PH262 Rapid Relaxation: Part I - RC and RL Circuits

I.       BACKGROUND INFORMATION

Exponential behavior in electrical circuits is frequently referred to as "relaxation", particularly for circuits which make use of some exponential property for control of timing, eg. a "relaxation oscillator" makes use of the exponential charging of an RC circuit to control some repetitive process such as the flashing of warning lights at construction sites.

In an earlier experiment, the exponential behavior of a resistor-capacitor combination was investigated by making time measurements with a stopwatch, using fairly large values of both R and C in order to get a sufficiently slow variation.  In this connection, it should be recalled that the time constant for an RC circuit is given by the product RC, and hence can be increased by increasing either of these quantities.  In a resistance-inductance combination, on the other hand, the time constant is given by the ratio L/R.  Attempts to obtain L values large enough to allow direct measurement of exponential behavior by the stopwatch technique are therefore likely to fail because of the simultaneous increase in resistance that will occur as more windings are added to the inductor.  (It is possible to obtain very long RL time constants by using superconducting technology to greatly reduce the resistance, but that is somewhat beyond the scope of this lab.)

In this experiment, the ability to display rapidly varying electrical signals on an oscilloscope will be   used to allow the investigation of rapid exponential behavior of both RC and RL circuits.  These procedures will then be extended to investigate the relaxation behavior of a series circuit containing all three components -- R, L, and C.

 

The graph in Figure 1 shows an            exponential     function starting   at         a          vertical coordinate of 8 and decaying to zero.  If we assume the horizontal axis to represent time, the half-life can be measured as the time required to fall from 8 to 4 on the

vertical scale, which is 2 divisions on this axis.  It is extremely important to note that it takes this same amount of time to get from any vertical value on the graph to a point which is 1/2 of the starting value.  From 6 to 3, 4 to 2, 2 to 1, 1 to 1/2 … all require 2 horizontal divisions.  Thus, in order to measure half-life, any starting point will do, but it is absolutely necessary to be able to see enough of the curve to locate the asymptote in order to establish the “zero level”.

 

The phrase "zero level" above is in quotes because of the possibility that the circuit under investigation might be asymptotically approaching some constant value other than zero.  For example, it has been previously shown that when an RC circuit is initially connected to a battery, the charge on the capacitor will grow toward its equilibrium value with an exponential time constant identical to that which governs its decay, ie. the product RC.  In this experiment both growing and decaying signals will be investigated, and it will be discovered that all such signals within the same circuit share the same half-life.

For a complete theoretical treatment you should consult your textbook.  The following is a brief summary of the essential points.  

The (V, I, t) and (V, Q, t) dependencies for the three types of circuit components are shown below.

 

                                  

^dQ                     V           ^Q

            VR = RI = R   _{dt           C} = _C ;  I = C ^dV_dt^C                        V_L = ^{L dI}_dt = ^{L d}_dt^2Q₂

In most physics textbooks, the preferred variable is the set (V, Q, t).  Unfortunately, most of the rest of the world uses the set (V, I, t) since V and I and t are readily measured with DMM’s or oscilloscopes.  In this lab we are going to investigate both an RC and an RL circuit driven by a square wave source.

 

THE METHOD

This discussion will be presented specifically for the RC circuit, and the variations that are needed for the RL case will be mentioned at the end.   Rather than the batteries and switches customarily shown in textbook illustrations of RC circuits, a square wave generator will be used in order to accomplish the alternation between charging and discharging rapidly enough to provide a continuous signal for display on the oscilloscope.  A square wave "instantaneously" switches back and forth, at a steady frequency, between a plus voltage and zero voltage (or ground).  When connected in series with an RC pair, this changing voltage has the effect of trying to instantaneously reverse the charge on the capacitor at this same frequency but, since the charge can move only as fast as allowed by the resistor, it can't keep up with the square wave.  The voltage across the capacitor will therefore resemble Figure 2, showing exponential behavior first in one direction and then in the other.

     

(

B)

(

B

)

(

A

)

(

A

)

 

 

 

 

1. RC Circuit Theory 

 

1. 1. If V_s = 0 (square wave at zero value) then dV_C/dt + V/R_C = 0 and V_C(t) = V_oe^-t/RC

         

Remember that the initial condition is V_C(t=0)=V₀, due to the previous charge cycle.

 

1. 2. If V_s = V_o (square wave at V_o value) then dV_C/dt + V/R_C = V_o/R_C  and               V_C(t) = V_o[1-e^-t/RC] 

 Recall that the initial condition for the capacitor voltage is V_C(t = 0) = 0, due to the previous discharge cycle.

 

1. 3. Sketch of both solutions 

 

 

 

 

 

 

 

 

2. RL Circuit Theory KVL → -Vs + VL +VR = 0

VL +VR = Vs

L(dI/dt) + IR = Vs

so dI/dt + (R/L)I = Vs/L

 

1. 1. When Vs = 0 then I(t) = Ioe^-(R/L)t

 

1. 2. When Vs = Vo then I(t) = Io[1-e^-(R/L)t]  

But with an oscilloscope you measure V(t), not I(t).  So multiply both sides of the I(t) equations by R and:  

         V_R(t) = Voe^-(R/L)t (discharge)

         V_R(t) = Vo[1-e^-(R/L)t] (charge)

 

1. 3. Sketch of both solutions

Just like part (C) in section 1 above, except you are looking at V_S(t) and V_R(t).

 

 

 

 

 

 

 

 

 

 

 

 

 

  

II. EXPERIMENTAL PROCEDURE

 

Go to every

c

ircuit.com

.

You can choose the components from the top. Move the cursor to the desired component and

left click on it. The component will pop up in the main work area. You can move the component

Electrical

components

 

 

to any location by dragging it.

You can rotate the componen

t in the main area. First select the component and click the rotation

sign.

Adjust tool

Rotate

Delete

Cut

 

 

 

 

Highlight the battery with the left click. Go to   adjust tool on the bottom left corner. Click on the square wave. You will find the pop-up window at the bottom left corner where you see the different parameters of simulation. You have to adjust these values (Low 0V, High 8V, Period 10ms, Width 50%, Rise 0%, Fall 0%, Delay 0%) to observe input and output waveform during circuit analysis. Highlighting the particular parameter. Go to the flywheel on the bottom right corner of the eveycircuit window, rotate the wheel to adjust the value of the parameter.

Square wave

Rotate the flywheel to

change the value

Highlight this part of

the circuit to have

input waveform

If you are using

oscilloscope you

are connecting CH1

here

Click here to display

input waveform

 

 

 

Click here to display

output waveform

Click on

this part of

the circuit to have

output waveform

.

If you are using

oscilloscope you

are connecting CH2

here

Click here to run

simulation