Class Activity (Week 12)

        Instantaneous, effective, and average power were discussed.

v(t) = Vm cos(ωt + θv) ,  i(t) = Im cos(ωt + θi)  or v(t) = Vm cos(ωt + θv - θi ) ,  i(t) = Im cos(ωt)

Instantaneous power is   P(t) = v(t)*i(t)  ,  P(t) = Pavg + Pavg cos(2ωt) - Pavg sin(2ωt) 

Average (Real) Power Pavg =  [(Vm*Im)/2] cos(θv - θi) = Vrms Irms cos(θv - θi) = S cos(θv - θi)

S is apparent power. Pf = Pavg / S = cos(θv - θi)

Reactive Power Q = [(Vm*Im)/2] sin(θv - θi)   

For resistive loads Q = 0 (Pf = 1) , For capacitive loads Q ˂ 0 (leading Pf) , 

For inductive loads Q  ˃ 0 (lagging Pf)

Figure 1: Effective or RMS Power

Vrms = Vm / √2   ,  Irms = Im / √2

Figure 2: Average Power

Figure 3: Apparent Power

S is apparent power. Pf = Pavg / S = cos(θv - θi)

Power Factor  Pf = cos(θv - θi)  ,  Reactive Power rf = sin(θv - θi) 

Lagging power factor shows that current lags voltage - hence an inductive load. 

Leading power factor shows that current leads voltage - hence a capacitive load.

Figure 4: Complex Power

Complex Power S (VA) = P (W) + jQ(VAR)  

P = (Ieff)^2 . R = (Veff)^2  / R   , Q = (Ieff)^2 . X = (Veff)^2  / X 

S = (Vrms).(Irms)* S = (V.I*)/2    ,    S = (Vrms * Irms) ˂(θv - θi) 

Figure 5: Power Triangle
(θv - θi) ˃ 0  Lagging power factor  ,  (θv - θi) ˃ 0  Lagging ˂ 0 Leading power factor

Figure 6: Impedance and Angel Phase

The power factor angle is equal to the angle of the load impedance, if V is the voltage across the load 

and I is current through the load.

Z = V˂θv  / I˂θi = (Vm / Im) ˂(θv - θi)  ,    Z = (Vrms / Irms) ˂(θv - θi)  

Figure 7: An Inductor Load (Lagging Power Factor)

Q = ((0.707*210)^2) / (2π*50*0.5) = 140.4 VAR

Figure 8: An Example

FREQUENCY DEPENDANCE AND TRANSFER FUNCTIONS

        The frequency response of a circuit may also be considered as the variation of the gain and phase with

frequency. The transfer function H(ω) (network function) is a useful analytical tool for finding

the frequency response of a circuit. In fact, the frequency response of a circuit is the plot of the circuit’s

transfer function H(ω) versus ω, with ω varying from ω = 0 to ω = ∞. 

H(ω) = Voltage gain = Vo(ω) / Vi(ω)    ,  H(ω) = Current gain = Io(ω) / Ii(ω) 

H(ω) = Transfer Impedance = Vo(ω) / Ii(ω)   , H(ω) = Transfer Admittance = Io(ω) / Vi(ω)  

 H(ω) = H(ω) φ. H(ω) = N(ω) / D(ω) . 

The roots of transfer function are determined when N(ω) = 0.  

 The poles of transfer function are determined when D(ω) = 0.

Figure 9: Zeros and Poles of a Series RLC

Figure 10: Zeros and Poles of a Parallel Circuit

Figure 11