Consider a power system consisting of $N$ number of buses. Buses in this power system are categorized into slack bus. $P V$ buses and $P Q$ buses for load flow study. The number of $P Q$ buses is $N_L$. The balanced Newton-Raphson method is used to carry out load flow study in polar form $H, S, M$ and $R$ are sub-matrices of the Jacobian matrix $J$ as shown below:

$$ \left[\begin{array}{l} \Delta P \\ \Delta Q \end{array}\right]=J\left[\begin{array}{l} \Delta \delta \\ \Delta \gamma \end{array}\right] \text {, where } J=\left[\begin{array}{ll} H & S \\ M & R \end{array}\right] $$

The dimension of the sub matrix $M$ is

Suppose $I_A, I_B$ and $I_C$ are a set of unbalanced current phasors in a three-phase system. The phase-B zero-sequence current $I_{B 0}=0.1 \angle 0^0$ p.u. If phase-A current $I_A=1.1 \angle 0^0$ p.u and phase- $C$ current $I_C=\left(1 \angle 120^0+0.1\right)$ p.u., then $I_B$ in p.u is

The causal signal with $z$-transform $z^2(z-a)^{-2}$ is ( $u[n]$ is the unit step signal)

Let $f(t)$ be an even function, i.e., $f(-t)=f(t)$ for all $t$. Let the Fourier transform of $f(t)$ be defined as

$F(\omega)=\int_{-\infty}^{\infty} f(t) e^{-j \omega t} d t$. Suppose $\frac{d F(\omega)}{d \omega}=-\omega F(\omega)$ for all $\omega$ and $F(0)=1$. Then