This post is dedicated to *RF Power* theory. In today's busy world Power is consumed by various smart devices such as PC's and smartphones. Whether you connect your Phone to a cellular tower or your PC to your Wi-Fi router, the Power plays a huge role in designing Radio Frequency or Microwave AC circuits.

**Power in RF Design:**

- Avoid loss
- Maximum Power transfer

**Instantaneos Power:**

- p(t) = v(t).i(t) OR v(t)² / R

- Imagine having a simple closed circuit with voltage source and a resistor connected. Then the instantaneous power is the power at a specific time. For example, v(t=100) = 2 v, and R = 50 ohm. Then, p(t=100) = v(100)² / R = 2² / 50 = 4 / 50 W.

**Average Power:**

- Pavg = 1 / T ∫0T. . p(t) dt = (1 / R) . (1 / T) ∫0Tv(t)²

- Usually, we deal with periodic waveforms. Hence, T is used as a time period of the periodic waveform.

- The average v(t)² over time = < V(t)² > = (1 / T) ∫0Tv(t)²

- Vrms = √< V(t)² > = root mean square or RMS voltage.

- Thus, the average power dissipated in the resistor R is P = < V(t)² > / R or V²rms / R.

- Example Problem: RMS value? of V(t) = 2cos(wt) [voltage source], and average power dissipated in 50 ohm resistance?
- Pavg = 1 / T ∫0T. . p(t) dt = (1 / R) . (1 / T) ∫0Tv(t)²

since, V(t) = A cos (wt) => < V(t)² > = (1 / T) ∫0Tv(t)²

- (1 / T) ∫
_{0}^{T }A^{2}cos^{2}(wt) =(since avg. of cos from 0 to T = 0 [zero] ) - => < V
^{2}(t) > = A^{2}^{ }/ 2 - Pavg = 1/R . A
^{2 }/ 2 = 1 / 50 . 4 / 2 = 2 / 50

- Vrms = √< V(t)² > = A / √2 = 2 / √2

**Power and Phasor:**

- 2sin(wt) = 2cos(wt - 90)
- Same Phase
- e
^{-j90 }= sin (wt) = cos (wt -90) = I, current - Z = R; I = V / Z = | V | e
^{-j90 }/ R e^{j0 }

- Z = jLw; I =V / Z = | V | e
^{j0}/ ( Lw )e^{j90}= (| V | / Lw)e^{-j90}

- Pure imaginary phase difference of 90
- 90 deg phase difference between I and V
- Z = jLw = ( Lw )e
^{j90}

**Power**

- p(t) = v(t) i(t) = |V| |I| cos (wt + θv) cos (wt + θ1) = ( |V| |I| ) / 2 . {cos (2wt + θv + θ1) + cos(θv - θ1)}, where yellow highlighted part of the equation is = 0 mathematically.

- Average power: Pavg = ( |V | | I | )/2 . cos (θv - θ1), where ϕ = θv - θ1

- Pavg = ( |V | | I | )/2 . cos ϕ

- Vrms = | V | / √2; Irms = | I | / √2

- Thus, Pavg = Vrms Irms cos(θv - θ1)

- Example, Pavg = ( |V | | I | )/2 . cos (θv - θ1) = 0, since (θv - θ1) = 90deg.

**Advanced concepts on power to be covered in another post.*

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