Network Circuit Analysis Week 12 Complex Power Complex Frequency

Example Find i(t) ω = 2

Use superposition i(t) can be found from current divider

i(t) can be found from current divider

Average Power in AC circuits From instantaneous power In AC circuits, voltage and current values oscillate. This makes the power (instantaneous power) oscillate as well. However, electric power is best represented as one value. Therefore, we will use an average power. Average power can be computed by integration of instantaneous power in a periodic signal divided by time.

Let v(t) in the form Change variable of integration to θ We got Then find the instantaneous power integrate from 0 to 2π (1 period)

Compare with power from DC voltage source AC

Root Mean Square Value (RMS) In DC circuits, In AC, we define Vrms and Irms for convenience to calculate power. Vrms and Irms are defined such that, Note: Vrms and Irms are constant all the time For sine wave,

3 ways to tell voltage V (volts) 311V t (sec) V peak (Vp) = 311 V V peak-to-peak (Vp-p) = 622V V rms = 220V

Reactive Power Capacitors and inductors have average power = 0 because they have voltage and current with 90 degree phase difference. Change variable of integration to θ Then integrate from 0 to 2π (1 period)

Capacitors and Inductors do not have average power although there are voltage and current. Therefore, reactive power (Q) is defined

Complex Power Power can be divided into two parts: real and imaginary Complex power S = P + jQ P = real power Q = Reactive power Inductor has no real power P =0 But it has complex power, computed by V, I that are perpendicular to each other.

Q S Complex power = S is a vector θ P where θ is the angle difference of (V-I) or (S-P) for sine waves

Phasor Diagram of an inductor Phasor Diagram of a resistor v v i i Power = (vi cosθ)/2 = 0 Power = (vi cosθ)/2 = vi/2 Note: No power consumed in inductors i lags v

Phasor Diagram of a capacitor Phasor Diagram of a resistor i v v i Power = (vi cosθ)/2 = 0 Power = (vi cosθ)/2 = vi/2 Note: No power consumed in capacitors i leads v

Power Conservation Theorem “The sum of real power and reactive power in a circuit is equal to zero” This is because power cannot disappear. It can only change form.

Example Find i(t), vL(t)

Phasor Diagram VL V θ = 77.47 I VR 1. Resistor consumes power 2. Inductor consumes no real power P = 0 but it has reactive power

3. Voltage source supplies power Conservation of energy holds Resistor 2 ohm Inductor 3H Voltage source

Power Factor Power factor of a source is the ratio of real power to the complex power Consider the voltage source, Q S Complex Power Diagram Power factor = 0.217 θ = 77.47 P The voltage source supplies 0.292W real power and 1.318VAR reactive power.

Complex Frequency The theory of phasor can be extended to support signal is a fundamental waveform of electrical engineering where s can be a complex number in a rectangular form It is composed of real and imaginary parts. Consider only real part

If σ = 0 The voltage source will be in the format that we already know. If σ is not 0, we get a new form of signal. Define s is called “complex frequency”

Summary of Procedures Change voltage/current sources in to phasor form Change R, L, C value into phasor form Use DC circuit analysis techniques normally, but the value of voltage, current, and resistance can be complex numbers Change back to the time-domain form if the problem asks.

Example Find i(t), ic(t)

We then substitute s = -2+j4 and got

ic(t) can be computed from current divider

Complex Frequency Characteristics 1 1 1