Phasor Analysis and AC Responses
Presentation of AC voltage
� = � cos(� + �)
RMS Voltage
� =
1
�
� 2
� 0
� =� 2
= 1
�
� 2
� 0
� = 1� � � 2
0
=� 2 When � = 2�
Euler’s Formula
� = cos � + (�)
Complex Impedences
Capacitors
� = [� � +� ] = �= � � ( )
��=�� � � = �� = ��∠ −�
- The voltage across the capacitor is led by the current into the capacitor by 90o - This is because capacitor takes time to charge up!
- Note that these phenomena only occurs in AC supply condition
- During DC connection, the capacitor will not have this effect because it will act as a block due to the two insulating plates.
- Note that the impedance of the capacitor ZC decreases when the capacitance increases
Inductors
= [ � +� ]
�� = � = ��
� = �� = ��∠�
- The voltage in the inductor now LEADS the current in the inductor by 90o (It is a voltage driven current device)
- Note that this phenomenon occurs because there is damping effect increases when the induced voltage is getting faster
- The impedance ZL increases as the inductance increases Impedance: = + �
- The impedances in series:
= 1+ 2+⋯ + - In parallel:
1 = 1 + 1 +⋯ + 1
Capacitor and Inductors in Bridge Circuits
Pressure sensor
- Capacitor can be placed in a Bridge Circuit consist of Wheatstone bridge - This can be used as a pressure sensor
- Pressure can be detected due to the capacitances and - The wheatstone bridge then can be used to vary the voltage
change as the pressure changes
- If the undisplaced distance of the capacitors is d and the diaphragm is displaced in either one direction by x, given 1 = 2
- �0 � =� � �
2 (Potential Divider)
- Assume the diaphragm move the the right by x,
= �
− � =
� +� - This can be applied to the condenser microphone
Inductor-position transducer
The position of the inductor and the winding also can be used to used as a transducer to track the voltage change
Here, Wheatstone bridge is again used to anaylse the change.
First order filters
Low pass filter circuit
Vout = Zc + �
Vout Vin =
1 jωC 1 jωC + R
= 1
ωRC 2+ 1−
jωRC ωRC 2+ 1
H f = Vout Vin =
1
1 + ωRC 2=
1 1 + ωω
B 2=
1 1 + ff
B 2
where
� = �� This can be used as a low pass filter whereas the high frequency is filtered out due to the large impedance caused by the capacitance
If f = fB, then the output voltage is 1
2 of the maximum voltage, so it is sometimes called half-power frequency In this case, if DC voltage is applied, the capacitor will act as a short, so the output voltage will be equal to the input.
Low past filter by Inductor
- Inductor also can be used to build a low pass filter - The output voltage is taken across the resistor
- At high frequency, the inductor has large impedance, hence blocking the voltage
Vout Vin =
R R + ZL =
1 1 + � �
�
�� =
+ ���
=
+ ��
where
fB = R 2πL
Applications of Low Pass Filters
- Noise filter which will result in disrupting the circuit by interference.
- To choose a suitable capacitance or resistance, set the fB as the desired filtering frequency and then solve the equation
High Pass filters
On the other hand, by changing the voltage output terminal by taking the output across the resistor, we got high pass filter
This is because at high frequency, the capacitor is effectively a short, so it will make the potential drop small, and hence, performing the job of a high pass filter should be doing.
Vout
Vin = + =
�
� + 1=
+ 1
Vout Vin =
+ 1
cot θ = f
fB = tan(90 − θ) θ = 90 − tan−1
Similar high pass filter can also be formed by using inductor by taking the output voltage across the inductor
Vout
� =
�� 2 �� 2+ 1
Applications of high pass filters
- High pass filters can be used to block DC from one part of a circuit
- DC filters in an audio pre-amplifier to block out the DC variations which causes degradation
- Again the values can be solve be letting fB as the frequency which we want to block when it decreases.
Decibels (Measure of Loudness)
H f dB = 20 log H f
- Bode plot can be used to see how the frequency changes with the change in dB - A linear function can be plotted to ease the analysis of the effect of the filters.
Cascaded Networks
H f = H1(f) × H2(f) � � � = � � � + � � �
RLC Filters
Z = R + jX = R + j − 1
ωC+ωL For the circuit to be purely resistive,
−ωC1 +ωL = 0
f0= 1 2π LC
f0 is known as the resonant frequency, the value is independent of the resistance of the resistor - At this point, the impedance of the circuit is at minimum
Qs =2πf0L
R =
1 2πf0CR
- Qs(Quality factor) is defined as the ratio of the inductor impedance to the resistance of the circuit Zs f = R 1 + jQS
f f0−
f0 f
Band-Pass filter
Consider the output voltage is taken across the resistor VR
VS = R Zs =
R
R + j ωL −ωC 1
= 1
1 +
0− 0
At resonant frequency, the gain is one, elsewhere, the output will be attenuated
To investigate the phase of this filter, Z = R + jX
tan θ =X R
The phase is 0o when ω = ω0, otherwise, it is tending to 90o in phase when the ratio is increasing and tending to -90o in phase when the ratio is decreasing
- This kind of filter can be used effectively in radio frequency receiver by using a tuned circuit
- Bandwidth is used to determine the quality of how well the frequency is received
B = f0
Qs = − �
Similarly, when the RLC circuits are all in parallel, the result will
Notch filter
If the series RLC circuit is now being modified by taking the output voltage across the LC junction, it will result in a Notch filter.
That is, when f = f0, the gain will go to ZERO!
This can be used to block the noise only at that particular frequency.
Similarly, when the LC are in parallel and are connected in SERIES with the resistor, notch filter will be formed too.
+
- Vs1
V
R1 R L1
L C1
C
Vout
