# Task 6.9: Measuring RMS Values > [!table] Table I: Triangle Wave RMS and Frequency > > | Measurement | Value | > | --- | ---: | > | RMS Voltage [V] | 1.699 | > | Frequency [kHz] | 1.000 | > [!calculation] Triangle Wave RMS Calculations > $V_{AC,RMS}=\frac{A}{\sqrt{3}}=\frac{3}{\sqrt{3}}=\boxed{1.732V}$ > > $V_{DC,RMS}=\sqrt{V_{AC,RMS}^2+\beta^2}=\sqrt{1.732^2+1^2}=\boxed{2.000V}$ > > Measured RMS voltage: > > $V_{meas}=1.699V$ > > Percent error: > > $\%\text{ error}=\left|\frac{1.699-1.732}{1.732}\right|\times100=\boxed{1.91\%}$ > [!table] Table II: Calculated vs. Measured RMS Voltage > > | Measurement | Voltage [V] | > | --- | ---: | > | Calculated AC RMS | 1.732 | > | Calculated DC RMS | 2.000 | > | Measured RMS | 1.699 | > | Percent Error (AC) | 1.91% | > [!figures] Figure I: AC and DC Coupled Triangle Waves > ![[Pasted image 20261019161037.png|center|450]] ## Step 9: RMS Voltage vs. Frequency > [!calculation] Theoretical AC RMS Values > $V_{RMS_{1}}=\frac{A}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\boxed{1.414V}$ > $V_{RMS_{2}}=\frac{A}{\sqrt{3}}=\frac{2}{\sqrt{3}}=\boxed{1.155V}$ > $V_{RMS_{3}}=A=\boxed{2.000V}$ > > [!table] Table III: Sine Wave RMS Measurements > > | Frequency [Hz] | Theoretical RMS [V] | Scope RMS [V] | DMM RMS [V] | Scope Error | DMM Error | > | ---: | ---: | ---: | ---: | ---: | ---: | > | 100 | 1.414 | 1.415 | 1.413 | 0.07% | 0.07% | > | 250 | 1.414 | 1.417 | 1.407 | 0.21% | 0.50% | > | 500 | 1.414 | 1.413 | 1.410 | 0.07% | 0.28% | > | 1000 | 1.414 | 1.412 | 1.404 | 0.14% | 0.71% | > | 5000 | 1.414 | 1.409 | 1.392 | 0.35% | 1.56% | > | 10000 | 1.414 | 1.406 | 1.374 | 0.57% | 2.83% | > [!table] Table IV: Triangle Wave RMS Measurements > > | Frequency [Hz] | Theoretical RMS [V] | Scope RMS [V] | DMM RMS [V] | Scope Error | DMM Error | > | ---: | ---: | ---: | ---: | ---: | ---: | > | 100 | 1.155 | 1.154 | 1.138 | 0.09% | 1.47% | > | 250 | 1.155 | 1.157 | 1.143 | 0.17% | 1.04% | > | 500 | 1.155 | 1.156 | 1.145 | 0.09% | 0.87% | > | 1000 | 1.155 | 1.153 | 1.134 | 0.17% | 1.82% | > | 5000 | 1.155 | 1.149 | 1.121 | 0.52% | 2.94% | > | 10000 | 1.155 | 1.142 | 1.099 | 1.13% | 4.85% | > [!table] Table V: Square Wave RMS Measurements > > | Frequency [Hz] | Theoretical RMS [V] | Scope RMS [V] | DMM RMS [V] | Scope Error | DMM Error | > | ---: | ---: | ---: | ---: | ---: | ---: | > | 100 | 2.000 | 2.002 | 1.990 | 0.10% | 0.50% | > | 250 | 2.000 | 1.998 | 1.982 | 0.10% | 0.90% | > | 500 | 2.000 | 2.004 | 1.976 | 0.20% | 1.20% | > | 1000 | 2.000 | 2.003 | 1.972 | 0.15% | 1.40% | > | 5000 | 2.000 | 1.992 | 1.942 | 0.40% | 2.90% | > | 10000 | 2.000 | 1.984 | 1.910 | 0.80% | 4.50% | > [!figures] Figure II: RMS Voltage vs. Frequency > ![[Pasted image 20261019161137.png|center|450]] > [!note] Step 9: Observations > - The RMS voltage remained relatively constant as frequency increased, consistent with theoretical expectations. > - The measurements were compared against theoretical AC RMS values, excluding the 4 V DC offset. > - The DMM showed larger deviations at higher frequencies, particularly for triangle and square waves. > - At 10 kHz, the triangle-wave DMM percent error reached 4.85%. > - The oscilloscope was more consistent, with a smaller average error and standard deviation. # Task 6.10: Hi-Z vs. 50Ω on Function Generator > [!table] Table VI: RMS Voltage at Different Loads and Output Modes > > | Load Resistance | Hi-Z RMS [V] | 50Ω RMS [V] | > | --- | ---: | ---: | > | $1k\Omega$ | 0.71 | 1.40 | > | $47\Omega$ | 0.33 | 0.67 | > [!calculation] Function Generator Voltage Division > > $V_{out}=V_{source}\frac{R_L}{R_s+R_L}$ > > For a $1k\Omega$ load: > > $\frac{R_L}{R_s+R_L}=\frac{1000}{50+1000}=0.9524$ > > For a $47\Omega$ load: > > $\frac{R_L}{R_s+R_L}=\frac{47}{50+47}=0.4845$ > > Therefore, the lower-resistance load experiences greater voltage attenuation. > > Switching from Hi-Z to 50Ω mode approximately doubles the generated voltage for the same displayed setting. > [!note] Step 5: Observations > - Switching from Hi-Z to 50Ω mode approximately doubled the RMS voltage across both resistors. > - The $47\Omega$ resistor produced a lower voltage than the $1k\Omega$ resistor. > - This occurs because a greater fraction of the source voltage drops across the function generator's internal resistance when load resistance decreases. > - The output mode changes the function generator's assumed load impedance and corresponding output voltage calibration. # Task 6.11: Superposition of Waves > [!figures] Figure III: Superposition Circuit Schematic > ![[Pasted image 20261019161224.png|center|450]] > [!table] Table VII: Superposition Circuit Parameters > > | Parameter | Value | > | --- | ---: | > | Transmission Signal | 50 Hz sine wave | > | Transmission Amplitude | 1 V | > | Transmission DC Offset | 2 V | > | Data Signal | 1 Hz triangle wave | > | Data Amplitude | 2.5 V | > | $R_t$ | $10k\Omega$ | > | $R_d$ | $5.6k\Omega$ | > | $R_L$ | $56k\Omega$ | > [!calculation] Theoretical Output RMS Voltage > The output is determined by superposition of the two input signals. > > $V_{trans}(t)=2+\sin(2\pi50t)$ > > $V_{data}(t)=2.5\,tri(2\pi t)$ > > $\frac{V_{out}-V_{trans}}{R_t}+\frac{V_{out}-V_{data}}{R_d}+\frac{V_{out}}{R_L}=0$ > > $V_{out}(t)=\frac{V_{trans}(t)/R_t+V_{data}(t)/R_d}{1/R_t+1/R_d+1/R_L}$ > > Since the two AC waveforms have different frequencies, their mean-square contributions can be added along with the DC offset contribution. > > $V_{out,RMS}=\sqrt{V_{DC,out}^2+V_{sine,out,RMS}^2+V_{triangle,out,RMS}^2}$ > > The calculated result reported in the prelab is: > > $\boxed{V_{out,RMS}=1.1261V}$ > [!figures] Figure IV: Superimposed Output Waveform > ![[Pasted image 20261019161237.png|center|450]] > [!calculation] Theoretical and Measured Output Power > Load resistance: > > $R_L=56k\Omega$ > > Theoretical power: > > $P_{theoretical}=\frac{V_{RMS}^2}{R_L}=\frac{1.1261^2}{56,000}=\boxed{ 22.65\mu W }$ > > Measured RMS voltage: > > $V_{measured}=1.122V$ > > Measured output power: > > $P_{measured}=\frac{1.122^2}{56,000}=\boxed{ 22.48\mu W }$ > > Voltage percent error: > > $\%\text{ error}_V=\left|\frac{1.122-1.1261}{1.1261}\right|\times100=\boxed{ .37\% }$ > > Power percent error: > > $\%\text{ error}_P=\left|\frac{22.48-22.65}{22.65}\right|\times100=.73\%$ > [!table] Table VIII: Calculated vs. Measured Superposition Output > > | Measurement | Theoretical | Measured | Percent Error | > | --- | ---: | ---: | ---: | > | RMS Voltage [V] | 1.1261 | 1.122 | 0.37% | > | Output Power [$\mu W$] | 22.65 | 22.48 | 0.73% | > [!note] Step 2: Observations > - The oscilloscope displayed high-frequency oscillations superimposed on a slower waveform. > - The 50 Hz sine wave formed the higher-frequency component. > - The 1 Hz triangle wave formed the slower variation. > - The output waveform demonstrated the superposition of the two signals. > - Only the combined waveform was captured in the report.