8.4.1. SERIES CONNECTION OF R AND C, OR R AND L

If a series connection of a resistor and a capacitor, or of a resistor and an inductor is connected across an ac-voltage they stand for two different impedances. Leaving aside that the capacitor and the inductor have always a small ohmic resistance, we can simplify the situation by looking at them at first as solely capacitive or inductive reactances Xc or X1.

Recollecting our knowledge about phase relations at L and C, we find that the voltages appearing in the circuits shown in fig. 69 must have special relations. As we know:

fig. 69

In a series connection of resistances the curent in both components is equal.

Intending to draw a phasor diagram we start therefore with the phasor of the current Itot. We know in both circuits the voltage at the resistor Vr must be exactly in phase with that current.

While the voltage at the capacitor must be lagging for 90 degrees in relation to the current and the voltage at the inductor must be leading for 90 degrees. As we know too: phasors are added geometrically.

fig. 70

Therefore the overall voltage Vtot will be found by shifting the start of Vr up to the end of Vc or V1 and by drawing a line from the noughtpoint up to the end of Vc or V1 we get the overall voltage Vtot necessary to let the current Itot flow through the circuit.

The voltages found at those components are depending on Ohm's Law, therefore Vr = I x R, Vc = I x Xc, and V1 = I x X1

These formulas demonstrate too: the relation between the voltages is equal to the relation between the reactances. In order to get an imagination of the behaviour of one of those circuits we can therefore draw instead of the voltage-triangle a triangle made up from the resistance, the reactance and showing the overall impedance.

fig. 71

This triangle shows very clearly: the impedance of the circuit can be calculated by using the old formula of PHYTHAGORAS. This combination introduced here can be used for so called PASSES.

fig. 72

If we observe how the OUTPUT VOLTAGE is changing while the frequency of the INPUT VOLTAGE is increased over a certain range we observe that the output voltage is:

- either changing from low to high values (HIGH-PASS)
- or from high values to low values (LOW-PASS)

The combination of R and C or R and L offers four different possibilities depending on where the components are positioned.

fig. 73

fig. 75a

fig. 75b

As we can see from the graphs showing the outputvoltage is not suddenly cut off totally at a special frequency, but the outputvoltage is fading out over a wide range of frequency.

We can calculate the output-voltages at various frequencies at each PASS-combination by the following formulas:

Nevertheless for technicians it is necessary to compare different passes in relation to their ability to pass or to cut off the input signal.

Therefore there was defined a certain “LIMITTING FREQUENCY” which is considered as the frequency from which on the output-voltage is defined as “cut-off”. This limitting frequency is reached if the output-voltage is equal or lower than 70.7% of the input-signal. This limiting frequency can be calculated by the following formulas:

CHECK YOURSELF.

1. What does the term PASS mean 7

2. What is the difference between a HIGH - and a LOWPASS.

3. An RL Highpass should have a limitting frequency of 120 Hz. You have a coil with 150 mH. What is the resistance necessary for this pass.

4. What is the limitting frequency of a Low pass which is consisting of a resistance R=120 Ohms and a capacitor of 2 mikroFarad?

5. The limitting frequency of an amplifier should be 28 Hz. The coupling-capacitor. The coupling-capacitor has a capacity of 4.7 nF. Which resistance must have the resistor?