sinusoidal ac circuit measurements
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Circuits Lab/JUST
December 5, 2007
Exp. 6: Sinusoidal AC Circuit Measurements
EE 316 – Section #4
Dr. Omar Qasaimeh
Report Prepared By:
Dana Mohammed Tubishat – 20052171048
Mahmoud Hassan Al-Qudsi – 20062171017
Page 2
Lab Objectives
1. To become familiar with the design and analysis of AC electric circuits; in
particular to observe and measure the phase angles present by means of an
oscilloscope.
2. Applying the Sinusoidal Average Power Formula.
3. Analyzing and testing the Thevenin equivalent circuit with AC open-circuit
voltage and short-circuit current values.
4. Determining the AC circuit parameters for maximum power transfer.
Page 3
Experiment Theory & Procedure
Measuring the Phase Angle θ
In any given AC RLC circuit, there is an unavoidable shift phase shift in the voltage
and current outputs due to the inherent non-real impedance present in these circuit
elements.
Given any particular RLC circuit, like the one shown above, it is possible to plot the
voltage across any two nodes on an oscilloscope display; setting the first channel to
display the input (source) voltage from the function generator, along with a second
channel showing the voltage across any element in the remainder of the circuit.
Page 4
Δt is the time difference (the “lag”) between the two signals, while T is the total period
for the sinusoidal wave.
From this generic circuit and the drawing of its voltage readings from an oscilloscope,
it is possible to determine the value of the Phase Angle (θ) in degrees by means of the
following equation:
𝜃 =∆𝑡
𝑇× 360°
It is also possible to determine the Phase Angle θ by other method. By setting the
oscilloscope to operate in X-Y mode instead of Voltage-Time display mode, it becomes
possible to plot the two signals against one-another, making it possible to directly
obtain the Phase Angle, θ, as follows:
Page 5
In the oscilloscope display above, there are four parameters that need to be taken into
consideration to correctly measure the phase angle: Xi (X-intercept), Xm (X-
maximum), Yi (Y-intercept), and finally Ym (Y-maximum). Θ can be obtained from
either the X or Y values, using one of the two formulas below:
𝜃 = sin−1 𝑋𝑖
𝑋𝑚
𝜃 = sin−1 𝑌𝑖𝑌𝑚
Page 6
It is important to note that the value of the phase angle θ obtained in all three
equations should be the same.
Calculating Average Power Dissipation
It is possible to calculate the average power dissipation in both the RL and RC
branches by means of these two equations:
𝑃 =1
2𝑉𝑠𝑚 𝐼𝑚 cos 𝜃
𝑃 =1
2𝐼𝑚
2𝑅
Vsm and Im respectively represent the maximum source voltage (v-peak) and the
current passing through the branch in question (generally,𝑉
𝑅). Θ is the phase angle
(also known as the power-factor) for the branch for which the average power
dissipation is being calculated.
In an ideal circuit composed of reactive elements (capacitors and inductors), the
average power dissipation over any given period of time is zero; as a result of their
intrinsic behavior which simply converts power (and therefore energy) from one form
to another, without actually using any of it. For that reason, the average power
dissipation in a given RL or RC branch is equal to the power dissipated by the non-
reactive elements present, in this case, the resistors. In other words, the presence of
reactive elements in a branch is ignored in the measurement of average power
dissipation in an ideal circuit.
Page 7
Determining the Thevenin Equivalent Circuit
All basic current and voltage laws that apply to DC circuit analysis are also fully-
applicable to AC circuits as well. Chief of these is the Thevenin equivalent circuit,
which can be used to replace any linear circuit, no matter how complex, with a simple
circuit consisting of no more than a voltage source and a resistor. In an AC circuit, the
only difference is that instead of a voltage source and resistance, a voltage source and
impedance will be used.
For the circuit above, it is possible to determine the Thevenin equivalent circuit as
seen by the general impedance “Z” (or the circuit between points ‘a’ and ‘b’).
The same steps used to determine the parameters for the DC Thevenin equivalent
circuit are also used to find its AC counterpart:
1. Find Vth by removing “Z” from the circuit and attaching an oscilloscope or
DMM across points ‘a’ and ‘b’. The RMS value will be presented on the DMM’s
display, and the frequency will be the same as the original input frequency.
2. Find ISC by checking the current across points ‘a’ and ‘b’ with a DMM.
3. Calculate Zth as follows:
𝑍𝑇𝐻 =𝑉𝑡ℎ𝐼𝑆𝐶
Page 8
The final Thevenin circuit will be as follows:
Finding the Maximum Power Transfer
According to the Maximum Power Transfer Theorem, or Jacobi’s law, the power that
can be transferred in a circuit with a fixed internal resistance is when the external
(load) resistance was equal to the internal resistance.
This same law is applicable to all AC circuits in a similar manner. When the load in a
RLC circuit has the same impedance as the internal impedance, maximum power
transfer is achieved. As such, this process is known as impedance matching.
As discussed in the previous section, the internal (fixed) impedance in an AC RLC
circuit is Zth; as was calculated above. In this case, the maximum power transfer can be
achieved when the impedance Z (in the circuit above) or, more generically, Zload is
equal to Zth.
Page 9
Results
Following the procedures above, the following values were determined:
Part 1
For the following circuit:
The 5.7 volt power supply is a 16V peak-to-peak function generator set at 400Hz.
Measured values:
R1: 1.24 kΩ
R2: 1.23 kΩ
Rdc: 152.8 Ω (inductor resistance)
Page 10
RL (Inductance) Branch Readings
The oscilloscope reading for this circuit (ideal simulation with Multisim, not the actual
reading!):
Actual measured values (from the lab) for the RL (inductance) branch:
𝐼𝐿 =𝑉1
𝑅1= 2.66 mA
∆t: 0.44 ms
T: 2.5 ms
𝜃 =∆𝑡
𝑇× 360° = 63.36°
Page 11
And in X-Y Mode:
𝜃 = sin−1 𝑋𝑖
𝑋𝑚 = 60.27°
𝜃 = sin−1 𝑌𝑖
𝑌𝑚 = 61.0°
RL (Capacitance) Branch Readings
And for the RC (capacitance) branch:
𝐼𝐶 =𝑉2
𝑅2= 3.33 mA
∆t: 0.4 ms
T: 2.5 ms
𝜃 =∆𝑡
𝑇× 360° = 57.6°
Page 12
And when the oscilloscope is operating in X-Y mode:
𝜃 = sin−1 𝑋𝑖
𝑋𝑚 = 60.0°
𝜃 = sin−1 𝑌𝑖
𝑌𝑚 = 63.82°
Average Power Dissipation
For the RL branch:
𝑃𝐿 =1
2𝑉𝑠𝑚 𝐼𝑚 cos𝜃 =
1
2(8)(2.66 × 10−3) cos 60 = 5𝑚𝑊
𝑃𝐿 =1
2𝐼𝑚
2𝑅 = 1
2 2.66 × 10−3 2 1200 = 4.2𝑚𝑊
For the RC branch:
𝑃𝐶 =1
2𝑉𝑠𝑚 𝐼𝑚 cos 𝜃 =
1
2(8)(3.33 × 10−3) cos 60 = 6.66𝑚𝑊
𝑃𝐶 =1
2𝐼𝑚
2𝑅 = 1
2 3.33 × 10−3 2 1200 = 6.6𝑚𝑊
Page 13
Part 2
For the following circuit:
Measured resistances for this circuit:
R1: 1.24 kΩ
R2: 470 Ω
R3: 3.24 kΩ
Rdc: 50 Ω
Using the same methods employed in the previous section (oscilloscope in X-Y and
Voltage-Time modes) for the measurement of the voltage across each element in the
Page 14
circuit combined with the phase angle for that measurement, the following values
were obtained:
Element Voltage (V) Phase Angle (°)
1 4.4 39.5
2 1.6 -55.7
3 4.4 -43
ab 4 25.1
L 3.4 61
C 2.15 51.8
According to Kirchhoff’s Voltage and Current laws (which, like all other core circuit
analysis techniques, are applicable in both AC and DC circuits) state the voltages
measured balance one-another out, particularly:
𝑉𝑎𝑏 = 𝑉𝐿 + 𝑉2 = 𝑉𝑐 + 𝑉3
𝑉1 + 𝑉2 + 𝑉𝐿 − 𝑉𝑠 = 0
The readings obtained in the lab:
Vab VL + V2 VC + V3 Vs V1 + V2 + VL V1 + V3 + VC
4 5 6.55 8 9.4 10.95
I1 = V1/R1 I2 = V2/R2 I3 = V3/R3 I1 – I2 – I3
3.6 mA 3.4 mA 1.3 mA -1.1 mA
Page 15
Part 3
For the following circuit:
Measured values in the process of determining the Thevenin equivalent circuit for the
original circuit shown above:
VOC θOC ISC θSC ZTH= VOC/ISC ZTH*
6 5.8° 1.6 mA 53.13° 3.75K 47.33 3.75K -47.33
Maximum power transfer is achieved when Z is equal to ZTH, or at approximately 3.75
kΩ.
Page 16
Discussion of Results
In this particular experiment, the measured results as taken in the laboratory were not
particularly close to the expected theoretical results that were obtained a simulation of
the circuits involved.
In dealing with reactive circuit components, such as inductors and capacitors, there is
a relatively-large discrepancy between the idealized theoretical results and the actual
outcome of the experiment due to the non-reactive resistance present in these real-life
components, which adds a fairly significant amount of excess power dissipation and
influences the actual impedance of the circuit.
All circuit drawings and depicted outputs/results in this report were obtained by
replicating the actual experiment in a simulator, and indicate the idealized, theoretical
output. The simulations performed (and the screen captures of the oscilloscope output
displays) were conducted using Multisim 10.1.
With regards to the expected outcome as determined by theoretical calculations and
idealized simulations in Multisim, there were several discrepancies. In particular, the
results obtained for the phase shifts in the first section show considerable deviation
from the expected output. In particular, the results obtained in the balancing of
current and voltage in the second section is not in-line with the expected outcome.
Page 17
Conclusion
This experiment served to verify the laws governing the analysis of AC circuits, and
successfully resulted in the analysis of the phase shift angles in multiple circuits (in
the first and second portions of the experiment procedure) and the determining of the
correct Thevenin equivalent circuit (and with it, the maximum power transfer
parameters) in the final part of the experiment.
In the first section, the presence of a systematic error that offset our results by a
consistent value was observed. In all six measurements of the phase angle,
approximately the same phase angle was determined (at or around 60 degrees);
though that value is not consistent with the theoretical measurements. This indicates
a constant factor that led to incorrect but consistent results, in keeping with the core
definition of a systemic error.
Through this experiment, it was possible to perform impedance matching leading to
the successful determination of the maximum power transfer values, which is an
important calculation to be able to obtain the highest quality of a large output signal
for any given AC linear circuit with a fixed internal impedance/resistance.
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