Chapter 5

Steady-State Sinusoidal Analysis

Where are we going?

A look ahead:High Pass FilterLow Pass FilterBand Pass FilterThree phase Source

5-1 Sinusoidal Currents and Voltages

v(t) = Vmcos(t + )

Vm = peak value

= angular frequency

= phase angle

T = periodf = Frequencyf = 1/T = 2f = 2/T

Definitions Vrms = root mean square voltage

Vpp = peak-to-peak voltage

Vave = average voltage

Pave = average power

2

VV mrms

mpp V2V

0Vave

rmsrmsave VIP

Example:Find v(t) and i(t)Vrms = root mean square voltageVpp = peak-to-peak voltageVave = average voltagePave = average power

R=50

Plot v versus t Label T

Plot v versus tLabel

Useful Trig Identities

sin(z) = cos(z - 90°)

Example: v(t) = 10sin(t + °)

(a) Write v(t) as a cosine function.

(b) Find Vm, , f, T and .

What are the following for a wall outlet?

f

Vrms = VDCEquivalent

Vm

Vave

Vpp

Notation Summary

Sinusoidal

v1(t) = V1 cos(t + 1)

Polar Phasor

V1 = V1 L 1

Complex Phasor

V1 = V1 cos(1) + j V1 sin(1)

1j

Real

Imaginary

C=A + jB

= phase angle

What is the magnitude of C?

What is the phase (or direction) of C?

22 BAC

A

Btan 1

Math

Example Problems

Convert the following voltages to phasors in polar form and complex form.

v1(t) = 20 cos(t - 45°)

v2(t) = 20 sin(t + 60°)

Example ProblemsConvert the following from complex

phasors to polar form.

V1 = 30 + j40

V2 = 4 - j20

V1 + V2

Phasor Math in Polar Form

(C11) (C22) = C1 C2 (1 + 2)

(C11) /(C22) = C1/C2 (1 - 2)

Ohm’s Law for AC Circuits

V = I Z

Impedance

Resistors

Suppose that v(t) = Vmcos(t)

What is i(t) ? Hint: v = i R

What is the phase relationship between i and v?

Capacitors

Suppose that v(t) = Vmcos(t)

What is i(t) ? Hint: v = q/C

What is the phase relationship between i and v?

Inductors

Suppose that i(t) = Imcos(t)

What is v(t) ? Hint: v(t) = L di/dt

What is the phase relationship between i and v?

Reactance, Impedance and Phasors

Go to notes…

Phasor Diagrams for V and I

Impedance Diagrams for R, L and C circuits

The R, L, and C Elements Ohm’s Law for Peak Values

Resistors: Vp=IpR

Capacitors: Vp=IpXC

Inductors: Vp=IpXL

Go to notes...

Example Problems

Example 3.15:

Convert the following from polar to rectangular form.

1053.1316-3025120

Example Problems

Example 3.15:

Convert the following from rectangular to polar form.

30 + j40

4 - j20

-3 - 4j

Impedance Diagrams

Resistor

ZR = R0

Capacitor

ZC = XC-90

Inductor

ZL = XL90

RL Circuit Example

Connect at AC power supply in series with an inductor and a resistor.

How does VR vary with the input frequency?

RC Circuit Example

Connect at AC power supply in series with an capacitor and a resistor.

How does VR vary with the input frequency?

RLC Circuit Example

Connect at AC power supply in series with an inductor, capacitor and a resistor.

How does VR vary with the input frequency?

3.18 Tuned Resonant Networks

RLC Series circuits are used in radios.Series RLC networks have a resonant

frequency that depends on C and L only.

LC2

1fs

What capacitance do you need to listen to 107.7 MHz on a radio with a 1H inductor?

Chapter 5

Steady-State Sinusoidal Analysis