Internal resistance

It is important to understand the characteristics of your instruments, since they become part of any circuit they are attached to. Knowing the limitations of your instruments can help you avoid embarrassing mistakes.

An ideal power supply would have zero internal resistance and could output infinite current. Real power supplies have limits due to internal resistance, internal impedance, and internal controls intended to prevent damage to the supply.

• Real instruments and devices all have non-zero capacitance and inductance, but these can be neglected for DC measurements.
• These electrical characteristics can depend on the instrument settings.

Shunt resistor circuit

• The internal and external resistances form a resistive divider, where the voltage across each resistance is proportional to the relative size of the resistance.
• The internal resistance is small, so to see a voltage drop you’ll want to use smallish shunt resistors and measure the voltage drop with the best precision possible to see small changes.
  • WARNING!!! If you put too much voltage across a resistor it may start smoking, so DO NOT put more voltage across a resistor than it can handle. (Remember that \(P=V^2/R\).)
• Use a resistor colour code chart or calculator to confirm the resistor values and determine their tolerances.
  • Never assume that all components in a drawer match the label on the drawer.
  • You can check the value of a resistor or capacitor with your multimeter.
• -25V refers to the port, not necessarily the voltage you're using. Pick the voltage/voltages that you think make the most sense.
• Consider using multiple shunt resistors if that helps you reducing the uncertainties.

AC Impedance

Create an RC divider with a 22nF ceramic capacitor and a 10k resistor in series.

• This can act as a high-pass or low-pass filter, depending on whether the Output is taken across the resistor or the capacitor.
• Before creating the circuit, use your multimeter to precisely measure the capacitance and the resistance.

Apply a 2V Peak-to-Peak sine wave across the divider and observe the peak-to-peak outputs across the whole divider (Channel 1) and across the resistor (Channel 2). Channel 1 shows the Input to the circuit and Channel 2 shows the Output voltage across the resistor. In a linear circuit such as this, the output will have the same frequency as the input, but may have a different amplitude and relative phase.

• If one of the outputs is fuzzy/thick, you may want to use the BW Limit that limits the bandwidth to filter out high frequency noise.

The attenuation or amplitude frequency response is

\[ A(f) = \frac{|V_{output}(f)|}{|V_{input}(f)|}\]

• The 3 dB frequency is where the attenuation is \(1/\sqrt{2} \approx 0.7\).
• In an RC circuit, the 3 dB radial frequency corresponds is the inverse of the RC time constant.
• Unless otherwise specified, “frequency” means the ordinary frequency \(f\), not radial frequency \(\omega=2\pi f\).

Both the amplitude and phase can change between the input and output of a circuit, so we are more generally interested in the transfer function:

\[H(f)=\frac{V_{output}(f)}{V_{input}(f)}=|H(f)|e^{i\phi(f)}\]

Frequency Response: Bode Plots

The above transfer function plot of the amplitude and phase frequency response of a circuit is known as a Bode plot. There are several ways to do it faster.

Before the lab, watch this video on Bode Plots on an Oscilloscope: The Old Way vs. The New Way from the Keysight Oscilloscope 101 Playlist.

Use the scope’s analysis functions to make a Bode plot of the same circuit.

• Press Analyze
• Rotate Entry until Frequency Response Analysis (FRA) is selected
• Setup
  • Start Freq to 10.0 Hz
  • Stop Freq to 20.00 MHz
  • Amplitude 1.00Vpp
  • Output Load High-Z
  • Points 36 (Need to use Fine)
• Back
• Run Analysis

RL Circuit

Capacitors are usually much cheaper, more compact, and radiate less electromagnetic noise than inductors, so when possible capacitors are usually preferred. Inductors, however, are often the best choice for high power applications and are found in almost every power supply.

Replace the nominal 22nf capacitor with a nominal 12mH inductor. Taking the output across the inductor, make a Bode plot using whichever of the above methods you prefer.

Band-pass Filter

in Lab S you combined low-pass and high-pass RC filters to design a band-pass filter that allows only a range of frequencies to pass. Build this filter and see if it lights up a Green LED only when the circuit is driven by a 6Vpp input frequency is (roughly) between \(\sim 0.5 \mathrm{kHz}\) and \(\sim 5 \mathrm{kHz}\).

• The LED may not be very bright. You can actually go a bit above 6V if it helps you see the LED light.
  • Use the LEDs with green plastic covers.
• You may need to revise your design:
  • If we don’t have the values of R and C that you chose. e.g. If the capacitor in your design is more than \(1\mu F\), be prepared to reduce the capacitance and increase the resistance to get the same RC values.
  • If the design does not work. e.g the resistor is so small that \(V^2/R\) heating fries it.