Second Order PLL's

The second order phaselock loop is widely used. One reason for this will be described in section 3.4.4. A second order phaselock loop can utilise either a passive or active loop filter. Typical passive and active filters are depicted in figures 3.3 and 3.4.

 

Figure 3.3:   Passive filter used in second order PLL's

 

Figure 3.4:   Active filter used in second order PLL's

The filter transfer functions are  

$$F_P(s)=\frac{s\tau_2 +1}{s\tau_1 +1}$$ (18)

where

$$\tau_1=(R_1+R_2)C\qquad \tau_2 =R_2C$$

for a passive filter as depicted in figure 3.3. For an active filter as shown in figure 3.4,  

$$F_A(s)=-\frac{-A(s\tau_2+1)}{s\tau_1+1+(1+A(sCR_1))}$$ (19)

where

$$\tau_1=R_1C\qquad \tau_2=R_2C.$$

Omitting the filter altogether, so that F(s)=1 results in a first order phaselock loop. Such a loop has a 3-dB bandwidth of  

$$B=K_o K_d\mathrm{[rad/sec]}.$$ (20)

An important phaselock loop attribute which will be utilised later is known as the open loop gain , K. It is detailed for various common loop types in table 3.2.

 

Loop Description Open Loop Gain, K(radian frequency)

First Order K_oK_d

$$\frac{K_oK_d\tau_2}{\tau_1}$$ Second order