INC 112 Basic Circuit Analysis Week 12 Complex Power Complex Frequency

Example Find i(t) ω = 2

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

Power in AC circuits 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 DC AC

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

V (volts) t (sec) 311V V peak (Vp) = 311 V V peak-to-peak (Vp-p) = 622V V rms = 220V 3 ways to tell voltage 0

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 powerQ = Reactive power Inductor has no real power P =0 But it has complex power, computed by V, I that are perpendicular to each other.

Example Find i(t), v L (t)

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

S P Q Complex Power Diagram Consider the voltage source, The voltage source supplies 0.292W real power and 1.318VAR reactive power. Definition: Power factor = cos θ Power factor = 0.217

Complex Frequency is a fundamental waveform of electrical engineering What if s is a complex number? Let s be a complex number composed of real and imaginary parts.

Phasor Frequency domain Time domain Euler’s Identity

Complex Frequency 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