AC (Alternating Current) Power Analysis
Electric power destroyed by a resistance in an AC circuit is different from the power destroyed by a reactance, as reactances do not dissipate energy.
In a DC circuit, the power consumed is simply the product of the DC voltage times the DC current, given in watts. Still, for AC circuits with reactive factors, we’ve got to calculate the consumed power elsewhere.
Also, the power absorbed or supplied by a circuit element is the product of the voltage, V, across the element and the current, I, flowing through it. So if we had a DC circuit with a resistance of “R” ohms, the power dissipated by the resistor in watts would be given by any of the following generalized formulas:
[Tex]P = V \cdot I= \frac{V^2}{R}= I^2 \cdot R [/Tex]
Electric Power in AC Circuit
In a DC circuit, the voltages and currents are generally constant; they aren’t varying with time as there’s no sinusoidal waveform associated with the force. Still, for power in AC circuits, the immediate values of the voltage, current, and thus power are constantly changing due to the force. So we can’t calculate the power in AC circuits in the same manner as we can in DC circuits, but we can still say that power( p) is equal to the voltage( v) times the amperes( i).
So, in this diagram, the term:
The AC power source defines the source of alternating current while providing a voltage V(t).
AC load defines the electrical load through which current I(t) flows.
This diagram is a fundamental representation and doesn’t include details such as power factor, phase angle, or the relationship between real power and reactive power.
P(t) =V(t) x I (t) that is instantaneous namaste power.
Where
[Tex]V = V_m \sin(wt + \theta_v) [/Tex]
[Tex]i = I_m \sin(wt + \theta_i) [/Tex]
Then
[Tex]P = V_m I_m \sin(wt + \theta_v) \sin(wt + \theta_i) [/Tex]
Apply this,
[Tex]\sin(A) \sin(B) = \frac{1}{2} \left[ \cos(A – B) – \cos(A + B) \right] [/Tex]
[Tex]P = \frac{V_m I_m}{2} \left[ \sin(\theta) – \cos(2wt + \theta) \right] [/Tex]
Now,
[Tex]\frac{V_m I_m}{2} = \frac{V_m}{\sqrt{2}} \cdot \frac{I_m}{\sqrt{2}} = V_{\text{rms}} \cdot I_{\text{rms}} [/Tex]
Instantaneous Equations:-
[Tex]P = VI \cos(\theta) – VI \cos(2wt + \theta) [/Tex]
[Tex]P = V \cdot I \cdot \cos(\theta)[/Tex]
[Tex]P = V^2 Z \cos(\theta) [/Tex]
[Tex]P = I^2 Z \cos(\theta) [/Tex]
So,
[Tex]Z = \sqrt{R^2 + (X_l – X_c)^2} [/Tex]
Real Power
Real power is defined as the actual power consumed by a circuit and is measured in watts.
[Tex]P = \frac{1}{T} \int_{0}^{T} p(t) \, dt[/Tex]
where T is time period of one cycle
Reactive Power
Reactive Power is defined as the power that oscillates between the source and load.
[Tex]Q = \frac{1}{T} \int_{0}^{T} v(t) \cdot i(t) \, dt[/Tex]
Sinusoidal Steady State Analysis – Electric circuits
In steady state (the fully charged state of the cap), current through the capacitor becomes zero. The sinusoidal steady-state analysis is a key technique in electrical engineering, specifically used to investigate how electric circuits respond to sinusoidal AC (alternating current) signals. This method simplifies the intricate details involved in time-varying AC circuits by representing voltages and currents as phasors—complex quantities that succinctly convey both amplitude and phase information.
Table of Content
- Sinusoidal Steady State Analysis
- Sinusoidal Source
- Derivation
- V-I Relation for an Inductor
- V-I Relationship for a Capacitor
- Frequency Response
- Bode Plots
- Examples