Addition and Subtraction of Continuous-Time SignalsThe sum of two continuous time signals 𝑥1(𝑡) and 𝑥2(𝑡) can be obtained by adding their values at every instant of time. Likewise, the difference of two continuous time signals 𝑥1(𝑡) and 𝑥2(𝑡) can be obtained by subtracting the values of one signal (say 𝑥2(𝑡)) ... Read More

Even SignalA signal which is symmetrical about the vertical axis or time origin is known as even signal or even function. Therefore, the even signals are also called the symmetrical signals. Cosine wave is an example of even signal.Continuous-time Even SignalA continuous-time signal x(t) is called the even signal or ... Read More

Causal SignalA continuous time signal 𝑥(𝑡) is called causal signal if the signal 𝑥(𝑡) = 0 for 𝑡 < 0. Therefore, a causal signal does not exist for negative time. The unit step signal u(t) is an example of causal signal as shown in Figure-1.Similarly, a discrete time sequence x(n) ... Read More

Causal SystemA system whose output or response at any time instant (t) depends only on the present and past values of the input but not on the future values of the input is called the causal system. For a causal system, the output or response does not begin before the ... Read More

A sinusoidal function or sinusoidal signal is a function that describes a smooth periodic oscillation.Continuous-Time Sinusoidal SignalA sinusoidal signal which is defined for every instant of time is called continuous-time sinusoidal signal. The continuous time sinusoidal signal is given as follows −𝑥(𝑡) = 𝐴 sin(𝜔𝑡 + 𝜑) = 𝐴 sin(2𝜋𝑓𝑡 ... Read More

What is Time Shifting?Time shifting or Shifting of a signal in time means that the signal may be either delayed in the time axis or advanced in the time axis.Time Shifting of Continuous-Time SignalThe time shifting of a continuous time signal x(t) is represented as, 𝑦(𝑡) = 𝑥(𝑡 − 𝑡0)The ... Read More

What is Time Reversal of a Signal?The time reversal of a signal is folding of the signal about the time origin (or t = 0). The time reversal or folding of a signal is also called as the reflection of the signal about the time origin (or t = 0). ... Read More

Discrete Time SignalsThe signals which are defined only at discrete instants of time are known as discrete time signals. The discrete time signals are represented by x(n) where n is the independent variable in time domain.Representation of Discrete Time SignalsA discrete time signal may be represented by any one of ... Read More

Consider a synchronous motor is operating at lagging power factor. The voltage equation of a synchronous motor is given by, $$\mathrm{V=E_{f}+I_{a}Z_{S}\:\:\:\:\:\:...(1)}$$Where, $$\mathrm{V=V\angle 0°\:and\:E_{f}=E_{f}\:\angle-δ}$$$$\mathrm{\therefore\:I_{a}=\frac{V-E_{f}}{Z_{S}}\:\:\:\:\:\:...(2)}$$$$\mathrm{\Longrightarrow\:I_{a}=\frac{V\angle 0°-E_{f}-δ}{Z_{S}\angleθ_{Z}}=\frac{V}{Z_{S}}\angle-θ_{Z}-\frac{E_{f}}{Z_{S}}\angle-(δ+θ_{Z})}$$$$\mathrm{\therefore\:I^{*}_{a}=\frac{V}{Z_{S}}\angleθ_{Z}-\frac{E_{f}}{Z_{S}}\angle(δ+θ_{Z})\:\:\:\:\:\:...(3)}$$Complex Power Output per Phase of a Synchronous MotorThe complex power output of a synchronous motor is given by, $$\mathrm{S_{o}=E_{f}I^{*}_{a}=P_{o}+jQ_{o}\:\:\:\:\:\:...(4)}$$$$\mathrm{\Longrightarrow\:S_{o}=E_{f}\:\angle-δ\left(\frac{V}{Z_{S}}\angleθ_{Z}-\frac{E_{f}}{Z_{S}}\angle(δ+θ_{Z})\right)}$$$$\mathrm{\Longrightarrow\:S_{o}=\left(\frac{VE_{f}}{Z_{S}}cos(θ_{Z}-δ)+j\frac{VE_{f}}{Z_{S}}sin(θ_{Z}-δ)\right)-\left(\frac{E^{2}_{f}}{Z_{S}}cosθ_{Z}+j\frac{E^{2}_{f}}{Z_{S}}sinθ_{Z}\right)}$$$$\mathrm{\therefore\:S_{o}=\left(\frac{VE_{f}}{Z_{S}}cos(θ_{Z}-δ)-\frac{E^{2}_{f}}{Z_{S}}cosθ_{Z}\right)+j\left(\frac{VE_{f}}{Z_{S}}sin(θ_{Z}-δ)-\frac{E^{2}_{f}}{Z_{S}}sinθ_{Z}\right)\:\:\:\:\:\:...(5)}$$Real Power Output per Phase of the Synchronous ... Read More

The phasor diagram at lagging power factor and the equivalent circuit diagram of a cylindrical synchronous motor are shown in Figure-1 and Figure-2, respectively.The terminal voltage (V) is taken as reference phasor and the excitation voltage (Ef) lags the terminal voltage (V) by an angle δ so that$$\mathrm{V=V\angle0°\:and\:E_{f}=E_{f}\:\angle-δ}$$Applying KVL in ... Read More