System function

An electrical circuit's system function, also known as the transfer function, relates the input and output voltage in terms of impedances. Besides simply relating the two voltages, the system function has one other major point of interest: its magnitude. These two quantities let you obtain a lot of information about the circuit's properties.

How does one determine the system function? Turn all of the elements into impedance boxes. Then it will be easy to relate the output voltage to the input voltage, because you can just treat every element like you would a resistor. Here's an example circuit:

                        C
             R        |   |
     -----/\/\/\/\----|   |-----------------O
     |                |   |       |
     |                            |         +
    /+\                           |
   /   \                          |        
  |  V  |Vi: Volt               -----       Vo        
   \   / voltage                      C
    \-/  source                 -----
     |                            |         -
     |                            |
     ---------------------------------------O
                        |
                      -----  Ground
                       ---
                        -

The impedance of the resistor is R, and the impedance of a capacitor is (1/(S * C)). So we can treat the relation of V_o and V_i as a voltage divider. Here's the relation:

                       1
        Vi     *     -----
                     S * C
Vo =  ----------------------
          1       1
        ----- + ----- + R
        S * C   S * C

Multiplying through by S * C, that simply becomes:

V_o = V_i / (2 + RSC)

Since S = j * omega, another way to write that is:

V_o = V_i / (2 + jRC(omega))

In that relation, j is the imaginary number that is usually represented in mathematics by i, and omega is the frequency. The system function itself is the ratio of output voltage over input voltage, and is represented as a function, H, of j(omega). Here it is in its proper form:

V_o/V_i = H(j*omega) = 1 / (2 + jRC(omega))

Now that we have that relation, we can derive its magnitude and phase. For a complex number a + bi, the phase angle is tan^-1(b/a). The time delay of the circuit can be derived from the phase angle. The magnitude, |H|, is the square root of (a² + b²). Since the function is in fraction form, we can take the magnitudes of the numerator and denominator separately and still get the correct magnitude. The magnitude of the numerator is just 1. The magnitude of the denominator is only a little more complicated: a = 2, so a² = 4. b = RC(omega), so b² = R²C²(omega²). Thus the magnitude is:

               1
|H| = -------------------
      (4 + R2C2(omega2))1/2

This gives us the magnitude of the system function as a function of frequency. If the magnitude is small, the output voltage will be small compared to the input voltage. When the magnitude is large, the output voltage will be comparatively large. For small values of omega, our example's magnitude will be large. For larger frequencies, it will be small. You can test this by plugging in numbers like .001 and 1000 for the frequency, and seeing the resulting magnitude.

If the magnitude is small, the output voltage will be small compared to the input voltage. When the magnitude is large, the output voltage will be comparatively large. Since the system function depends on frequency, certain circuits will effectively block or pass certain frequencies. If, at a certain frequency, the magnitude is small, that frequency is blocked by the circuit. Constructing circuits in different ways allow us to design circuits that pass certain values of frequency. Our circuit above is an example of a low pass filter, because it rejects all of the high frequencies and passes the low ones. Other examples of common filters are high pass filters, band pass filters, and notch filters.