Chapter 5 - Multistage RC and Single RLC Circuits

Overview¶

Introduction¶

These experiments will continue using a FeelTech FY3200S signal generator and a Tektronix TBS 1072B-EDU digital oscilloscope to measure the waveforms connected to resistor, incductor, and capacitor circuits.

Part 1 of this lab is to measure and compare to a theoretical model the gain and phase of multistage RC circuits as shown in Figure 1.

Part 2 of this lab is to measure and compare to a theoretical model the gain, phase, and resonance of RLC circuits like the one shown in Figure 2.

Experiment¶

Part 1 - Theory of Multistage RC Circuits¶

We saw in the RC lab that the impedance of an RC circuit is

In this lab we will use two different resistor/capacitor values for each stage, and we want the cross-over frequency ($f=1/RC$) to be the same for each stage to keep our analysis simple. We also want the first stage to define the current. To do this we choose $R_1\ll R_2$ or $R_2/R_1>10$. Since $1/R_1C_1 = 1/R_2C_2$, we will need to choose capacitors accordingly. More details are in the Measurements section.

In general, each loop’s current can be written

For simplicity, let’s assume we can separate each stage for analysis. See Figure 3. This shows that the voltage across $R_1$ is the input voltage to the second stage.

According to Ohm’s Law, the voltage across the resistor is

Similarly, for R2

We are interested in how the output relates to the input of the circuit. We can define a ratio of the output voltage to the input voltage.

This is a complex function, and in the real world we can only measure real values. Therefore, we need to calculate the gain of the circuit by calculating the magnitude of this complex function.

When calculating the magnitude of a complex function, we complex square it and take the square root. First, the two terms need to be written in the form

Then, the magnitude is

After some algebra, you should find

The phase for this circuit can be expressed as

Part 1 - Measurements¶

As observed in Part 3 - Signal Integrity, the function generators work well from 1 Hz to 10 MHz. Choose the cross-over frequency such that it will be in the middle of this range on a log-scale. That is, choose components such that

and

Set up your circuit and measure the gain and phase as a function of frequency

import matplotlib.pyplot as plt
import numpy as np

# theoretical data
omega = np.linspace(0,100000, 100001)
R1 = 1000
R2 = 10000
C1 = 1e-7 #100 nF
C2 = 1e-8 #1nF
print(1/R1/C1, 1/R2/C2)
gain_theory = omega*R1*C1/np.sqrt(1+(omega*R1*C1)**2) * omega*R2*C2/np.sqrt(1+(omega*R2*C2)**2)
phase_theory = np.atan(1/(omega*R1*C1)) + np.atan(1/(omega*R2*C2))

# experimental data
omega_exp = np.array([])
gain_exp = np.array([])
phase_exp = np.array([])

fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)

ax1.semilogx(omega, gain_theory, '-k', label='Theoretical Gain')
ax1.semilogx(omega_exp, gain_exp, 'ob', label='Experimental Gain')
ax1.set_ylabel('Theoretical Gain')
ax1.legend()
ax2.semilogx(omega, phase_theory, '-k', label='Theoretical Phase')
ax2.semilogx(omega_exp, phase_exp, 'ob', label='Experimental Phase')
ax2.set_ylabel(r'Phase, $\phi$ (rad)')
ax2.legend()
plt.xlabel(r'log($\omega$)')
plt.show()

Part 2 - Theory of RLC Circuits¶

Inductors are a circuit component you likely have not encountered. It is essentially a coil of wire with a magnetic core to enhance its strength. See .

A voltage applied across an inductor creates a magnetic field along the inductor. The strength of the field or magnetic flux is related to the voltage, cross-sectional area, length, and number of turns.

We define inductance, $L$, as

Where Φ is the magnetic flux and $I$ is the current through the coil. The relationship for an inductor that is similar to Ohm’s Law comes from Faraday’s Law of Induction

This equation says that changes in current through an inductor cause a counter emf ($V$). It should be immediately obvious that the voltage and current will be phase-shifted for an AC signal in an inductor circuit. The impedance of an inductor can be found similarly to what we did with capacitors. Assume we have a sinusoidal current.

Thus, the inductor equivalent of Ohm’s Law gives a maximum impedance of

In this lab, we will look at an RLC circuit that acts like an electronic damped oscillator. To analyze this circuit Figure 2, we consider Kirchoff’s voltage law

To put this in terms of a common variable ($I$) we differentiate with respect to time to obtain

This is a second order differential equation that applies to any driven harmonic oscillator, where the left hand side is the driving “force” and the right hand side is the dissipative ($R$), accelerative ($L$), and restoring ($C$) “force” terms. I use quotes because this is not a force equation, but there is an analogous force equation for a spring mass system. We can guess a solution of the form

where ϕ is a phase that may be caused by differences in voltage and current like we saw with capacitors. Plugging this solution into the differential equation and doing some algebra, we obtain

The time dependent functions are orthogonal to one another, i.e., the integral of their product over all space is zero. This means the only way for the above equation to be true is if the coefficients are equal to zero. We will assume $I_o\neq 0$.

From the first equation, we obtain the phase

$L$, $R$, and $C$ are fixed from the components used. The frequency will always be greater than 1. As ω increases, the dominant term changes as shown in Figure 5. This cross-over point changes the phase from negative to positive, i.e., a lead to a lag in the current.

From the second equation, we obtain the amplitude of the current.

Consider the impedance triangle shown in Figure 7.

Based on the triangle we can rewrite the current using the magnitude of the impedance.

The denominator happens to be the magnitude of the complex impedance

and therefore, the current is simply voltage divided by impedance, very similar to Ohm’s Law for a series circuit. A graph of the current vs. log() is shown in Figure 8.

All driven oscillators will display resonance when driven at the resonant frequency. The resonant frequency for an RLC circuit is

The resistor dissipates energy in the circuit making the circuit a damped oscillator. The damping coefficient is

A final parameter describing an oscillator is its quality factor. The quality factor is ratio of energy stored to energy dissipated over one cycle of the AC input. The higher the quality factor the sharper the resonance. For an RLC series circuit, the quality factor is described as

As an example, the quality factor for Figure 8 is

Figure 9 shows how changing the resistor by a factor of 100 affects the quality factor.

A graphical way to think about the quality factor is the frequency of the resonance divided by the width of the resonance.

An example of estimating this from a graph would be to observe that the center resonance of the $R=1000~\Omega$ circuit is at $\omega_o=10^5~{\rm rad/s}$. The width of the resonance, which is calculated at $I_{max}/2=0.707\times 10^{-5}~{\rm A}$. Therefore, $\Delta\omega=0.65\times 10^7-1.5\times 10^5 = 6.35\times 10^6~{\rm rad/s}$. See Figure 10 where the width $\Delta\omega$ is shown by a horizontal arrow at the appropriate height on the graph.

Part 2 - Measurements¶

Set up the circuit shown in Figure 2. Use values of $R$, $L$, and $C$ such that the resonant frequency is $10 < f < 100~{\rm kHz}$. You will be using a $L= 230~{\rm \mu H}$ inductor because that’s what we have. Using a signal generator with a 10 Volt peak-to-peak sine wave and oscilloscope measure the phase and current as a function of frequency. You will need to split the input signal so that you can connect a BNC to BNC for phase measurements. Note: You will measure current on the oscilloscope as $I = V_R/R$.

Put a smaller resistor of approximately 10 Ω. Change the input to a square wave and see if you can find the damped oscillations on the oscilloscope. It is interesting that a square wave which is a +/- DC signal would create a sinusoidal response. That is the resonant oscillator in action. Damped oscillations look like the signal in Figure 11.

#code for exercise 5