## Objectives

The objective of this post is to further understand and observe the transient responses of RL and RC circuits via experimentation. Additionally, another goal of this post is to learn how to measure the time constant of first order circuits using the ORCAD PSPICE software utility.

## Theoretical Background

It is useful to be aware of the inverse laplace transform operation an operation whereby you can switch between the time domain and the frequency domain in circuits. It is oftentimes helpful to know how to use the partial fraction expansion technique, so you can transform equations inside of an inverse laplace operation into ones that are more wieldy and easily used, or can be looked up in a table of known equivalents.

You should also know that a capacitor has the ability to store electrical charge and energy. Therefore, voltage across the capacitor is related to the charge by the equation for steady state values.

In this post, we cover first order RC and RL circuits. For RC circuits, there are some equations that are required to know. For instance, the time constant of a very simple series RC circuit will be , and the natural response, which is derived from the differentials of voltage, which would be . **In this case, the L^{-1} operator is the inverse laplace transform.** Poles would be located at . For a series-connected RC circuit, view the table with equations below for help.

Voltage Across the Capacitor | Voltage Across the Resistor |

Those equations can be obtained from the Laplace transform. You can obtain a natural response by performing an inverse laplace transform on the voltage equation. Additionally, since the circuit is in series, the current would not change, and therefore would be governed by the equation below.

For the parallel-connected RC circuit, the output voltage *V*_{out} is equal to the input voltage *V*_{in}, which means the circuit doesn’t act as a filter on the input signal. You should also have some familiarity with RL circuits. For series-connected RL circuits, we can obtain the following equations.

Voltage Across the Inductor | Voltage Across the Resistor |

Since the circuit is in a series, the current would be the same everywhere. This means it can be governed by the equation below at every point on the circuit, assuming an ideal or simple RL circuit.

Additionally, it would have a single pole located at . The impulse response of this circuit would be derived from the inverse laplace of the transfer function, and would be modeled by the equations below.

Impulse Response for Inductor Voltage | Impulse Response for Resistor Voltage |

The parallel-connected RL circuit has a similarity to it’s RC cousin in that the output voltage *V*_{out} is equal to the input voltage *V*_{in}, which means the circuit doesn’t act as a filter on the input signal.

## Experimental Procedures

Construct the circuit shown below on PSpice, with a resistor of 1 kΩ and a capacitor of 1𝜇F. 𝑉𝑔 is an independent voltage source generating square waveforms. Set 𝑇𝑅 = 𝑇𝜇 =*TD* = 0, 𝑉1 = 0,and 𝑉2 = 10 𝑉. Following the construction of the circuit, set the period of the waveform so that the natural and step responses 𝑣(𝑡) of the RC Circuit can clearly be observe.

Once the circuit has been constructed, measure 𝑖(𝑡) and 𝑣(𝑡), then plot them. The results will help fill the simulation results in Table 1.

Let 𝑅 = 1 𝑘Ω and 𝑇𝑅 = 𝑇𝜇 = *TD *= 0, 𝑉1 = 0, 𝑉2 = 10𝑉, and *PER*=2𝑚s, select the capacitance of the capacitor so that the response 𝑣(𝑡) of the circuit is a triangle waveform, as shown below. Construct another circuit with a resistor of 10 Ω and an inductor of 10 *mH*. Set 𝑇𝑅 = 𝑇𝜇 = *TD *= 0, 𝑉1 = 0, 𝑉2 = 10 𝑉 for the pulse voltage source. Wisely select its pulse width so that the responses of the RL circuit can be observed.

Measure the current 𝑖(𝑡) in the circuit and voltage 𝑣(𝑡) across the inductor then plot the results. You can find the simulation results in Table 2.

Resistance (kΩ) | Capacitance (µF) | Calculated Time Constant | Measured Time Constant |

1 | 0.5 | .5ms | .516ms |

1 | 1 | .001 | 1.0065ms |

1 | 2 | .002 | 2.0016ms |

Resistance (kΩ) | Inductance (mH) | Calculated Time Constant | Measured Time Constant |

1 | 10 | .001 | 1.0065ms |

1 | 20 | .002 | 2.0016ms |

1 | 30 | .003 | 3.0116ms |

## Analysis

In this experiment, we explored RC and RL circuits in PSPICE along with calculating the time constant from the voltage graphs of these circuits. We used what is referred to as VPULSE in PSPICE because we wanted the voltage to have a waveform so we can observe the natural and step responses of the RC circuit. For Circuit 1, we varied the capacitance to calculate the different time constants as shown in Table 1. After running the simulations with the different capacitances we notice the graphs tend to look very similar as they are based of the function which produces the reflection of an exponential graph. We were able to measure the time constant on the graph by checking at what time the system reached 6.321V as instructed to.

This is because tau for RC circuits is the time the current takes to reach 63.21% of the maximum voltage when charging the capacitor. We were able to find the time constant easily because for the first duration of tau the graph is essentially linear. Our measured values for the time constant were very similar to our calculated values because the measurement could not be exact.

Next we were tasked with selecting a capacitance so that the voltage produces a waveform graph. After some trial and error, we found that the waveform graph starts taking the shape of a triangle around 0.4uF but becomes a recognizable triangle waveform at 0.8uF.

Lastly, we constructed Circuit 2 in PSPICE in accordance with the parameters specified in the procedures. This circuit is an RL circuit instead of an RC circuit which means it has an inductor instead of a capacitor. This time we vary the inductance instead of capacitance and measure the time constant with the help of the waveform graphs. After graphing all the waveform graphs with varying inductance it is clear that the graphs follow the function which produces an exponential decay graph.

To measure the time constant we find the time corresponding to 3.68V as shown in the lab handout. This is because tau for RL circuits is the amount of time the current takes to reach 36.8% of the maximum voltage. The measured time constants are relatively the same as the calculated values while taking into account measuring errors. Also, the time constants increased as the capacitance and inductance increased.

In conclusion, the experiment consisted of first order RC and RL circuits and we observed their transient responses including their time constants. From the oscilloscope, we noticed that the time constants changed as we connected the capacitors in series and parallel. This is the case as the capacitance changes as a result of different configurations. Comparing the RC and RL circuits that we constructed, the RL circuit had more current flowing than in the RC circuit.

## Acknowledgements

Huge thanks to Nodebechukwu Paully-Umeh and Michael Rodriguez for their help with this lab experiment back in November 20, 2020.

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