INC 112 Basic Circuit Analysis Week 7 Introduction to AC Current

Meaning of AC Current AC = Alternating current means electric current that change up and down When we refer to AC current, another variable, time (t) must be in our consideration.

Alternating Current (AC) Electricity which has its voltage or current change with time. Example: We measure voltage difference between 2 points Time1pm2pm3pm4pm5pm6pm DC: 5V5V5V5V5V5V AC:5V3V2V-3V-1V2V

Signals Signal is an amount of something at different time, e.g. electric signal. Signals are mentioned is form of 1.Graph 2.Equation

Graph Voltage (or current) versus time V (volts) t (sec) v(t) = sin 2t 1 st Form 2 nd Form

V (volts) t (sec) DC voltage v(t) = 5

Course requirement of the 2 nd half Students must know voltage, current, power at any point in the given circuits at any time. e.g. What is the current at point A? What is the voltage between point B and C at 2pm? What is the current at point D at t=2ms?

Periodic Signals Periodic signals are signal that repeat itself. Definition Signal f(t) is a periodic signal is there is T such that f(t+T) = f(t), for all t T is called the period, where when f is the frequency of the signal

Example: v(t) = sin 2t Period = πFrequency = 1/π v(t+π) = sin 2(t+π) = sin (2t+2π) = sin 2t (unit: radian) Note: sine wave signal has a form of sin ωt where ω is the angular velocity with unit radian/sec

Sine wave Square wave

Fact: Theorem: (continue in Fourier series, INC 212 Signals and Systems) “Any periodic signal can be written in form of a summation of sine waves at different frequency (multiples of the frequency of the original signal)” e.g. square wave 1 KHz can be decomposed into a sum of sine waves of reqeuency 1 KHz, 2 KHz, 3 KHz, 4 KHz, 5 KHz, …

Implication of Fourier Theorem Sine wave is a basis shape of all waveform. We will focus our study on sine wave.

Properties of Sine Wave 1. Frequency 2. Amplitude 3. Phase shift These are 3 properties of sine waves.

Frequency sec volts Period ≈ 6.28, Frequency = Hz period

Amplitude sec volts Blue 1 volts Red 0.8 volts

Phase Shift Period=6.28 Phase Shift = 1 Red leads blue 57.3 degree (1 radian)

Sine wave in function of time Form: v(t) = Asin( ω t+ φ) Amplitude Frequency (rad/sec) Phase (radian) e.g. v(t) = 3sin( 8 πt+π/4 ) volts Amplitude 3 volts Frequency 8π rad/sec or 4 Hz Phase π/4 radian or 45 degree

Basic Components AC Voltage Source, AC Current Source Resistor (R) Inductor (L) Capacitor (C)

AC Voltage Source AC Current Source Voltage Source Current Source เช่น Amplitude = 10V Frequency = 1Hz Phase shift = 45 degree

What is the voltage at t =1 sec ?

Resistors Same as DC circuits Ohm’s Law is still usable V = IR R is constant, therefore V and I have the same shape.

Find i(t) Note: Only amplitude changes, frequency and phase still remain the same.

Power in AC circuits In AC circuits, voltage and current fluctuate. This makes power at that time (instantaneous power) also fluctuate. Therefore, the use of average power (P) is prefer. Average power can be calculated by integrating instantaneous power within 1 period and divide it with the period.

Assume v(t) in form Change variable of integration to θ We get Then, find instantaneous power integrate from 0 to 2π

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 convenient in calculating power Note: V rms and I rms are constant, independent of time For sine wave Asin( ω t+ φ)

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

Inductors Inductance has a unit of Henry (H) Inductors have V-I relationship as follows This equation compares to Ohm’s law for inductors.

Find i(t) from

ωL is called impedance (equivalent resistance) Phase shift -90

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

DC Characteristics When stable, L acts as an electric wire. When i(t) is constant, v(t) = 0

Capacitors Capacitance has a unit of farad (f) Capacitors have V-I relationship as follows This equation compares to Ohm’s law for capacitors.

Find i(t) Impedance (equivalent resistance) Phase shift +90

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

DC Characteristics When stable, C acts as open circuit. When v(t) is constant, i(t) = 0

Combination of Inductors

Combination of Capacitors

Linearity Inductors and capacitors are linear components If i(t) goes up 2 times, v(t) will also goes up 2 times according to the above equations

Transient Response and Forced response

Purpose of the second half Know voltage or current at any given time Know how L/C resist changes in current/voltage. Know the concept of transient and forced response

Characteristic of R, L, C Resistor resist current flow Inductor resists change of current Capacitor resists change of voltage L and C have “dynamic”

I = 1A I = 2A Voltage source change from 1V to 2V immediately Does the current change immediately too?

Voltage Current time 1V 2V 1A 2A AC voltage

I = 1A I = 2A Voltage source change from 1V to 2V immediately Does the current change immediately too?

Voltage Current time 1V 2V 1A 2A Forced Response Transient Response + Forced Response AC voltage

Unit Step Input and Switches Voltage time 0V 1V This kind of source is frequently used in circuit analysis. Step input = change suddenly from x volts to y volts Unit-step input = change suddenly from 0 volts to 1 volt at t=0

This kind of input is normal because it come from on-off switches.

PSPICE Example All R circuit, change R value RL circuit, change L RC circuit, change C

I am holding a ball with a rope attached, what is the movement of the ball if I move my hand to another point? Movements 1.Oscillation 2.Forced position change Pendulum Example

Transient Response or Natural Response (e.g. oscillation, position change temporarily) Fade over time Resist changes Forced Response (e.g. position change permanently) Follows input Independent of time passed

Forced response Natural response at different time Mechanical systems are similar to electrical system

connect i(t) Stable Changing Transient Analysis Phasor Analysis

Transient Response RL Circuit RC Circuit RLC Circuit First-order differential equation Second-order differential equation