1. Alternating Current (AC) R, L, C in AC circuits
Chapter 33
Alternating Current (AC)
R, L, C in AC circuits

2. AC, the description
A DC power source, like the one from a battery, provides a potential difference (a voltage) that does not change its polarity with respect to a reference point (often the ground)
An AC power source is sinusoidal voltage source which is described as
Here t
is the instantaneous voltage with respect to a reference (often not the ground).
is the maximum voltage or amplitude.
is the angular frequency, related to frequency f and period T as
Symbol in a circuit diagram: or
The US AC system is 110V/60Hz.
Many European and Asian countries use 220V/50Hz.

3. Resistors in an AC Circuit, Ohm’s Law
The voltage over the resistor:
Apply Ohm’s Law, the current through the resistor:
The current is also a sinusoidal function of time t. The current through and the voltage over the resistor are in phase: both reach their maximum and minimum values at the same time.
The power consumed by the resistor is
We will come back to the power discussion later.
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ACTIVE FIGURE

4. Phasor Diagram, a useful tool.y
The projection of a circular motion with a constant angular velocity on the y-axis is a sinusoidal function.
To simplify the analysis of AC circuits, a graphical constructor called a phasor diagram is used. A phasor is a vector whose length is proportional to the maximum value of the variable it represents
The phasor diagram of a resistor in AC is shown here. The vectors representing current and voltage overlap each other, because they are in phase. R O x
The projection on the y-axis is

5. The power for a resistive AC circuit and the rms current and voltage
When the AC voltage source is applied on the resistor, the voltage over and current through the resistor are:
Both average to zero.
But the power over the resistor is
Pmax
Pav P
And it does not average to zero. The averaged power is:

6. The power for a resistive AC circuit and the rms current and voltage
So the averaged power the resistor consumes is
If the power were averaged to zero, like the current and voltage, could we use AC power source here?
The averaged power can also be written as:
Pmax
Pav P
Define a root mean square for the voltage and current: or
One get back to the DC formula equivalent:

7. Resistors in an AC Circuit, summary
Ohm’s Law applies. Te current through and voltage over the resistor are in phase.
The average power consumed by the resistor is
From this we define the root mean square current and voltage. AC meters (V or I) read these values.
The US AC system of 110V/60Hz, here the 110 V is the rms voltage, and the 60 Hz is the frequency f, so

8. Inductors in an AC circuit, voltage and current
The voltage over the inductor is
To find the current i through the inductor, we start with Kirchhoff’s loop rule: or
Solve the equation for i or

9. Inductors in an AC circuit, voltage leads current
Examining the formulas for voltage over and current through the inductor:
Voltage leads the current by ¼ of a period (T/4 or 90° or π/2) . Or in a phasor diagram, the rotating current vector is 90° behind the voltage vector.
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ACTIVE FIGURE

10. Inductive Reactance, the “resistance” the inductor offers in the circuit.
Examining the formulas for voltage over and current through the inductor again:
This time pay attention to the relationship between the maximum values of the current and the voltage:
This could be Ohm’s Law if we define a “resistance” for the inductor to be:
And this is called the inductive reactance. Remember, it is the product of the inductance, and the angular frequency of the AC source. I guess that this is the reason for it to be called a “reactance” instead of a passive “resistance”. The following formulas may be useful:

11. Capacitors in an AC circuit, voltage and current
The voltage over the capacitor is
To find the current i through the capacitor, we start with Kirchhoff’s loop rule: or
Solve the first equation for q, and take the derivative for i or
Here I still like to keep the Ohm’s Law type of formula for voltage, current and a type of resistance.

12. Capacitors in an AC circuit, current leads the voltage
Examining the formulas for voltage over and current through the capacitor:
Current leads the voltage by ¼ of a period (T/4 or 90° or π/2) . Or in a phasor diagram, the rotating voltage vector is 90° behind the current vector.
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ACTIVE FIGURE

13. Capacitive Reactance, the “resistance” the capacitor offers in the circuit.
Examining the formulas for voltage over and current through the capacitor again:
This time pay attention to the relationship between the maximum values of the current and the voltage:
This could be Ohm’s Law if we define a “resistance” for the capacitor to be:
And this is called the capacitive reactance. It is the inverse of the product of the capacitance, and the angular frequency of the AC source. The following formulas may be useful:

14. The RLC series circuit, current and voltage
The voltage over the RLC is
Now let’s find the current. From the this equation, write out each component:
Apply to both sides, and remember
That we have
Phase angle between current and voltage
“Simply” solve for the current i :
Overall resistance
Phase angle
Where:

15. The RLC series circuit, current and voltage, solved with Phasor Diagrams
The RLC are in serial connection, the current i is common and must be in phase: i
So use this as the base (the x-axis) for the phasor diagrams:

16. The RLC series circuit, current and voltage, solved with Phasor Diagrams
Now overlap the three phasor diagrams, we have:

17. The RLC series circuit, current and voltage, solved with Phasor Diagrams
Now from final phasor diagram, we get the voltage components in x- and y-axes:
From:
or:
We have:
Here Z is the overall “resistance”, called the impedance.
From the diagram, the phase angle is
We have:
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ACTIVE FIGURE

18. Determining the Nature of the Circuit
If f is positive
XL> XC (which occurs at high frequencies)
The current lags the applied voltage
The circuit is more inductive than capacitive
If f is negative
XL< XC (which occurs at low frequencies)
The current leads the applied voltage
The circuit is more capacitive than inductive
If f is zero
XL= XC (which occurs at )
The circuit is purely resistive and the impedance is minimum, and current reaches maximum, the circuit resonates.
Often this resonant frequency is called