Properties of Z transform

Linearity $latex \displaystyle x_1(n) \rightleftharpoons X_1(z) &s=2 &bg=ffffff$ ROC : $latex \displaystyle a_1 < |z| < b_1 &s=2 &bg=ffffff$ $latex \displaystyle x_2(n) \rightleftharpoons X_2(z) &s=2 &bg=ffffff$ ROC : $latex \displaystyle a_2 <|z| < b_2 &s=2 &bg=ffffff$ According to property of Linearity $latex \displaystyle a_1 x_1(n) + a_2 x_2(n) \rightleftharpoons a_1 X_1(z) + a_2 X_2(z) &s=2 … Continue reading Properties of Z transform

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Rank of a Matrix

Rank   $latex \rightarrow &s=2 &bg=ffffff$  No. of linearly independent rows or columns in a matrix $latex A &s=2 &bg=ffffff$ is called the Rank of $latex A &s=2 &bg=ffffff$ The rank is commonly represented by Rank (A) = rk (A) = $latex \displaystyle \rho (A) &s=2 &bg=ffffff$ $latex \Rightarrow &s=2 &bg=ffffff$  A fundamental result in linear algebra is that the … Continue reading Rank of a Matrix

Steady State Stability Analysis

$latex \Rightarrow &s=2 &bg=ffffff$  The steady state stability limit is defined as the maximum power that can be transmitted to the receiving end without loss of synchronism. $latex \displaystyle P_m - P_e = M \frac{d^2\delta}{dt^2} &s=2 &bg=ffffff$ All quantities are in Per Unit $latex \displaystyle P_e = P_{max} sin(\delta) &s=2 &bg=ffffff$ $latex \Rightarrow &s=2 &bg=ffffff$  Let the system be … Continue reading Steady State Stability Analysis

Swing Equation

$latex \Rightarrow &s=2 &bg=ffffff$  The problem of stability can understand electrically as well as mechanically if there is an imbalance between mechanical input torque and electrical output torque, the rotor will either accelerate or deaccelerate. Accelerating Torque $latex \rightarrow &s=2 &bg=ffffff$  $latex T_a = T_m - T_e &s=2 &bg=ffffff$ $latex \displaystyle T_m - T_e = J \frac{d^2 \theta}{dt^2} &s=2 … Continue reading Swing Equation

Memoryless LTI System

Consider an LTI system $latex \displaystyle x(t) \rightarrow &s=2 &bg=ffffff$  input to the system $latex \displaystyle h(t) \rightarrow &s=2 &bg=ffffff$  impulse response of the system $latex \displaystyle y(t) \rightarrow &s=2 &bg=ffffff$  output to the system $latex \displaystyle y(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau  &s=3 &bg=ffffff$ $latex \Rightarrow &s=2 &bg=ffffff$ For an LTI system to be memory-less its … Continue reading Memoryless LTI System

Stability of LTI systems

$latex \Rightarrow &s=2 &bg=ffffff$  The system is said to be stable if it produces bounded output for every bounded input. Consider an LTI system whose output is $latex y(t) &s=2 &bg=ffffff$ and impulse response is $latex h(t) &s=2 &bg=ffffff$ $latex \displaystyle y(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau &s=3 &bg=ffffff$ $latex \displaystyle |y(t)| = | \int_{-\infty}^{\infty} x(\tau) h(t … Continue reading Stability of LTI systems

Causality of LTI system

$latex \displaystyle \Rightarrow &s=2 &bg=ffffff$  The output of the causal system depends only upon the present and past values of input (Output does not depend on the future values of input at any instant of time) In case of LTI system, output $latex y(t) &s=2 &bg=ffffff$ is given as convolution of unit impulse response $latex h(t) &s=2 &bg=ffffff$ … Continue reading Causality of LTI system

Properties of Convolution

Property 1  $latex \displaystyle x_1(t) * x_2(t) = x_2(t) * x_1(t) \rightarrow &s=2 &bg=ffffff$   Commutative Property   Property 2 $latex \displaystyle x_1(t) * [x_2(t) * x_3(t)] = [x_1(t) * x_2(t)] * x_3(t) \rightarrow &s=2 &bg=ffffff$  Associative Property Property 3 $latex \displaystyle x_1(t) * [x_2(t) + x_3(t)] = x_1(t) * x_2(t) + x_1(t) * x_3(t) \rightarrow &s=2 … Continue reading Properties of Convolution

Basics of Convolution

Representation of   $latex x(t) &s=2 &bg=ffffff$ in terms of Impulses Property 1 We know the property of impulse that area of the impulse function is equal to one. $latex \displaystyle \int_{-\infty}^{\infty} \delta(t) dt = 1 &s=3 &bg=ffffff$ Property 2 $latex \delta(t) &s=2 &bg=ffffff$   is defined only at $latex t = 0 &s=2 &bg=ffffff$ $latex \displaystyle x(t) \delta(t) = … Continue reading Basics of Convolution