Syllabus Sections:-

A.C. circuits Part 2

3h.1 Recall that for a resistor the p.d. and current are in phase.

This is the simple part !!!

In an AC circuit, such as that below, the current through the resistor is in phase with the voltage across the resistor. This being the case Ohm's Law can be applied with regards to the resistor.

V = I x R

Recall that current lags potential difference by 90° in an inductor and that current leads by 90° in a capacitor.

Re-writing the syllabus makes it easier to understand

Recall that potential difference leads current by 90°in an inductor and that current leads potential difference by 90° in a capacitor.

However the voltage across the capacitor will lag the current by 900 and the voltage can be determined by V = I x Xc

So you cannot just add the voltages but they have to be added using a vector diagram.

Vector diagram

As we are in a series circuit the current is common for the resistor and to the capacitor and is the reference vector. However as we are discussing phases then let's call that the Current phase.

As the voltage is in phase with the current it can be drawn along the same line as the current and its length according to the voltage in this case 15V

The voltage through the capacitor is out of phase by 900 so we can draw that downwards and again is the length corresponding to it value in this case 12V.

The supply voltage is thus the sum of the vectors which equals the length of the diagonal line which can be worked out by the use of Pythagoras's Theorem equation

"the square of the hypotenuse = the sum of the squares on the other two sides"

Thus V2 = VR2 + VC2 where V = VT


Memory aid: This section is quite difficult to grasp for some students so an old hand at Amateur Radio, at a recent BRATS club meeting, suggested the following was added to the course notes.

C I V I L

C I V......V I L

with a Capacitor the current leads the voltage the so I leads the V thus this gives us Capacitor IV

and with an Inductor voltage leads the current so V leads the I thus this gives VI L (Inductor)

With that in mine continue with the topic.


Capacitor - Current leads Potential Difference (C I V )

If the black curve now represents the current through a capacitor then the blue curve represents the potential difference across a capacitor. The current leads the potential difference by 900.

This capacitive reactance can be calculated by the formula XC= 1/2fC

f is in Hertz with C in Farads

Inductor - Potential difference leads the Current ( V I L )

If the black curve represents the potential difference across an inductor then the blue curve represents the current across an inductor. The potential difference leads the current by 900.

So let us explain further. When the Black curve is at position 0 the blue curve has not yet reached the same 0 position thus the black cure is said to lead the blue curve.

This inductive reactance can be calculated by the formula XL=2fL

f is in Hertz with L in Henrys

PD (voltage Wave forms)

Taking this further at the point of supply of AC the wave forms of PD potential difference (voltage) and Current are in phase but when it encounters a capacitor then from above the current leads the PD (voltage) CIV thus in fact across the capacitor the PD or voltage wave will lag behind the supply waveform.

Had it been an Inductor then the PD (voltage) leads the current VIL thus in fact across the Inductor the PD or voltage wave will lead the supply waveform.

Can you draw the appropriate wave forms ??

Recall that the term 'reactance' describes the opposition to current flow in a purely inductive or capacitive circuit where the phase difference between V and I is 90°.

Understand and apply the equations for inductive and capacitive reactance.

The equations are :-

Reactance

Reactance is the opposition to current flow in an AC circuit and can be loosely thought of as AC resistance and has values in ohms. This "AC resistance" is not actually resistance and it must be remembered that a capacitor stores energy electrostatically and an inductor stores energy magnetically.

Taking the case of the capacitor it can be loosely thought of as a very small rechargeable battery which accepts a charge and can then accept no more. On the next half cycle the capacitor then gives up that charge back to the supply and charges in the opposite direction. For this reason a "perfect capacitor" (one with no losses) does not get hot. It will be appreciated that the capacitor will charge and once charged can pass no more electricity hence it then apparently acts as an open circuit.

To differentiate from "normal" resistance we call that caused by the capacitor Capacitance Reactance and that by an inductor Inductive Reactance and both have values in ohms.

, in ohms, represents capacitive reactance in an AC circuit.

, in ohms, represents inductive reactance in an AC circuit.

In the formulae that follow they use certain symbols :-

= frequency in Hz = inductance in henries = capacitance in Farads

= 3.142

Inductive Reactance

If we draw a graph of inductive reactance , against Frequency, we get :-

Note: Inductive reactance is considered to be positive.

Capacitive Reactance

If we draw a similar graph for capacitive reactance ,against frequency we get:-

Note: capacitive reactance is considered to be negative.

Resonance

Now, if we have an LC circuit and draw both graphs on a common frequency line we get :-

Examination will show that only at one particular frequency are the reactances equal and opposite -> the resonance frequency (see the dotted line).

This diagram may make it a little easier to see that the XC = XL and thus cancel each other out as the XL is the positive  and XC a negative  reactance

Series Resonance

We have spoken about the reactance of the coil (inductor) which is like resistance but the coil still has "normal" resistance and for these calculations we call this and identify it in the circuit diagram as a resistor.

Let == 10k and = 10R

10mV RMS at the resonant frequency.

as and cancel each other out, so =

Thus =

Thus = = 1 mA

Voltage across or = x or

Thus Voltage across or =

Thus Voltage across or = 10 volts

Gain

The "gain" between the input and the output is known as magnification factor which in this case = 1000 (10mV rising to 10V = 1000 times) and thus the

magnification factor at resonance =

Parallel Resonance

By varying the frequency of the input current, the point where = approaches and a circulating current increases to a maximum in the parallel loop limited only by . The voltage across the circuit = x .

At the same time the overall resistance to the input current rises to a maximum and the through current is a minimum.

At resonance this circuit is called a rejecter circuit.

Resonance Curves

Assuming that we loosely couple a signal generator into the inductor of a parallel LC circuit and measure the voltage across with an RF voltmeter

Now let us sweep the input frequency and draw graph of the resulting voltage changes.

is the resonant frequency giving a peak voltage, .

The points either side are generally accepted as limits of usable frequency - This range to is the bandwidth [eg. 12kHz for speech] for a parallel LC circuit, which could be the load in a power valve RF amplifier.

Q Meter

This measurement, requiring only a voltmeter, is used in a common form of Q meter. Q meters give a measure of "Goodness" of a component or circuit.

In this case =

ie the Bandwidth compared to the resonant frequency = SELECTIVITY

In the exam the sheet of equations shows this formula as

fC = centre frequency fU = the upper frequency and fL = the lower frequency

There is also something called Carson's rule which is related to the above

This defines the approximate modulation bandwidth required for a carrier signal that is frequency-modulated by a spectrum of frequencies rather than a single frequency.

The Carson bandwidth rule is expressed by the relation Bw = 2(Af+Δf )

Where Bw is the bandwidth requirement,

Af is the highest modulating frequency

Δf is the carrier peak deviation frequency

For example, Carson's rule would say that the bandwidth or speech (300Hz to 3kHz) with a peak deviation of 5kHz would be.

Bw = 2(3kHz+5kHz)

Bw =2*8kHz

Bw =16kHz


Longwave Coils (for interest only)

If we build a receiver for say 120kHz, the RF stage (if we still require 12kHz speech bandwidth) will have a measured "MAX"

= = =

NOTE: a of is very low !!

(Quality Factor)

Providing is greater than a single figure the following basic statements are true with small and negligible errors.

= , the MAGNIFICATION FACTOR is =

The parallel resistance of a tuned circuit at resonance is x . This is known as the DYNAMIC resistance

An example of this is shown, a parallel tuned circuit in the "ANODE" of a RF amplifier.

Let = 1000 Ohms

and = 100

To tune to resonance we adjust until the DYNAMIC RESISTANCE is HIGHEST, ie for a max dip on the ammeter.

The coupling resistance to the next stage (Dynamic Resistance) is x = 100,000 Ohms

The series RF resistance of the coil

= = 10 Ohms

The general expression for Dynamic Resistance is :-

= which does not include any frequency component.

It is obvious from this that the larger compared to , the higher will be the Dynamic Resistance and .


3h.2 Understand that impedance is a combination of resistance and reactance and apply the formula for impedance and current in a series CR or LR circuit.

The formulae are :-

If there is a resistor and an inductor (and or a resistor and capacitor or even a resistor inductor and a capacitor) linked together in series in an AC circuit then the total opposition to the flow of current is known as impedance symbol Z.

Resistance and reactance CANNOT just be added together

The impedance is made up of both resistance R and reactance X, both are measured in ohms but cannot be added together ( as you can do for resistor in series ) but have to be added together like vectors :-

Ohms law can now be applied to the circuit and the current determined by the formula:-


3h.3 Understand the use of capacitors for coupling (d.c. blocking) and decoupling a.c. signals (including r.f. bypass) to ground.

It should be clear to you that from the construction of a capacitor (two separate plates with no link between them) provided no path for DC to pass. Thus it can be said that capacitors block DC and thus can be used for blocking DC in a circuit.

On the other hand in an AC circuit current appears to pass because of the build up and decay of the charge on one plate and then the other AC changes direction of its flow of electrons.

If therefore there is the possibility of an AC signal (and this include RF which is also AC) in a DC circuit then this can be channelled to ground via a capacitor and as such this is called decoupling hence the term "decoupling capacitor".




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