Welcome back to my workbench,

I would like to summarize what we have learned about the application of a pulse input to a RC circuit:

1. A capacitor does not receive its charge at a constant rate. It initially charges rapidly with the rate decreasing as the capacitor voltage grows.

2. Since the capacitor voltage is continuously increasing, the voltage drop across the resistor must be decreasing. Less charging current creates a longer charging time.

3. The product of R in ohms and C in farads it is called the ‘time constant’ of a circuit.

4. You will often see the time constant represented by the Greek letter (tau).

5. In an RC circuit, one time constant will be required to charge VCAP to 63.2% of the supply voltage, three time constants will be required to charge to 95% of the supply voltage, and five time constants will be required to charge to approximately 99% of the supply voltage.

6. The rise time (in sec) of an output pulse from an RC circuit will be = 2.2 X RC

7. ‘e’ is a numerical constant equal to approximately 2.71828.. . It is a non-periodic irrational number. An expression incorporating a power of ‘e’ is often called an exponential function.

8. Functions incorporating exponential growth or decay can be used to model systems where constant change in the independent variable results in the same proportional change in the dependent variable

9. Commercial software such as TINA-TI is useful for creating quick simulations to model and study circuit response. This allows us to ‘what if’ ideas before we commit to a prototype.

10. If we apply a square wave to an RC circuit, the resulting maximum and minimum output voltage after a few cycles of operation will be less than the applied voltage.

11. This maximum and minimum voltage can be calculated using the applied voltage, the resulting output, and an exponential function. Using these methods, we can achieve reasonable agreement between manual calculations and a simple spice simulation.

12. If we apply a square wave to an RC circuit, the shape of the resulting output waveform will be a function of the applied pulse width and the circuit RC time constant.

13. When the circuit RC time constant is greater than ten times the applied pulse width, and the output is measured across the capacitor, the voltage will be proportional to the area of the applied pulse. In math speak we are generating a voltage signal that is proportional to the ‘definite integral’ of the bounded area under a curve. Because of this response, you will often see this circuit called an ‘integrator’.

15. When the circuit RC time constant is less than 0.1 times the applied pulse width, and the output is measured across the resistor, the voltage will be proportional to the rate of change of the applied voltage. In math speak we are generating an output signal that is proportional to the ‘differential’ of the applied excitation. Because of this response, you will often see this circuit called a ‘differentiator’.

16. Components and circuits attached to an RC circuit will influence the output waveform shape. If a BJT is attached, the resulting waveform across the output resistor or capacitor will be clipped at the forward conduction voltage of the transistor’s base emitter junction. This influence can be minimized by the addition of a suitable series resistor in the base circuit. In practice, emitter or voltage followers are often used to isolate the attached parts.