Chelmsford Amateur Radio Society Advanced Course (3) Technical Aspects Part-4 - AC Circuits

1Chelmsford Amateur Radio SocietyAdvanced Licence Course

Carl Thomson G3PEM Slide Set 4: v1.2, 20-Aug-2006(3) Technical Aspects - AC Circuits

Chelmsford Amateur Radio Chelmsford Amateur Radio Society Society

Advanced CourseAdvanced Course(3) Technical Aspects(3) Technical Aspects

Part-4 - AC CircuitsPart-4 - AC Circuits



AC Generation

• Consider a rotating coil in a magnetic field

• Voltage is induced when the ‘magnetic flux’ lines are cut

• As the coil rotates, the Output is a Sine Wave

+V

Time

One Rotation

-V

N S

AC Volts OutputBrush

Slipring



Period & Frequency

• In the last courses we just described the shape of a sine wave

• The Period, T of one cycle, in seconds is equal to 1/f, where f is in Hertz

Frequency, f = 1 / T or Period, T = 1 / f

Amplitude

Time

One Cycle



Phase

• Another way of looking at the sine wave is as a cycle of 360 degrees

• The voltage or current has a complete rotation as in the generator;

• This indicates the phase of the signal at any part of the cycle

• Phase difference can be used to describe the delay between two signals.

• Phasor diagrams also describe the phase difference - See Handbook

180° 360°

0°

Vmax

Vmin

90°

270°

Time



R.M.S. Value

• RMS = Root Mean Square

• The RMS value of any varying shaped waveform is the equivalent of the constant DC Voltage that would have the same power or heating effect

• For a sine wave, the RMS value is equal to 1/2 of the peak value.

Vrms = 0.707 . Vpeak and Irms = 0.707 . Ipeak

Vpeak

Vrms

Time

One Period, T



AC with Pure Resistance

RF, Hz

VPhasor Diagram

V I

• Voltage and Current are in Phase• Standard Ohms Law Applies



AC with Pure Inductance

Phasor DiagramV

I

• THE CURRENT LAGS 90° BEHIND THE VOLTAGE• The magnitude of the current depends upon;

a) the inductanceb) the frequency of the applied ac current.

• These two factors influence the Back EMF.• The current, I equals Volts divided by L - a form of Ohms law• This unusual form of conductor resistance is the opposition due to the

Back EMF and is known as REACTANCE and given the symbol XL

XL = 2FL = L Note: is just common shorthand for 2F

L

F, Hz

VI



AC with Pure Capacitance

I

V

Phasor Diagram

• The CAPACITIVE REACTANCE is the ratio of voltage to current

V / I = Xc = 1/(2.F.C) = 1/(.C)

• So the Current LEADS the Voltage by 90°

• Reactance and therefore the current is dependent upon the frequency as well as the C or L

Remember the word: CIVIL

C

F, Hz

VI



Resistance & Inductance in Series

• Impedance is the vector sum of the resistance and reactance.

• A definition is the ratio of the RMS EMF in a circuit, to the RMS current

VLV

VR

IL

• R represents the 'total' circuit resistance.

• The Voltage is made up of two parts; a PD across the resistance VR with the voltage and current in phase, and a PD across the inductance VL leading the current by 90°.

• The resultant is the applied voltage V, which is the vector sum given by:-

• Impedance, Z = ( R2 + XL2) The current in the circuit is I = V / Z

RL

V



Resistance & Capacitance in Series

I

V

VR

VC

• To maintain a current of I the applied voltage provides two components;a) A voltage VR = I.R across the resistance, in phase with the current, and

b) A voltage VC = I.C = I.1/(2FC) which lags the current by 90°.

• The resultant is V which is the vector sum of these two components.

• The impedance of the circuit is Z = ( R2 + XC2 )

R C

V



Tuned CircuitsSeries Resonance

• The applied voltage has three components;VR = IR across R and in phase with the current I

VL = I.L across the inductance and leading the current by 90°

VC = I.1 /C across the capacitance and lagging the current by 90°

• VL and VC being 180° out of phase.

• At resonance VL = VC therefore I.L = I.1 /C so XL = XC

• The particular frequency when XL = XC is known as the resonant frequency

• The formula is F = 1 / LC or F = 1 / 2(LC) or in terms of • L = 1 / 4 2 F2 C or in terms of C = 1 / 4 2 F2 L

The series resonant circuit gives maximum current and minimum impedance at resonance and is known as an acceptor circuit

R CL

V



Tuned CircuitParallel Resonance

The active current has three components;

• IR = V / R in phase with the voltage.

• IC = CV which leads the voltage by 90°

• IL = V / L which lags the voltage by 90°

When we consider IL = IC then V / L = CV • F = 1 / 2 LC or alternatively . . .• L = 1 / 4 2 F2 C or C = 1 / 4 2 F2 L

• A parallel circuit tuned to resonance is known as a rejector circuit.

• It offers maximum impedance to the resonant frequency.

• At resonance the supply current, I = IL - IC and as they are equal and thus are zero, the impedance Z = V / I = V / 0

• Thus impedance is infinitely great. In practice the R modifies this.

LC

F, Hz

V R

IR IC IL



Magnification Factor ‘Q’

• At resonance the voltage across the inductance or capacitance can be several times greater than that supplied.

• The current is determined by the value of R but the voltage across the circuit is determined by the current multiplied by the reactance.

• This gives a voltage greater than that applied.

• The ratio of the volts across the resistor to that across the reactance is called the Magnification factor, Q.

• If the current at resonance is I for the inductance:

Q = IXL / IR = 2FL / R or Q = IXC / IR = 1/ 2FCR

• Q can be constrained by the inductance as good quality capacitors have very little loss.



Dynamic Resistance

• Practical Parallel Tuned circuits do not have infinite impedance at resonance due the finite resistance, r of the Inductor

• The effective value of the impedance of a parallel tuned circuit at resonance is called the Dynamic Resistance, RD

• For a high RD the ratio of L to C should be high and r small.

• Note: If a resistance is connected in parallel with RD then the circuit is damped and the Q is lowered - used to shape the response of tuned circuits in amplifiers.

r

C

L

V

RD RD=L/(C.r)



Bandwidth

• Bandwidth is defined as the width of the resonance curve at a specified point from the peak, normally at 3 dB down.

• Note that for 3dB down from the peak, decibel calculations give this as the ½ power point, or 1/2 which is 0.707 of the peak value.

• The bandwidth can be altered by changing the Q of the circuit, eg damping resistors value or if coupling factors.

• Bandwidth is also be related to Q:

-3dB

0dB

f0 f2f1

0.707V

1.0V

Q = f0 / (f2 - f1)



Shape Factor

• Shape Factor: Resonant and Filter responses have a shape to them

• The better the shape factor the better the rejection of unwanted signals.

-6dB

-60dB

Shape Factor is defined as: Bandwidth at -6dB Bandwidth at -60 dB



Circulating Currents

Parallel Tuned Circuits

• These have high impedance and low current across the circuit

• Internally within the tuned circuit the current sees a series circuit and therefore a low impedance

• This can cause very high currents and the danger of over heating.

Series Tuned Circuits

• Because of the high reactance's the voltage can be very high,though with relatively little current present.



Quartz Crystals

• Quartz is natural material which vibrates due to the piezo-electric effect

• Quartz Crystals are slabs of quartz clamped between two metal plates.

• They are equivalent to a series tuned circuit with a very high Q

• There is also a parallel circuit, C2.

• The series resonance is a low impedance acceptor circuit and the parallel resonance is a high impedance rejector circuit.

Circuit Symbol C1

C2

R

LEquivalent Circuit



Filters

Amplitude

Frequency

Amplitude

Frequency

Low Pass

PI Section T Section

High Pass

PI Section T Section



Band Pass Filters

Amplitude

Frequency

Crystal Filters

• Quartz Crystals can be configured to form a half lattice filter.

• Two crystals are chosen so their frequencies differ by the amount of bandwidth required.

T Section PI Section



Band Stop / Notch Filters

Series LC to Ground

• Low Impedance at resonance

• Stops a given band of frequenciesat resonance.

• Passes others outside of resonance

Parallel LC in Signal Path

• High Impedance at resonance

• Blocks the unwanted signal

• Passes others outside of resonance

Notch Filter

• When response is sharp they are callednotch filters removing a spot frequency.

VoutVinL

C

VoutVinL

C



Notch Filter Response

Frequency

0

Pass Band

Stop Band

Pass Band

fc

Pass Band Loss

10

20

Loss (dB)

Chelmsford Amateur Radio Society Advanced Course (3) Technical Aspects Part-4 - AC Circuits

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