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Chelmsford Amateur Radio Society
Advanced Course
(3) Technical Aspects
Part-4 - AC Circuits
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
1
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
N
S
-V
Slipring
Brush
Chelmsford Amateur Radio Society
Advanced Licence Course
AC Volts Output
Carl Thomson G3PEM
One Rotation
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
2
Period & Frequency
• In the last courses we just described the shape of a sine wave
Amplitude
Time
One Cycle
• The Period, T of one cycle, in seconds is equal to 1/f, where f is in Hertz
Frequency, f = 1 / T
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or
Carl Thomson G3PEM
Period, T = 1 / f
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
3
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;
Vmax
270°
0°
90°
Time
Vmin
180°
360°
• 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
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
4
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
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
5
AC with Pure Resistance
Phasor Diagram
V
R
I
V
F, Hz
• Voltage and Current are in Phase
• Standard Ohms Law Applies
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
6
AC with Pure Inductance
V
I
V
L
Phasor Diagram
F, Hz
I
• THE CURRENT LAGS 90° BEHIND THE VOLTAGE
• The magnitude of the current depends upon;
a) the inductance
b) 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 = 2FL = L Note:  is just common shorthand for 2F
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
7
AC with Pure Capacitance
I
V
I
C
Phasor Diagram
F, Hz
V
• 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
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
8
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
L
R
V
VL
V
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)
Chelmsford Amateur Radio Society
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The current in the circuit is I = V / Z
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
9
Resistance & Capacitance
in Series
R
VR
C
I
V
VC
V
• 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/(2FC) 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
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Z = ( R2 + XC2 )
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
10
Tuned Circuits
Series Resonance
R
C
L
V
The series resonant circuit gives maximum
current and minimum impedance at resonance
and is known as an acceptor circuit
• 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
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
11
Tuned Circuit
Parallel Resonance
V
F, Hz
R
IR
L
C
IC
IL
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.
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
12
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 = 2FL / R or
Q = IXC / IR = 1/ 2FCR
• Q can be constrained by the inductance as good quality capacitors
have very little loss.
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
13
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
L
r
RD
C
RD=L/(C.r)
V
• 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.
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
14
Bandwidth
• Bandwidth is defined as the width
0dB
of the resonance curve at a
specified point from the peak,
-3dB
normally at 3 dB down.
1.0V
0.707V
• 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:
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f1
f0
f2
Q = f0 / (f2 - f1)
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
15
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:
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Bandwidth at -6dB
Bandwidth at -60 dB
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
16
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.
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
17
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
Equivalent
Circuit
L
C1
R
C2
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
18
Filters
Low Pass
Amplitude
PI Section
Frequency
T Section
High Pass
Amplitude
Frequency
PI Section
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T Section
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
19
Band Pass Filters
T Section
PI Section
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.
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
20
Band Stop / Notch Filters
Series LC to Ground
• Low Impedance at resonance
• Stops a given band of frequencies
at resonance.
• Passes others outside of resonance
L
Vin
C
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 called Vin
notch filters removing a spot frequency.
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Vout
C
L
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
Vout
21
Notch Filter Response
0
Stop Band
Pass Band Loss
Loss (dB)
10
20
Pass Band
Pass Band
Frequency
fc
Chelmsford Amateur Radio Society
Advanced Licence Course
Carl Thomson G3PEM
Slide Set 4: v1.2, 20-Aug-2006
(3) Technical Aspects - AC Circuits
22