[\begin{array}{ccc} 1 & 2 & -1 \\ 1 & 0 & 1 \\ 4 & -4 & 5 \end{array}\right] $$
1. Solve the system: x' =3x+5y, y' =-x-y2. Find the general solution to
Solve the system: \(x^{\prime}=3 x+5 y, y^{\prime}=-x-y\)Find the general solution to$$ \vec{x}^{\prime}=\left(\begin{array}{ll} 2 & 1 \\ 0 & 2 \end{array}\right) \vec{x} $$Find the general solution to$$ \vec{x}^{\prime}=\left(\begin{array}{ccc} 3 & 0 & -2 \\ 0 & 5 & 0 \\ 2 & 0 & 3 \end{array}\right) \vec{x} $$
Use A-1 to solve the following system of linear exuations
Let \(A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 1 & -5 & 1 \\ 2 & -7 & 1\end{array}\right]\)a) Compute \(A^{-1} .\)b) Use \(A^{-1}\) to solve the following system of linear exuations:$$ \begin{array}{r} 2 x_{1}+-x_{3}=3 \\ x_{1}-5 x_{2}+x_{3}=1 \\ 2 x_{1}-7 x_{2}+x_{3}=4 \end{array} $$
3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).4. (3pts) Let \(T(x)=A x\) be the linear transformation given by$$...
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
op-amp & capacitorplease solve this problem6. 76 Given the network in Fig. \(\mathrm{P} 6.76 .\)(a) Determine the equation for the closed-loop gain \(|\mathrm{G}|=\left|\frac{v_{0}}{v_{i}}\right|\)(b) Sketch the magnitude of the closed-loop gain as a function of frequency if \(R_{1}=1 \mathrm{k} \Omega, R_{2}=10 \mathrm{k} \Omega\), and \(C=2 \mu \mathrm{F}\).
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