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
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