Admin:• Assignment 6 is posted. Due Monday.

• Thevenin hint – the current source appears to be shared between meshes…. Until you remove the load...

• Assignments make up 20% of your grade, and are relatively easy points – keep up with them (and if you miss any, complete them with the late penalty)

• Collect your exam at the end of class

• Discussion sections as usual this week.

• Moving onto AC circuits – tricky part. Shout at me if I go too fast!

Alternating Current

• pure direct current = DC

• Direction of charge flow (current) always the same and constant.

• pulsating DC (this is unusual)

• Direction of charge flow always the same but variable.

• AC = Alternating Current

• Direction of Charge flow alternates

DC

AC

V

V

-V

t

Why use AC? The "War of the Currents"

• Late 1880's: Westinghouse backed AC, developed by Tesla, Edison backed DC (despite Tesla's advice). Edison killed an elephant (with AC) to prove his point.

• Turning point came when Westinghouse won the contract for the Chicago Worlds fair

• Westinghouse was right:

• Lowest transmission loss uses High Voltages and Low Currents (less resistive heating)

• In DC, difficult to transform high voltage to more practical low voltage efficiently

• AC transformers are simple and extremely efficient - see later.

• Nowadays, distribute electricity at up to 765 kV

AC circuits: Sinusoidal Waves

• Fundamental wave form

• Fourier Theorem: Can construct any other wave form (e.g. square wave) by adding sinusoids of different frequencies

• x(t)=Acos(ωt+)

• ω=2πf (rad/s) angular frequency

• f=1/T (cycles/s = Hz) frequency

• =2π(Δt/T) rad/s phase

• =360(Δt/T) deg/s

RMS quantities in AC circuits

• What's the best way to describe the strength of a varying AC signal?

• Average = 0; Peak=+/-

• Sometimes use peak-to-peak

• Usually use Root-mean-square (RMS)

• (DVM measures this)

rmsrmsavep

rmsp

rms VIPV

VI

I ,2

,2

i-V relationships in AC circuits: Resistors

Source vs(t)=AsinωtvR(t)= vs(t)=Asinωt

tR

A

R

tvti R

R sin)(

)(

vR(t) and iR(t) vary in a similar way over time

They are in phase with each other

Play animation

Complex Number Review

A complex number consists of a Real part (a) and an Imaginary part (b).

a + jb

It represents a point on the complex plane.

Note that, in electronics, “j” is usually used to denote the imaginary unit ( j2 = -1 ), since “i” is more commonly used for current.

AC current or Voltage signals can be expressed as complex numbers. The Real part of the complex number is equal to the actual (physical) current, or Voltage.

The imaginary part is just that – imaginary. It’s simply a mathematical trick which makes calculating the real current and Voltage easier

A complex number can be written in a few different forms.

These are all equivalent:

is often used in electronics

A is the modulus of the complex number, equal to distance from the origin

A= θ=tan-1(b/a)

An example:

1 + 2 j

An example:

1 + 2 j

θ = tan-1(2/1) = 63.4 degreesA=sqrt(12+22) = 2.24

Phasor form = 2.24 /_ 63.4= 2.24 e j63.4

= 2.24(cos[63.4] + sin[63.4])

Real part = 1 = A cosθImaginary part = 2 = A sinθ