Fundamentals of Electric Power, Formula and Types of Electric Power

Electric Power

In general power is defined as the change in energy (expanding and absorbing) i.e. transfer of energy with respect to time. The SI unit of power is watt that is also expressed in joules per second.

p  =  dw/dt  J/s (Joule per Second )  ……………………..(1)

In equation (1)

• p represents the power

• dw represents the change in energy

• dt represents the change in time

 

Similarly, Electric Power is defined as the transfer of electrical energy with respect to time. The SI unit of electric power is watt.

Electric Power Formula

 

Multiply dq/dq in equation (1)

 

p(t)  =  (dw/dq).(dq/dt)

 

p(t)  =  v(t).i(t)  Watts  

P= V.I   Watts     ………………  (2)   

 

Equation (2) represents the formula for electric power in consumed by the load in circuit that is connected with a source having voltage V and drawn a current I.

 

Equation (2) can be combined with ohm’s law ( V = I.R) to derive the power dissipated in case of resistive load or in case of power dissipated in DC circuit.

Put the value of V in equation (2)

P  =  (I.R).I     =    I².R   =    V²/R …………(3)

Equation (3) represents the power dissipated in resistive load in case of power dissipated in DC circuit.

 

So, the electric power is also defined as product of voltage applied to the load terminal and current drawn by it.

In the above figure there is a load element X potential difference across this element is V and current passing through this element is I. So, power across this element is

 

P  =  V.I Watts     …………………..(2)

The polarity of applied voltage and direction of current in the circuit are important for determining whether the power is delivered or absorbed by the circuit or element.

• If P is positive in equation (3) the element absorb power or we can say power is dissipated in the circuit element.

• If P is negative in equation (3) then the circuit element delivers the power, or we can say it is generated the power in circuit.  

How can we assure that the Power is absorbed or delivered?

If the direction of current in such a way that it enters the positive terminal of voltage across through the element then the sign of power is positive and the power is absorbed by the element.

If the current enters negative terminal of voltage across through the element, then the sign of p is negative and the is delivered by the element.

Let us understand by the circuit, In the given figure direction of current in such a way that it enters the positive terminal of Z and negative terminal of Y. So, from both of above statement we can say the element Z is load and it absorb power and element Y is source and it delivers power.

DC Circuit Power Analysis

In Dc circuit energy storage elements inductor and capacitor does not store energy, they behave like short circuit and open circuit simultaneously. In Dc circuit only resistive load are taken into account. So, in dc circuit equation (2) & (3) are valid.

P  =  V.I Watts     …………………..(2)

P  =  (I.R).I     =    I².R   =    V²/R …………(3)

 

 

AC Circuit Power Analysis

Unlike DC, AC current changes its direction periodically and changes its magnitude continuously with time. The waveform of AC voltage and current are sinusoidal in nature and represented as in the given figure.

                          

In AC magnitude of current and voltage continuously changes with time, therefore we have to calculate instantaneous power.

 

Instantaneous Power in AC Circuit

The power absorbed by the element in ac circuit at any instant of time is called instantaneous power. Let us understand by a circuit.

Let us say voltage v(t) is applied to an element Z and current drawn by the element Z is i(t) as shown in given circuit.

v(t)  = V_m sin(wt + θ_v)

i(t)    =  I_m sin(wt ± θ_i)       …………    (4)

 

p(t)   =   v(t).i(t)

p(t)   =    I_m.V_m. sin(wt + θ_v). sin(wt ± θ_i).

p(t)   =    I_m.V_m /2 . (cos(wt+ θ_v)-(wt + θ_i))-cos((wt+ θ_v)-(wt + θ_i))-)))

p(t)   =    I_m.V_m /2 . (cos(2wt ± θ_v -  θ_i)   +  cos(θ_v -  θ_i))

 

Let us say   θ    =    θ_v -  θ_i    

 

p(t)   =    I_m.V_m /2 . cos(2wt ± θ)     +     I_m.V_m /2 cos(θ))      ……………….   (5)

(i)                                                       (ii)

 

In equation (5) θ is the phase angle difference between voltage v(t) and current i(t)

• If θ is positive then it means current is lagging with respect to voltage (inductive load).

• If θ is negative then it means current is leading with respect to voltage (capacitive load).

• If θ is zero then it means current is phase with voltage (resistive load).

 

The instantaneous power is the sum of two terms. The first term is time dependent and the second term is time independent.

The instantaneous power changes with time, therefore it is difficult to measure. There is more convenient way to measure power in ac circuit i.e. average power.

 

Average Power in AC Circuit

The concept of average power is used in AC circuit because the magnitude and direction of the voltage and current in an AC circuit are changed continuously. Unlike in DC circuits they oscillate sinusoidaly. As a result, instantaneous power also varies with time as shown in equation (5). By calculating the average power over a complete cycle, we can determine the actual power dissipated in the circuit.

 

The average power provides a measure net power dissipated or delivered over a complete cycle, which is useful for determining the actual power consumption in AC circuits.

Therefore, the average power corresponding to the p(t) can be calculated by integrating the instantaneous power shown in equation (5) over one period of sinusoidal signal.

 

P_av  =   1/T ₀ʃ^T p(t).dt

P_av  =   1/T ₀ʃ^T (I_m.V_m /2 . cos(2wt ± θ)     +     I_m.V_m /2 cos(θ))

The average of the sinusoid over a complete cycle is zero. Hence the integral of first part is zero. The Second term is constant so the average power.

P_av     =     I_m.V_m /2 cosθ

P_av     =     (I_m/√2).(V_m/√2).cosθ

P_av     =      V_rms.I_rms cosθ     ………………………………. (6)

 

In equation (6) cosθ is power factor which is defined as the cosine of phase angle difference between voltage and current.

Θ is zero in case of pure resistive load so, average power consumed by pure resistive load

P_av     =      V_rms.I_rms cos0      =      V_rms.I_rms                                    (  cos0   =   1  )

 

Θ is +90⁰ and -90⁰ in case of pure inductive and capacitive load respectively. So, average power absorbed by pure inductive and capacitive load

 

P_av     =      V_rms.I_rms cos90⁰      =       0                                (cos90⁰    =     0) 

 

Types of electric Power in AC Circuit

In AC circuit analysis we deal with three types of power these are

1. Complex power or Apparent power

2. Active Power or Real Power or True Power

3. Reactive Power

 

 

Apparent Power

In an AC (alternating current) circuit, apparent power is a measure of the total power supplied to the circuit or element, considering both the real power and the reactive power. It is expressed as the product of V_rms and  I_rms and its unit volt-amperes (VA).

 

Apparent Power Formula

 

Consider a network

In above circuit V∟θ_v is applied voltage to the element a and I∟θ_i is the current drawn by element X.

S   =   V_rms.I^*_rms    =     V_rms∟θ_v.I_rms∟θ_i                   (I^*   means Conjugate of I)

     =    V_rms.I_rms∟θ_v - θ_i                                             ……………………………… (7)

Converting the equation (7) from polar form to rectangular form gives

S    =    V_rms.I_rms (cos(θ_v- θ_i)     +   j.sin(θ_v- θ_i))    

S    =     V_rms.I_rms cos(θ_v- θ_i)     +    j. V_rms.I_rms.sin(θ_v- θ_i)  

 

Let us say    θ   =   θ_v- θ_i 

S    =     V_rms.I_rms cosθ      +    j. V_rms.I_rms.sinθ     ………………….     (8)

(i)                                                             (ii)

 

 

The apparent power consists of two parts, in which part (i) is real and part (ii) is imaginary. The real part of S is called real power or active power and it is similar to the average power that we are calculated previously it is denoted by P, and imaginary part of the S is called reactive power and is denoted by Q.

 S   =   P  +  j Q       …………………………       (9)

 

Equation (9) shows that the apparent power (S) in an AC circuit is the vector sum of the real power (P) and the reactive power (Q).

The practical significance of apparent power is as a rating unit of generators and transformers that is in a KVA or MVA.

 

 

Reactive Power

The reactive power is the portion of the complex power that is exchange between reactive load (inductor and capacitor) and source. Let us understand by the given circuit.

In the given circuit voltage v(t) = Vm.sinwt is applied to an inductor. In this circuit inductor stores the energy when positive half of v(t) is applied to inductor means the flow of energy is from source to load. When negative half v(t) is applied to inductor, inductor releases the stored energy during positive half. This oscillation of energy between inductor and source is going on until the voltage is applied. This energy (power) is called reactive power and it is measured in VARs (Volt Ampere Reactive).

 

If the inductor is ideal, means it is purely inductive load then 100% of complex power oscillates between load and source and there is no active power absorbed by inductor. (We discussed this Inductor).

 

 

Reactive power has no effect on net power absorbed by circuit or element. But it has its own significance.

As we know that the different electrical machines like electric motor, transformer etc establish a magnetic domain in the form magnetic flux for performing their operation. To established this magnetic domain, electric machines use a small fraction of complex power, this fraction of complex power is known reactive power. It is denoted by Q.

Q  =   V_rms.I_rms.sinθ                      ………………………    (10)

Θ is the phase angle difference between applied voltage and current drawn

 

 

Active power

Active power or real power is the fraction of complex power which is used to fulfill the load demand of consumers. It is the actual power which is used or dissipated by the load in circuit. It is also called true power or real power and is measured in watts. It is similar with the average power dissipated in DC circuit.

P  =   V_rms.I_rms.cosθ          ………………………           (11)

 

In equation (11) cosθ is the power factor of the circuit.

 

The relation between these three types of power is studied with the help of Power Triangle.