Transcript RC Circuits - McMaster University

AC Circuits (Chapt 33)
- circuits in which the currents vary in time
- differential equations
AC Voltage
The current in any AC circuit
is driven by an AC source.
This alternating current varies
sinusoidally with time:
Δv = ΔVmax sin ωt
* Δv is the instantaneous voltage
* ΔVmax is the maximum output
voltage of the source
* ω is the angular frequency of the
AC voltage
AC Voltage
• The angular frequency is
2π
ω  2π ƒ 
T
– ƒ is the frequency of the source
– T is the period of the source
• The voltage is positive during one half of the cycle
and negative during the other half
• Commercial electric power plants in Canada/US use a
frequency of 60 Hz
– This corresponds with an angular frequency of 377 rad/s
Resistors in an AC Circuit
• Consider a circuit
consisting of an AC source
and a resistor
• The AC source is
symbolized by
• Δv = ΔvR = Δvmaxsin wt
• ΔvR is the instantaneous
voltage across the resistor
Resistors in an AC Circuit
• The instantaneous current in the resistor is
vR Vmax
iR 

sin ωt  I max sin ωt
R
R
• The instantaneous voltage across the
resistor is also given as:
ΔvR = Imax R sin ωt
Resistors in an AC Circuit
• The graph shows the
current through and the
voltage across the resistor
• The current and the
voltage reach their
maximum values at the
same time
• The current and the
voltage are said to be in
phase
Resistors in an AC Circuit
• For a sinusoidal applied voltage, the current
in a resistor is always in phase with the
voltage across the resistor
• The direction of the current has no effect on
the behavior of the resistor
• Resistors behave essentially the same way
in both DC and AC circuits
RMS Current and Voltage
• The average current in one cycle is zero
• The rms current is the average of importance
in an AC circuit
• rms stands for root mean square
I rms
I max

 0707
.
I max
2
• Alternating voltages can also be discussed in
terms of rms values
Vrms
Vmax

 0707
.
Vmax
2
Example
Show that the RMS values for a sinusoidal function are 0.707
of the max.
Notes About RMS Values
• RMS values are used when discussing
alternating currents and voltages because:
– AC ammeters and voltmeters are designed to
read rms values
– Many of the equations that will be used have
the same form as their DC counterparts
Example
An AC power supply produces a
maximum voltage ΔVmax = 100 V. This
power supply is connected to a 24.0-Ω
resistor, and the current and resistor
voltage are measured with an ideal AC
ammeter and voltmeter, as shown below.
What does each meter read? Note that an
ideal ammeter has zero resistance and
that an ideal voltmeter has infinite
resistance.
Example
In the simple AC circuit shown,
R = 70.0 Ω and Δv = ΔVmax sin ωt.
a) If ΔvR = 0.250 ΔVmax for the first
time at t = 0.010 0 s, what is the
angular frequency of the source?
b) What is the next value of t for
which ΔvR = 0.250 ΔVmax?
Example
Show that the rms value for the sawtooth voltage
shown in the figure is ΔVmax /√3.
Solution