Alternating current circuits
Alternating current
1- Alternating current: an electric current that reverses
its direction many times a second at regular intervals,
typically used in power supplies.
2- The potential difference of an AC power (source) is
also alternating, and in this case, it follows a sine
function where:
v = Vmax sin (w t)
Set calculator to radiant not degrees.
Where:
W: is the angular frequency
(𝑤 =
2𝜋
𝑇
= 2𝜋𝑓)
T: is the period (sec)
f: is the frequency (Hz)
3- Resistors in AC circuits
a. The potential difference across a resistor
and the current passing through resistors
are both alternating, and they are in phase.
b. Ohms law is still applicable. And the current is
directly proportional to the potential differnce
across the resistor.
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4- The power through a resistor can be found using the following
equations:
(𝛥𝑉𝑡 )2
𝑃𝑡 = (𝐼𝑡 )2 𝑅 =
= (𝐼𝑡 )2 𝑅
𝑅
- However, unlike DC circuits in AC circuits the current and the
potential difference have a changing value over time which
also means that the Power will also have a changing value
over time. Thus, we care about the Average and maximum
power for AC circuits where:
(𝛥𝑉𝑚𝑎𝑥 )2
-
𝑃𝑚𝑎𝑥 = (𝐼𝑚𝑎𝑥 )2 𝑅 =
-
𝑃𝑟𝑚𝑠 = (𝐼𝑟𝑚𝑠 )2 𝑅 =
-
The power is always positive even if the current or
potential difference is negative.
RMS: stands for Root Mean Square.
NEVER MIX BETWEEN RMS VALUES AND MAX VALUES
-
𝑅
(𝛥𝑉𝑟𝑚𝑠 )2
𝑅
= (𝐼𝑚𝑎𝑥 )2 𝑅
= (𝐼𝑟𝑚𝑠 )2 𝑅
Created by Majd Al Sharqawi (Zayed educational complex- Al Barsha)
Where:
𝐼𝑟𝑚𝑠 =
1
𝐼
√2 𝑚𝑎𝑥
𝛥𝑉𝑟𝑚𝑠 =
1
√2
= 0 ⋅ 707𝐼𝑚𝑎𝑥
𝛥𝑉𝑚𝑎𝑥 = 0 ⋅ 707𝛥𝑉𝑚𝑎𝑥
LC circuits
1- the self-inductance (L) of a solenoid can be found using the
following equation:
Where:
N: is the number of turns
𝜇0 : the permeability of free space,
µo = 4  x 10-7 T.m / A
A: the Area of the solenoid = (π 𝒓𝟐 )
l: the length of the solenoid
-Note: the self-inductance of a solenoid is only dependent on its physical quantities
2- The potential difference developed due to self-inductance can be found using the following
equation, and its direction will always oppose the change in current.
𝜀𝑖𝑛𝑑 = −𝐿
ⅆ𝐼
ⅆ𝑡
1- LC circuits are circuits that are made of (C) a capacitor,
and (L) an inductor. The capacitor and the inductor keep
exchanging the energy between them resulting in an
Alternating current that keeps changing its direction and
magnitude as a sine function.
Created by Majd Al Sharqawi (Zayed educational complex- Al Barsha)
2- In our study we assume that we deal with and “ideal LC circuit” meaning that it doesn’t
have resistance, and as such in every oscillation the maximum amount of current remains
the same, however in reality there will always be some resistance leading to decrease in
the amount of current with each oscillation as some of the energy gets released outside
the circuit.
Created by Majd Al Sharqawi (Zayed educational complex- Al Barsha)
3- We can calculate the following quantities for an ideal LC circuit:
1-the period of oscillation can be obtained using the following
equation:
𝑇 = 2𝜋√𝐿𝐶
2- the frequency of the period can be found using this equation
𝑓=
3- the angular frequency in the LC circuit can be found using the
following equations:
𝜔=
𝐼𝑚𝑎𝑥 =
4- the maximum current of the LC circuit using this equation:
1
1
√
2𝜋 𝐿𝐶
2𝜋
1
= 2𝜋𝑓 =
𝑇
√𝐿𝐶
𝐶𝛥𝑉max
√𝐿𝐶
or
ⅈ(𝑡) = 𝐼𝑚𝑎𝑥 𝑠ⅈ𝑛(𝑤𝑡)
5- the current at any given time using this equation:
4- The total energy in the circuit in an ideal LC circuit is conserved which means:
𝐸𝑡𝑜𝑡𝑎𝑙 = 𝑈𝐿 + 𝑈𝑐 = 𝑈𝐿𝑚𝑎𝑥 = 𝑈𝑐𝑚𝑎𝑥
Where:
1
𝑈𝐿 = 𝐿𝐼
2
2
&
𝐼𝑚𝑎𝑥 =
1
𝑈𝐶 = 𝐶 (𝛥𝑉
2
)2
1
1 𝑄2
2
2 𝐶
= 𝑄𝛥𝑉 =
Created by Majd Al Sharqawi (Zayed educational complex- Al Barsha)
𝑄max
√𝐿𝐶
RLC circuits
1- LR circuits: circuits that have (L)
an inductor and (R) a resistor. If
you have more than one resistor
in LR circuit, you can
2- In and RLC
- The instantaneous voltage
across the resistor is in
phase with the current
- The instantaneous voltage
across the inductor leads
𝜋
the current by 90° or 2
-
rad.
The instantaneous voltage
across the capacitor lags
𝜋
the current by 90° or 2
rad.
3- The phase angle (𝜙) in an RLC circuit (Resistor, Inductor, Capacitor) refers to the
difference in phase between the input voltage and the current. In other words, it's a
measure of how much the current is leading or lagging the voltage in the circuit,
typically expressed in degrees or radians. Phasor diagrams
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RLC equations
Resistor
Capacitor
Reactance, Resistance, and
Impedance equation
𝛥𝑉𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
𝑅=
𝐼𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
1
1
𝑥𝐶 =
=
2𝜋𝑓𝑐 𝜔𝑐
Inductor
𝑥𝐿 = 2𝜋𝑓𝐿 = 𝜔𝐿
Source
𝑍 = √𝑅 2 + (𝑥𝐿 − 𝑥𝐿 )2
-
Current equation
Phase angle (𝜙)
𝛥𝑉𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
𝑅𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
𝛥𝑉𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
𝐼=
𝑥𝐶
𝛥𝑉𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
𝐼=
𝑥𝐶
0 (rad)
𝐼=
𝐼𝑠𝑜𝑢𝑟𝑐𝑒 = 𝐼𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
= 𝐼𝑖𝑛𝑑𝑢𝑐𝑡𝑜𝑟
= 𝐼𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟
𝜋
− 2 (rad)
𝜋
2
(rad)
𝑥𝐿 − 𝑥𝑐
)
𝑅
𝜙 = tan−1 (
OR
𝛥𝑉𝐿 − 𝛥𝑉𝐶
𝜙 = tan−1 (
)
𝛥𝑉𝑅
IF THE PHASE ANGLE IS POSITIVE THE VOLTAGE LEADS THE CURRENT, AND IF THE
PHASE ANGLE IS NEGATIVE THE CCURRENT LEADS THE VOLTAGE.
Created by Majd Al Sharqawi (Zayed educational complex- Al Barsha)
4- Phasor diagrams: In an RLC circuit, a phasor diagram is used to depict the phase
relationship between the voltage and current across each component. Here's a basic
way to interpret these diagrams:

The horizontal axis represents the resistive component (R), which is in phase
with the AC supply. The current and voltage across a resistor are always in phase,
meaning their peaks occur at the same time.

The vertical axis represents the reactive components (L and C). For an inductor
(L), the current lags the voltage by 90 degrees, so it's represented downwards on
the vertical axis. For a capacitor (C), the current leads the voltage by 90 degrees,
so it's represented upwards on the vertical axis.

The length of each phasor represents the magnitude of the voltage or current for
each component.

The total current in the circuit is represented by a phasor that is the vector sum
of the individual phasors. This is called the resultant phasor.

The angle between the total current phasor and the voltage phasor is the phase
angle for the entire circuit. It tells you whether the total circuit is more capacitive
(current leads voltage) or more inductive (current lags voltage).

Phasor diagrams are a very helpful tool for visualizing and understanding the
behavior of AC circuits. They can show you briefly what the phase relationships
are, how the magnitudes of the voltages or currents compare, and what the
overall behavior of the circuit is.
Created by Majd Al Sharqawi (Zayed educational complex- Al Barsha)
Transformers
1- Transformer is an apparatus used for reducing or increasing the voltage of an alternating
current.
2- Step up transformers increase the voltage AKA potential difference in the secondary coil.
3- Step down transformers decrease the voltage AKA potential difference in the secondary
coil.
4- Long-distance transmission of electrical energy is economical only if low currents and very
high voltages are used.
5- In an ideal transformer no power is lost, as such the power in the primary coil is equal to
the power in the secondary coil.
𝑃𝑃 = 𝑃𝑆
6- The loop with a larger number of turns will have the highest voltage but the lowest
current, and vice versa.
7- The equations needed to solve transformers questions are:
𝐼𝑝 𝑉𝑝 = 𝐼𝑠 𝑉𝑠 &
𝐼𝑠
𝐼𝑃
=
𝑉𝑝
𝑉𝑠
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=
𝑁𝑝
𝑁𝑠
Created by Majd Al Sharqawi (Zayed educational complex- Al Barsha)