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Type II compensators are widely used in the control loops for power converters. However, there are cases where the phase lag of a power converter can approach 180 degrees, while the maximal phase from a type II compensator at any frequencies is at most zero degree. Thus in these cases, the type II compensator cannot provide enough phase margin to keep the loop stable, and this is where a type III compensator is needed. A type III compensator can have a phase plot going above zero degree at some frequencies, and therefore it can provide the required phase boost to maintain a reasonable phase margin.
Although the concept of the type III compensator has been around for years, an indepth analysis on the compensator is not easy to find. There are some design procedures described in the literature [1,2,3,4]. However, these procedures are usually empirically derived, and the derivation processes are not provided, which make it difficult to follow and evaluate these procedures.
An analog implementation of type III compensators is shown in Fig. 1, where six passive circuit components are needed. The transfer function of the Type III compensator in Fig. 1. is given by:
where C_{12} is the parallel combination of C_{1} and C_{2},
The Type III compensator has three poles (one at the origin) and two zeros. In practice, it is usually arranged to have two coincident zeros and two coincident poles, and the loop crossover frequency is placed somewhere between the zeros and poles. For this kind of design, the transfer function in Equation (1) can be rewritten as:
where the zero's and pole's frequencies are given by:
and the constant gain K is given by:
The amplitude of the transfer function in Equation (3) at a given frequency ω can be calculated as:
The phase of the transfer function in Equation (3) at a given frequency ω can be calculated as:
As can be seen, the phase of C(jω) has two parts: a constant part of π/2 due to the pole at the origin, and a variable part as a function of frequency ω given by:
Equation (8) can be converted to:
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Equation (9) has a useful feature in that the function reaches its maximum value somewhere between ω_{z} and ω_{p}. This can be shown as follows. Note that the inverse tangent function is monotonically increasing. Therefore, to find the maximum value of Φ_{v}(ω), we can first search for the maximum of the following function:
The maximum value of F(ω) can be found through its derivative, which is calculated as:
Based on Equation (11), you find that F(ω), and hence Φ_{v}(ω), reaches their maximum values at the frequency defined by:
Equation (12) says that the maximum phase of Φ_{v}(ω) occurs at the geometric mean of ω_{z} and ω_{p}. Here, we call ω_{m} the maximum phase frequency of a type III compensator. By substituting Equation (12) into (9), you get the maximum phase of Φ_{v}(ω) as:
Define the ratio of the pole's frequency to the zero's frequency as:
From Equation (12) and (14), k can also be defined as:
Then the maximum phase of Φ_{v}(ω) can be written as:
And the maximum phase of the type III compensator is given by:
Note that k is a measure on the distance between the zero and pole, and hence we call it separation factor. With Equation (17), you can calculate the maximum phase boost of the type III compensator for a given separation factor, or vise versa.
The maximum value of an inverse tangent function is 90°. Base on Equation (17), the maximum phase boost from a type III compensator is 90°. Fig. 2 shows the maximum phase of a type III compensator vs. the separation factor. As can be seen, the phase increases quickly when the separation factor k is small, and it becomes more and more flat as the factor goes high. Thus, in the low range of k, it is more effective to adjust the phase boost from a type III compensator by changing the separation factor. It is worth to note that there is a range for the separation factor where the phase of the type III compensator is negative. Since the main purpose of using a type III compensator is to boost the control loopís phase, it is useful for the designer to know the value of the separation factor at which the phase is zero. From Equation (17) you can find that the phase is zero when the inverse tangent function meets:
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From Equation (18) we can see that k needs to meet:
By solving Equation (19), we get the value of k that gives zero phase boost for a type III compensator:
As a rule of thump, the separation factor of a type III compensator should be larger than 6 in order to provide a positive phase boost to the loop.
DESIGN PROCEDURES
Procedure I
With Procedure I, you place the loop crossover frequency (ω_{c}) at ω_{m}, and in this way you can reach the maximal loop phase margin with a given separation factor. In the following, a design procedure is derived to achieve this design goal.
Let the control plant's gain and phase at ω_{c} be Gp and Φ_{p}, and the desired phase margin be Φ_{m}. To meet the phase margin requirement, we should have:
Thus, we can get the phase Φ_{v}(ω_{c}) as follows:
Since we have chosen ω_{c} =ω_{m}, thus from Equation (16) and (22) we get:
Or equivalently, we have:
where b is defined by:
Define:
Then, from Equation (24) we get the following quadratic equation in terms of x:
The solutions to Equation (27) are given by:
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From Equation (26) you can see that x is positive, therefore the solution we need is given by:
Given ωm and k, we can get the zero ω_{z} and pole ω_{p} based on Equation (15):
From Equation (6), we can get the compensator's gain at the crossover frequency, ω_{c} = ω_{m}:
At the crossover frequency ω_{c}, the loop gain is equal to 1, that is:
From Equation (32) we get:
As shown in [2] and [3], the design of a type III compensator usually starts with choosing a value for R_{1}. With R_{1} chosen, the rest components values can be calculated as follows.
From Equation (5), we get:
Equation (4) is equivalent to the following four equations:
Equations (34) to (38) are the five equations that we need to determine the componentsí values R_{2}, R_{3}, C_{1}, C_{2} and C_{3}.
By subtracting Equation (37) from Equation (36), we get the value of C_{3}:
From Equation (35) we have:
Equation (38) can be rewritten as:
By Substituting Equation (34) and Equation (40) into Equation (41), we get:
From Equation (42), we get the solution for C_{1}:
From Equation (34), we get the solution for C_{2}:
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Now that we have determined the values for all of the capacitors. The resistor values can be obtained from Equation (37) and Equation (38):
Summarizing Procedure 1 for the Type III compensator, we find that:
Given desired crossover frequency ω_{c} and phase margin Φ_{m}, and the control plant's gain and phase at ω_{c} as Gp and Φ_{p}.
 Calculate the tangent value b using:
 Calculate the zero and pole separation factor k:
 Calculate the zero's and pole's frequency:
 Calculate the compensator's gain K:
 Select a resistor value for R_{1}.
 Calculate C_{3}:
 Calculate C_{1}:
 Calculate C_{2}:
 Calculate R_{2}:
 Calculate R_{3}:
 Verify the calculated compensator's values and frequency response.
 Check the closed loop's frequency response.
Procedure II
A type III compensator is usually used for the control plant that has a big phase lag around the loop crossover frequency range. For this type of plants, the control loop may end up to be conditionally stable, which is not desired in some applications. In the following, a design procedure is derived which takes unconditional stability into consideration.
A conditionally stable loop is the one whose phase plot goes more negative than 180°, but comes back above 180° again before the crossover frequency. This occurs usually around the frequency where the plant has the maximum phase lag. We name this frequency as ω_{mp}.
To make the loop unconditionally stable, some phase boost is needed at ω_{mp}. On the other hand, to make the loop stable, a certain amount of phase boost is also needed at the crossover frequency ω_{c}. To meet these requirements, one can place the maximum phase frequency ω_{m} (defined by Equation 12) somewhere between ω_{mp} and ω_{c}. The placement of ω_{m} can be described by the following:
where ∝is a number to be determined.
At the logarithmically scaled frequency axis, the geometric mean of ω_{mp} and ω_{c} has an equal distance to ω_{mp} and ω_{c}, as shown in Fig. 3. You can see that if ∝ < 1, then ω_{m} is closer to ω_{mp} than to ω_{c}. On the other hand, if ∝> 1, then ω_{m} is closer to ω_{c} than to ω_{mp}. Therefore, you can use ∝ to adjust the location of the maximum phase point of a type III compensator, and by taking a trialanderror search on ∝, you can eventually find the proper location of ωm that meets both of the phase margin and unconditional stability requirements.
As soon as ω_{m} has been selected, the type III compensator can be calculated as follows. Given the control plant's gain and phase at ω_{c}: Gp and Φ_{p}, and the desired phase margin Φ_{m}. Also given the frequency ω_{mp} at which the plant has the maximum phase lag. At the crossover frequency, based on Equation (9) we have:
To meet the phase margin requirement, we need to satisfy Equation (22), which in turn leads to:
Or, equivalently:
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Based on Equation (12) and Equation (59) we can get the following two equations about ω_{p} and ω_{z}:
where ω_{d} is defined by:
Note that ω_{d} is known with the given parameters Φ_{m}, Φ_{p}, and ω_{p}, and the selected frequency ω_{c}.
We can solve Equation (60) and (61) and get the compensator's zero and pole frequencies:
The separation factor can be calculated as:
With ω_{z}, ω_{p} and k determined based on the above equations, the compensator's components can be determined in the same way as in Procedure I. The design procedure that accounts for unconditional stability is summarized below.
Given desired crossover frequency ω_{c} and phase margin Φ_{m}, and the control plant's gain and phase at ω_{c} as Gp and ω_{p}. Also given the frequency ω_{mp} at which the plant has the maximum phase lag.
Based on Equation (56) determine the compensator's maximum phase frequency by choosing a value for ∝. Usually you can start with ∝ =1, and adjust it based on the loop's Bode plot resulted from the design procedure.

Calculate the difference between the zero's frequency and pole's frequency using Equation (62).

Calculate the zero's frequency ω_{z} and pole's frequency ω_{p} using Equation (63) and Equation (64).

Calculate the separation factor k using Equation (65).

From Equation (6) we have
At the crossover frequency:
Thus, we can calculate the compensator's constant gain K as:
With K determined, one can follow the steps 5 through 12 in Procedure I to finish the design.
The synchronous dctodc converter shown in Fig. 4 [2] will be used an example for applying the design procedures. In [2], the target bandwidth is set to 90kHz, and the phase margin is required to be larger than 45°. Here, the same bandwidth is targeted, and the phase margin is targeted at 60°. From Fig. 4, one can find that at 90kHz, the plant's gain is 29.14dB or 0.0349, and the phase is 109.1°.
First, Procedure I will be used to calculate the compensator. With this approach, we place the compensator's maximum phase boost frequency at the target crossover frequency, that is, ω_{m}=2×π×90×10^{3}rad/s. By choosing R_{1} = 2kΩ and following the procedure, we get the following component values:
R_{1} = 2kΩ
R_{2} = 34.7kΩ
R_{3} = 571Ω,
C_{1} = 31pF
C_{2} = 108pF
C_{3} = 1.5nF
Fig. 5. shows the compensator's and the resulting loop's Bode plots. The separation factor is 4.5 and the phase plot of the compensator is under zero degree as shown in Fig. 5. The loop bandwidth is 90kHz and phase margin is 60°. However, the phase plot goes more negative than 180°, thus making the loop only conditionally stable.
Utilizing Procedure II, the first step is to locate the compensator's maximum phase boost frequency. From Fig. 4, the maximum phase lag frequency is about 9kHz. Thus, ω_{m} should be somewhere between 9kHz and 90kHz. Based on Equation (56), it was found that with ∝=0.7 we can get a good unconditionally stable control loop. With this value of ∝, the following component values result:.
R_{1} = 10kΩ
R_{2} = 30.4kΩ
R_{3} = 568Ω,
C_{1} = 66pF
C_{2} = 1.2nF
C_{3} = 3.4nF
Fig. 6. shows the resulting loop's Bode plots. As you can see, the loop in Fig. 6 is unconditionally stable, as opposed to that in Fig. 5.
REFERENCES

Venable Technical Paper #3, Optimum feedback amplifier design for control systems.

Intersil Technical Brief TB417.1 by Doug Mattingly, Designing stable compensation networks for single phase voltage mode Buck regulators, 2003.

Sipex Application Note 16, Loop compensation of voltagemode buck converters, 2006.

Keng Wu, Switchmode power converters, design and analysis, Academic Press, 2006.
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