Lab3.md

Lab 3 - RLC Circuit

Objective

1. Verify the accuracy of theoretical phasor analysis on RLC circuit by comparing the result to real-world circuit.
2. Understand phase difference and how to find the phase difference from measurement in the time domain.

The RLC Circuit

Equipment

Simulation

• Cadence OrCAD for simulation

Real-World Measurement

• Breadboard
• Circuit Components from the schematics
• Oscilloscope
• Power Supply
• Signal Generator

Hand Calculation

Transform the RLC circuit into the phasor domain to get the imedance, then combine the impedance of the inductor and resistor in series, capacitor in parallel, and 1 kΩ resistor in series

Given:

$$ \begin{align*} R_1 &= 1000,\Omega \\ R_2 &= 240,\Omega \\ C &= 0.33 \times 10^{-6},F \\ L &= 100 \times 10^{-3},H \\ \end{align*} $$

The equivalent impedance, $Z_{eq}$, is calculated as:

$$ \begin{align*} Z_{eq} &= \frac{(\frac{-j}{\omega C})(R_2 + j\omega L)}{R_2 + j\omega L - \frac{-j}{\omega C}} + R_1 \\ &= \frac{R_2 + j\omega L}{1 + j\omega C(R_2 + j\omega L)} + R_1 \\ &= \frac{R_2 + j\omega L + R_1(1 + j\omega C(R_2 + j\omega L))}{1 + j\omega C(R_2 + j\omega L)} \\ &= \frac{R_2 + j\omega L + R_1 + jR_1R_2\omega C - R_1\omega^2 CL}{1 + j\omega C R_2 - \omega^2 LC} \\ &= \frac{(R_1 + R_2 - R_1 \omega^2 LC) + j(\omega L + R_1 R_2 \omega C)}{(1 - \omega^2 LC) + j(\omega CR_2)} \end{align*} $$

Simplify the $Z_{eq}$ to:

$$ \begin{equation*} Z_{eq} = \frac{a + jb}{c + jd} \end{equation*} $$

Here

$$ \begin{align*} a &= R_1 + R_2 - R_1 \omega^2 LC\\ b &= \omega L + R_1 R_2 \omega C \\ c &= 1 - \omega^2 LC \\ d &= \omega CR_2 \\ \end{align*} $$

To further simplify the complex fraction, multiply the numerator and the denominator by the conjugate of the denominator:

$$ \begin{equation*} Z_{eq} = \frac{(a + jb)(c - jd)}{(c + jd)(c - jd)} = \frac{ac + bd + j(bc - ad)}{c^2 + d^2} \end{equation*} $$

This results in a real part $Re$ and an imaginary part $Im$ of the equivalent impedance:

$$ Z_{eq} = Re + jIm $$

where

$$ Re = \frac{ac + bd}{c^2 + d^2} $$

$$ Im = \frac{bc - ad}{c^2 + d^2} $$

Frequency (Hz)	Angular Frequency (ω)	Z_eq	Phase (degrees)	Inductive or Capacitive?
100	200π	1245.7 + j51.3	2.36	Inductive
500	1000π	1464.6 + j294.5	11.4	Inductive
1000	2000π	1707.3 - j912.7	-28.1	Capacitive
2000	4000π	1012.8 - j295.4	-16.3	Capacitive

OrCAD Simulation

Frequency (Hz)	$V_1$ (V)	$I_1$ (A)	Time Difference (Δt) (µs)	Phase Difference (Φ) (Degree)	Re(Zeq) (Ω)	Im(Zeq) (Ω)	Inductive or Capacitive?
100	0.802	0.00802	65.5	2.4	1246.9	52.3	Inductive
500	0.670	0.00067	63.2	11.4	1492.5	301	Inductive
1000	0.515	0.000515	-77.8	-28	1941.7	-1032	Capacitive
2000	0.947	0.000947	-22.6	-16.3	1056	-309	Capacitive

Note:

1. $I_1 = \frac{V_1}{Z_{eq}}$

2. The circuit is inductive if $I_1$ lags $V_1$, and capacitive if $I_1$ leads $V_1$.

3. Since $V_1$ has no phase angle, if $Z_{eq}$ has a positive phase angle, $I_1$ will have a negative phase angle, vice versa.

4. $\phi = (\Delta t)(Frequency)(360 \space degrees)$, where $\Delta t$ is the time difference between waveforms, which were calculated by placing two cursors on the Time-intercept of the two waveforms and looking at the Y1 - Y2 time difference.

5. Y1 Cursor Always on Red $I_1$ Waveform, Y2 Cursor Always on Green $V_1$ Waveform

$f = 100 Hz$

$f = 500 Hz$

$f = 1000 Hz$

$f = 2000 Hz$

Real-World Measurement

Frequency (Hz)	Vs (V)	V1 (V)	I1 (A)	Time Difference (Δt) (µs)	Phase Difference (Φ) (Degree)	Re(Zeq) (Ω)	Im(Zeq) (Ω)	Inductive or Capacitive?
100	0.995	0.935	0.000935	68.0	2.45	1070	45.7	Inductive
500	0.995	0.730	0.00073	53.0	9.54	1370	230	Inductive
1000	0.995	0.875	0.000875	-76.0	-27.4	1143	-591	Capacitive
2000	1.025	0.975	0.000975	-23.0	-16.6	1026	-305	Capacitive

Note:

1. Cursor X2 measures the math waveform
2. Cursor X1 measures the chanel 1 waveform
3. $I_1$ is the purple waveform.
4. $V_1$ is the yellow waveform.

$f = 100 Hz$

$f = 500 Hz$

$f = 1000 Hz$

$f = 2000 Hz$

Summary

In conclusion, the RLC circuit behaves as an inductor at low frequency, and as a capacitor at high frequency.

OrCAD vs. Measurement

Frequency	Re(Zeq) Percent Difference (%)	Im(Zeq) Percent Difference (%)
100	14.2	12.6
500	8.2	23.4
1000	41.1	43.0
2000	3	1.3

The resulting waveform changes when we switch the inductors around or even twist it on the circuit board slightly. Therefore, the percent difference between our results are highly inconsistent.

Nevertheless, the experiment result generally follows the trend of the simulation and pre-lab calculation. At low frequency f < 1000, the capacitor acts as an open circuit and the total impedance is dominated by the inductor, so we get an inductive circuit. At high frequency f > 1000, the inductor acts as an open circuit and the total impedance is dominated by the capacitor, so we get a capacitive circuit.