AC Instantaneous and RMS:
Difference between AC AND DC:
RLC Series Circuit:
RC Series circuit:
Pure Inductive circuits
RMS Value:
Pure Resistive circuit:

__AC Instantaneous and RMS:__

**Instantaneous Value:**

The
Instantaneous value of an alternating voltage or current is the value of
voltage or current at one particular instant. The value may be zero if the
particular instant is the time in the cycle at which the polarity of the
voltage is changing. It may also be the same as the peak value, if the selected
instant is the time in the cycle at which the voltage or current stops
increasing and starts decreasing. There are actually an infinite number of
instantaneous values between zero and the peak value.

**RMS Value:**

The
average value of an AC waveform is NOT the same value as that for a DC
waveforms average value. This is because the AC waveform is constantly changing
with time and the heating effect given by the formula ( P = I ^{2}.R ),
will also be changing producing a positive power consumption. The equivalent
average value for an alternating current system that provides the same power to
the load as a DC equivalent circuit is called the "effective value".
This effective power in an alternating current system is therefore equal to: (
I^{2}.R. Average).

As power
is proportional to current squared, the effective current, I will be equal to √
I 2 Ave. Therefore, the effective current in an AC system is called the Root
Mean Squared or R.M.S.

__Pure Resistive circuit:__

Resistors
are “passive” devices that are they do not produce or consume any electrical
energy, but convert electrical energy into heat. In DC circuits the linear
ratio of voltage to current in a resistor is called its resistance. However, in
AC circuits this ratio of voltage to current depends upon the frequency and
phase difference or phase angle ( φ ) of the supply. So when using resistors in
AC circuits the term **Impedance**,
symbol **Z** is the generally used and
we can say that DC resistance = AC impedance, R = Z.

It is
important to note, that when used in AC circuits, a resistor will always have
the same resistive value no matter what the supply frequency from DC to very
high frequencies, unlike capacitor and inductors.

For
resistors in AC circuits the direction of the current flowing through them has
no effect on the behaviour of the resistor so will rise and fall as the voltage
rises and falls. The current and voltage reach maximum, fall through zero and
reach minimum at exactly the same time. i.e, they rise and fall simultaneously
and are said to be “in-phase” as shown below.

We can
see that at any point along the horizontal axis that the instantaneous voltage
and current are in-phase because the current and the voltage reach their
maximum values at the same time, that is their phase angle θ is 0^{o}.
Then these instantaneous values of voltage and current can be compared to give
the ohmic value of the resistance simply by using ohms law. Consider below the
circuit consisting of an AC source and a resistor.

The
instantaneous voltage across the resistor, V_{R} is equal to the supply
voltage, V_{t} and is given as:

V_{R}
= V_{max} sinωt

The
instantaneous current flowing in the resistor will therefore be:

I_{R}
= V_{R} / R

= V_{max}
sinωt / R

= I _{max}
sinωt

In purely
resistive series AC circuits, all the voltage drops across the resistors can be
added together to find the total circuit voltage as all the voltages are
in-phase with each other. Likewise, in a purely resistive parallel AC circuit,
all the individual branch currents can be added together to find the total
circuit current because all the branch currents are in-phase with each other.

Since for
resistors in AC circuits the phase angle φ between the voltage and the current
is zero, then the power factor of the circuit is given as cos 0^{o} =
1.0. The power in the circuit at any instant in time can be found by
multiplying the voltage and current at that instant.

Then the
power (P), consumed by the circuit is given as P = Vrms Ι cos Φ in watt’s. But
since cos Φ = 1 in a purely resistive circuit, the power consumed is simply
given as, P = Vrms Ι the same as for Ohm’s Law.

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This then
gives us the “Power” waveform and which is shown below as a series of positive
pulses because when the voltage and current are both in their positive half of
the cycle the resultant power is positive. When the voltage and current are
both negative, the product of the two negative values gives a positive power
pulse.

Then the
power dissipated in a purely resistive load fed from an AC rms supply is the
same as that for a resistor connected to a DC supply and is given as:

P = V rms
* I rms

= I 2 rms *
R

= V 2 rms /
R

__Pure Inductive circuits:__

This
simple circuit above consists of a pure inductance of L Henries ( H ),
connected across a sinusoidal voltage given by the expression: V(t) = V_{max}
sin ωt. When the switch is closed this sinusoidal voltage will cause a current
to flow and rise from zero to its maximum value. This rise or change in the
current will induce a magnetic field within the coil which in turn will oppose
or restrict this change in the current.

But
before the current has had time to reach its maximum value as it would in a DC
circuit, the voltage changes polarity causing the current to change direction.
This change in the other direction once again being delayed by the self-induced
back emf in the coil, and in a circuit containing a pure inductance only, the
current is delayed by 90^{o}.

The
applied voltage reaches its maximum positive value a quarter ( 1/4ƒ ) of a
cycle earlier than the current reaches its maximum positive value, in other
words, a voltage applied to a purely inductive circuit “LEADS” the current by a
quarter of a cycle or 90^{o} as shown below.

The
instantaneous voltage across the resistor, V_{R} is equal to the supply
voltage, V_{t} and is given as:

V_{L}
= V_{max} sin (ωt + 90)

I_{L}
= V / X_{L}

X_{L}
= 2πfL

__Pure Capacitive circuits:__

When the
switch is closed in the circuit above, a high current will start to flow into
the capacitor as there is no charge on the plates at *t = 0*. The sinusoidal supply voltage, V is increasing in a positive
direction at its maximum rate as it crosses the zero reference axis at an
instant in time given as 0^{o}. Since the rate of change of the
potential difference across the plates is now at its maximum value, the flow of
current into the capacitor will also be at its maximum rate as the maximum
amount of electrons are moving from one plate to the other.

As the
sinusoidal supply voltage reaches its 90^{o} point on the waveform it
begins to slow down and for a very brief instant in time the potential
difference across the plates is neither increasing nor decreasing therefore the
current decreases to zero as there is no rate of voltage change. At this 90^{o}point
the potential difference across the capacitor is at its maximum ( V_{max}
), no current flows into the capacitor as the capacitor is now fully charged
and its plates saturated with electrons.

At the
end of this instant in time the supply voltage begins to decrease in a negative
direction down towards the zero reference line at 180^{o}. Although the
supply voltage is still positive in nature the capacitor starts to discharge
some of its excess electrons on its plates in an effort to maintain a constant
voltage. These results in the capacitor current flowing in the opposite or
negative direction.

When the
supply voltage waveform crosses the zero reference axis point at instant 180^{o},
the rate of change or slope of the sinusoidal supply voltage is at its maximum
but in a negative direction, consequently the current flowing into the
capacitor is also at its maximum rate at that instant. Also at this 180^{o}
point the potential difference across the plates is zero as the amount of
charge is equally distributed between the two plates.

Then
during this first half cycle 0^{o} to 180^{o}, the applied
voltage reaches its maximum positive value a quarter (1/4ƒ) of a cycle after
the current reaches its maximum positive value, in other words, a voltage
applied to a purely capacitive circuit “LAGS” the current by a quarter of a
cycle or 90^{o} as shown below.

I_{C}
= I_{max} sin (ωt + 90)

I_{L}
= V / X_{C}

X_{C}
= 1 / 2πfC

__RL Series circuit:__

_{ }

In othe r words, an Inductor in an
electrical circuit opposes the flow of current, ( i ) through it. While this is
perfectly correct, we made the assumption in the tutorial that it was an ideal
inductor which had no resistance or capacitance associated with its coil
windings.

However,
in the real world “ALL” coils whether they are chokes, solenoids, relays or any
wound component will always have a certain amount of resistance no matter how
small associated with the coils turns of wire being used to make it as the
copper wire will have a resistive value.

Then for
real world purposes we can consider our simple coil as being an “Inductance”, L
in series with a “Resistance”, R. In other words forming an **LR Series Circuit.**

A **LR Series Circuit** consists basically of
an inductor of inductance L connected in series with a resistor of resistance
R. The resistance R is the DC resistive value of the wire turns or loops that
goes into making up the inductors coil

The above
*LR series circuit* is connected across
a constant voltage source, (the battery) and a switch. Assume that the switch,
S is open until it is closed at a time t = 0, and then remains permanently
closed producing a “step response” type voltage input. The current, i begins to
flow through the circuit but does not rise rapidly to its maximum value of Imax
as determined by the ratio of V / R(Ohms Law).

This
limiting factor is due to the presence of the self induced emf within the
inductor as a result of the growth of magnetic flux, (Lenz’s Law). After a time
the voltage source neutralizes the effect of the self induced emf, the current
flow becomes constant and the induced current and field are reduced to zero.

We can
use Kirchoffs Voltage Law, ( Kirchoffs Voltage Law, (KVL) to define the
individual voltage drops that exist around the circuit and then hopefully use
it to give us an expression for the flow of current.

Vt = VR +
VL

VR = I*R

VL = i dL
/ dt

V(t)
= I*R + i dL / dt

Since the
voltage drop across the resistor, V_{R} is equal to IxR (Ohms Law), it
will have the same exponential growth and shape as the current. However, the
voltage drop across the inductor, V_{L} will have a value equal to: Ve^{(-Rt/L)}.
Then the voltage across the inductor, V_{L} will have an initial value equal
to the battery voltage at time t = 0 or when the switch is first closed and
then decays exponentially to zero as represented in the above curves.

The time
required for the current flowing in the LR series circuit to reach its maximum
steady state value is equivalent to about **5
time constants** or 5τ. This time constant τ, is measured by τ = L/R, in
seconds, were R is the value of the resistor in ohms and L is the value of the
inductor in Henries. This then forms the basis of an RL charging circuit were
5τ can also be thought of as “5 x L/R” or the *transient time* of the circuit.

The
transient time of any inductive circuit is determined by the relationship
between the inductance and the resistance. For example, for a fixed value
resistance the larger the inductance the slower will be the transient time and
therefore a longer time constant for the LR series circuit. Likewise, for a
fixed value inductance the smaller the resistance value the longer the
transient time.

However,
for a fixed value inductance, by increasing the resistance value the transient
time and therefore the time constant of the circuit becomes shorter. This is
because as the resistance increases the circuit becomes more and more resistive
as the value of the inductance becomes negligible compared to the resistance.
If the value of the resistance is increased sufficiently large compared to the
inductance the transient time would effectively be reduced to almost zero.

__RC Series circuit:__

The
fundamental passive linear circuit elements are the resistor (R), capacitor (C)
and inductor (L). These circuit elements can be combined to form an electrical
circuit in four distinct ways: the RC circuit, the RL circuit, the LC circuit
and the RLC circuit with the abbreviations indicating which components are
used. These circuits exhibit important types of behaviour that are fundamental
to analogue electronics. In particular, they are able to act as passive
filters. This article considers the RL circuit in both series and parallel as shown
in the diagrams.

In
practice, however, capacitors (and RC circuits) are usually preferred to
inductors since they can be more easily manufactured and are generally
physically smaller, particularly for higher values of components.

Both RC
and RL circuits form a single-pole filter. Depending on whether the reactive
element (C or L) is in series with the load, or parallel with the load will
dictate whether the filter is low-pass or high-pass.

Frequently
RL circuits are used for DC power supplies to RF amplifiers, where the inductor
is used to pass DC bias current and block the RF getting back into the power
supply.

__RLC Series Circuit:__

_{ }

_{ }

**Difference between AC AND DC:**

Current
that flows continuously in one direction is called direct current . Alternating
current (A.C) is the current that flows in one direction for a brief time then
reverses and flows in opposite direction for a similar time. The source for
alternating current is called AC generator or alternator.

**Cycle:**

One
complete set of positive and negative values of an alternating quantity is called

cycle.

**Frequency:**

The
number of cycles made by an alternating quantity per second is called
frequency. The unit of frequency is Hertz(Hz)

**Amplitude or Peak value**

The
maximum positive or negative value of an alternating quantity is called
amplitude or peak value.

**Average value:**

This is
the average of instantaneous values of an alternating quantity over one
complete cycle of the wave.

**Time period:**

The time
taken to complete one complete cycle.

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Basic Electrical and electronics : Electric Circuits and Measurements : Introduction to AC Circuits |

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