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Wize University Physics Textbook (Master) > AC Circuits

Basics of AC Circuits

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AC Voltage Sources


As their name suggests, alternating current (AC) circuits have properties that change with time. We need to define some terms and tools in order to analyze these systems.

AC Voltage
  • The source voltage of an AC circuit must change with time.
  • This can be written as a periodic function:
V(t)=Vmax⁡sin⁡(ωt)\boxed{V(t)=V_{\max}\sin\left(\omega \\t\right)}
  • In this formula:
  • Vmax⁡V_{\max} is the maximum voltage of the source (sometimes calledVoV_oor VpeakV_{peak})
  • ω\omega is the angular frequency of the source (in rad/s).
Wize Tip
You may also encounter this equation written in terms of emf, ε(t)=εmax⁡sin⁡(ωt)\varepsilon\left(t\right)=\varepsilon_{\max}\sin\left(\omega \\t\right).

Write it Down
Angular frequency and frequency (number of cycles per second) are related by the equation ω=2πf\omega=2\pi f.

The period (time taken for one cycle) is the inverse of the frequency: f=1/Tf=1/T.

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RMS and other definitions
  • Usually, AC sources are described by their RMS voltage (root-mean-square):
Vrms=12Vmax⁡\boxed{V_{rms}=\frac{1}{\sqrt{2}}V_{\max}}
  • Peak-to-peak voltage(VPP)\left(V_{PP}\right)is defined as the voltage difference between the max and min points.
  • This is equal to 2Vmax⁡2V_{\max}.
  • The period (T) can be measured from graphs such as the one above.
  • For example, you can measure the time between two peaks (maximums) or two troughs (minimums), or double the time between a peak and a trough.

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Example: Household Voltages


In Canada, household voltage sources are delivered at 120 V and 60 Hz.
a) What is the maximum voltage delivered by such a voltage source?
b) What is the average voltage delivered by this source?
c) What are the angular frequency and period of such a voltage source?

Part a)

We use the standard RMS formula to convert from rms to maximum voltage:
Vrms=Vmax12Vmax=2VrmsVmax=2(120V)Vmax=169.7V\begin{aligned} V_{rms}&=V_{max}\frac1{\sqrt{2}} \\ V_{max}&=\sqrt2 V_{rms}\\ V_{max}&=\sqrt2 (120V)\\ V_{max}&=169.7V\\ \end{aligned}

Part b)

All AC voltage sources have an average delivered voltage of zero. That's why we use rms!

Part c)

We can convert frequency to angular frequency as follows:
ω=2πfω=2π(60 Hz)ω=377 rad/s\begin{aligned} \omega&=2\pi f \\ \omega&=2\pi (60~Hz)\\ \omega&=377~rad/s \end{aligned}
To find the period, we take the inverse of the frequency:
T=1fT=160 HzT=0.0167 s\begin{aligned} T&=\frac1f \\ T&=\frac1{60~Hz}\\ T&=0.0167~s \end{aligned}

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Resistive AC Circuits and AC Current


The simplest type of AC circuit has only an AC voltage source and a resistor.
  • Consider an AC source with a maximum voltage of VmaxV_{max} and an angular frequency of ω\omega connected to a resistance RR.

  • Let's apply Kirchhoff's Loop Rule (the sum of all voltage drops in a closed loop is zero):
ΔV=0Vsource−IR=0Vmaxsin⁡(ωt)−IR=0I(t)=VmaxRsin⁡(ωt)\begin{aligned} \Delta V&=0 \\ V_{source}-IR&=0 \\ V_{max}\sin(\omega t)-IR&=0 \\ \end{aligned}\\ \boxed{I(t)=\frac{V_{max}}{R}\sin(\omega t)}

Wize Concept
This tells us, for simple resistive AC circuits, that the current varies in phase with the voltage. That is, when the voltage is a max, the current is also a max; and when the voltage is a min, the current is also a min

Wize Tip
The above equation could also have been derived with Ohm's Law for ideal resistors, V=IRV=IR.

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  • Because the current is time-varying with the same frequency, we can write the AC current as follows:
I(t)=Imaxsin⁡(ωt)\boxed{I(t)=I_{max}\sin(\omega t)}
  • We can also define the rms current:
Irms=12Imax⁡\boxed{I_{rms}=\frac{1}{\sqrt{2}}I_{\max}}
  • The power dissipated also varies in time.
  • The average power dissipated can be found with rms current and voltage values:
Pavg=IrmsVrmsPavg=Vrms2RPavg=Irms2R\begin{aligned} P_{avg}&=I_{rms}V_{rms} \\ P_{avg}&=\frac{V^2_{rms}}{R} \\ P_{avg}&=I^2_{rms}R \\ \end{aligned}

Wize Tip
The one thing in this section that does NOT vary is resistance. Resistance is a material property of an object and does not depend on currents that may or may not be flowing through them.

Practice: Resistive AC Circuit


Consider a voltage source with maximum voltage Vo=200 VV_o=200~V and frequency f=300 Hzf=300~Hz connected to a resistor of resistance 50 Ω50~\Omega.
a) Write down a formula for the current through the resistor as a function of time.
b) What is the average power dissipated by the resistor?
Part a)