Pole splitting arises most famously in connection with Miller frequency compensation, where placing a capacitor across an inverting gain stage causes the poles associated with that stage’s input and output ports to split into opposite directions in the s plane, the former shifting to a *lower* frequency, and the latter shifting to a *higher* frequency. A byproduct of Miller compensation is the creation of a right half-plane zero (RHPZ).

To illustrate, let’s start out with the basic gain stage of Figure 1, where the dependent source $$G_mV_1$$ models gain, and the $$R_1-C_1$$ and $$R_2-C_2$$ networks model the input-port and the output-port poles.

**Figure 1. **AC model of a gain stage.

**Figure 1.**AC model of a gain stage.

By inspection,

$$V_1 =\frac{1/(sC_1)}{R_1+1/(sC_1)}V_i=\frac{1}{1+sR_1C_1}V_i \; \; \; \; V_o = - G_{m}V_{1} (R_2\parallel\frac{1}{sC_2} ) = \frac{-G_{m}R_2}{1+sR_2C_2}V_1$$

**Equation 1**

**Equation 1**

where *s* is the complex frequency. Eliminating $$V_1$$, we express the stage’s gain *a(s)* in the insightful form

$$a(s) =\frac{V_o}{V_i}=\frac{a_0}{(1+s/\omega _{10})(1+s/\omega_{20})}$$

**Equation 2**

where

$$a_0 = -G_mR_2$$

**Equation 3**

**Equation 3**

is the stage’s *dc gain*, which is negative, and

$$\omega_{10} = \frac {1}{R_1C_1} \; \; \; \; \; \; \; \; \omega_{20} = \frac {1}{R_2C_2}$$

**Equation 4**

**Equation 4**

are the *angular frequencies* associated with the input-port and output-port poles. (Recall that the angular frequency *ω*, in radians/sec, and the frequency *f*, in Hertz, are related as *ω = 2πf*).

The denominator of *a(s)* vanishes for $$s = –\omega_{10}$$ and $$s = –\omega_{20}$$, which are aptly called the *poles* of *a(s)* because they cause *a(s)* to blow up to infinity. In this case, both poles lie on the *negative real axis* of the *s* plane.

Let us now connect the capacitor $$C_f$$ across the gain stage as in Figure 2, and let us recalculate the gain *a*(*s*).

**Figure 2.** AC model of a gain stage with $$C_f$$ present.

**Figure 2.**AC model of a gain stage with $$C_f$$ present.

Applying KCL at the node to the left of $$C_f$$ gives

$$\frac{V_i-V_1}{R_1} = \frac {V_1}{1/(sC_1)}+\frac{V_1-V_o}{1/(sC_f)}$$

**Equation 5a**

**Equation 5a**

Likewise, KCL at the node to the right of $$C_f$$ gives

$$\frac{V_1-V_o}{1/(sC_f)} = G_mV_1 + \frac{V_o}{R_2}+\frac {V_o}{1/(sC_2)}$$

**Equation 5b**

**Equation 5b**

Eliminating $$V_1$$ and solving for the ratio $$V_o/V_i$$ we get, after suitable algebraic manipulations,

$$a(s) = \frac{V_o}{V_i} = a_0 \frac{1-sC_f/G_m}{1+s\left\{R_1[C_1+C_f(1+G_mR_2)]+R_2(C_f+C_2)\right\}+s^2R_1R_2(C_1C_f+C_1C_2+C_fC_2)}$$

*Equation 6*

*Equation 6*

with $$a_0$$ as in Equation 3. It is fair to say that the above expression doesn’t provide a tremendous amount of insight, yet we can still glean some interesting peculiarities.

- Despite the presence of three reactive elements, the circuit is only of the
*second-order*type. This is so because the capacitors form a loop, so once we know the voltages across two of them, the voltage across the third is also known, by KVL, so we have only two degrees of freedom. - The numerator vanishes for $$1 – sC_f/G_m = 0$$, indicating a gain of zero for $$s = ω_0$$, where

$$\omega_0 = \frac {G_m}{C_f}$$

**Equation 7**

**Equation 7**

and $$\omega_0$$ is aptly referred to as a *zero frequency*.

Physically, the current transmitted via $$C_f$$ towards the output node splits among $$R_2$$, $$C_2$$, and the dependent source. If we adjust *s* so that the current sourced by $$C_f$$ *equals* that sunk by the dependent source, then none will go to $$R_2$$ and $$C_2$$, thus implying $$V_o$$ = 0. When this happens, we have $$(V_1 – 0)/[1/(sC_f)] = G_mV_1,$$ or $$sC_f = G_m$$, or $$s = G_m/C_f = ω_0$$. This zero lies on the *positive real axis* of the *s *plane, so it is known as a right half-plane zero (RHPZ).

Based on the above observations, we stipulate a gain expression of the more insightful type

$$a(s) =\frac {V_o}{V_i}=a_0\frac {1-s/\omega_0}{(1+s/\omega_1)(1+s/\omega_2)}$$

**Equation 8**

**Equation 8**

with $$a_0$$ as in Equation 3, and $$ω_0$$ as in Equation 7. Our next task is to seek possibly insightful expressions for the pole frequencies $$\omega_1$$ and $$\omega_2$$. Given the negative-feedback action provided by $$C_f$$, we expect these frequencies to be quite different from $$\omega_{10}$$ and $$\omega_{20}$$ of Equation 4, where subscript zero was meant to signify “with $$C_f$$ = 0.” In particular, being already familiar with the Miller effect, we anticipate a *dominant pole* on the order of

$$\omega_1\cong \frac {1}{R_1[C_1+C_f(1+G_mR_2)]}$$

*Equation 9*

*Equation 9*

and such that $$\omega_1 << \omega_2$$. This last condition allows us to make the following approximation:

$$\frac{1-s/\omega_0}{(1+s/\omega_1)(1+s/\omega_2)}=\frac{1-s/\omega_0}{1+s/\omega_1 +s/\omega_2+s^2/(\omega_1 \omega_2)}\approx\frac{1-s/\omega_0}{1+s/\omega_1 +s^2/(\omega_1 \omega_2)}$$

**Equation 10**

**Equation 10**

Equating the coefficients of s in the denominators of Equations 6 and 10, we easily find

$$\omega_1 = \frac {1}{R_1[C_1+C_f(1+g_mR_2+R_2/R_1)]+R_2{C_2}}$$

**Equation 11**

**Equation 11**

Considering that both denominators in Equations 9 and 11 are usually dominated by the Miller capacitance $$C_M = C_f(1 + G_mR_2)$$, we have a good reason to trust that the approximation of Equation 9 is a fairly good one.

Equating the coefficients of $$s^2$$ in the denominators of Equations 6 and 10, we get

$$\omega_1\omega_2 = \frac{1}{R_1R_2[C_f(C_1+C_2)+C_1C_2]}$$

**Equation 12**

**Equation 12**

indicating that for a given $$C_f$$, the product $$\omega_1\omega_2$$ is constant, so the *downshift* $$\omega_{10}$$ → $$\omega_{1}$$ must be accompanied by an *upshift* $$\omega_{20} \rightarrow \omega_{2}$$.

### Pole-Splitting in PSpice

Let us use PSpice to observe the pole-splitting phenomenon. Consider Figure 3, below.

**Figure 3. **PSpice circuit to plot pole splitting for $$C_f = 2 pF$$.

**Figure 3.**PSpice circuit to plot pole splitting for $$C_f = 2 pF$$.

With the component values shown, we have (by Equation 4), $$\omega_{10} = 1/(100 \times10^3 \times 25 \times10^{–12})$$ = 400 krad/s and $$\omega_{20}$$ = 4.0 Mrad/s, corresponding to the frequencies

$$f_{10} = 63.66 kHz \; \; \; \; \; \; \; \; f_{20} = 636.6 kHz$$

**Equation 13**

**Equation 13**

which are separated by a decade. By Equation 11 we have

Note, incidentally, that had we used Equation 9, we would have gotten $$\omega_1$$ = 23.42 krad/s, a fairly good approximation. By Equation 12 we have

$$\omega_2=\frac{1/23.23 \times 10^3}{10^5\times50\times10^3[2\times10^{-12}(25+5)10^{-12}+25\times5\times10^{-24}]}=46.40 \; Mrad/s$$

Finally, Equation 7 gives $$\omega_0 = 4 \times 10^{–3}/(2 \times 10^{–12})$$ = 2.0 Grad/s. The above data yield the frequencies

f_{1} = 3.70 kHz f_{2} = 7.38 MHz f_{0} = 318 MHz

Our findings are confirmed by the plots of Figure 4.

**Figure 4. ** Gain magnitude and phase plots for the PSpice circuit of Figure 3.

**Figure 4.**Gain magnitude and phase plots for the PSpice circuit of Figure 3.

In summary, the presence of *C _{f}* has a triple effect:

- It
*downshifts*the first pole frequency by a factor of 63.66/3.70 = 17.2. - It
*upshifts*the second pole frequency by a factor of 7,380/636.6 = 11.6. - It establishes a
*transmission zero*at 318 MHz.

The pole/zero situation in the *s* plane is depicted in Figure 5.

**Figure 5. **S-plane illustration (not to scale) of pole splitting as well as RHPZ creation.

**Figure 5.**S-plane illustration (not to scale) of pole splitting as well as RHPZ creation.

### The Right Half-Plane Zero (RHPZ)

Let us conclude by taking a closer look at the right half-plane zero (RHPZ), which will be referenced abundantly in the next article on stability in the presence of a RHPZ.

**Figure 6.** PSpice circuit to contrast a RHPZ and a LHPZ.

**Figure 6.**PSpice circuit to contrast a RHPZ and a LHPZ.

Looking first at the $$C_f$$ = 0 plots of Figure 4, we note that from $$f_{10}$$ to $$f_{20}$$ magnitude rolls off at the rate of –20 dB/dec, and past $$f_{20}$$ at the rate of –40 dB/dec. The corresponding phase angle swings from 180° (due to the fact that $$a_0$$ is negative) to 90°, and then from 90° to 0°.

Turning next to the $$C_f$$ = 2 pF plots, we note that from $$f_1$$ to $$f_2$$ magnitude rolls off at a rate of –20 dB/dec, from $$f_2$$ to $$f_0$$ at a rate of –40 dB/dec, and past $$f_0$$ again at a rate of –20 dB/dec. The corresponding phase angle swings from 180° to 90° and from 90° to 0° because of the pole frequencies $$f_1$$ and $$f_2$$ , and then from 0° to –90° because of the RHPZ frequency $$f_0$$. In other words, a RHPZ introduces a phase delay just like an ordinary left half-plane pole! This delay may destabilize a negative-feedback circuit containing a RHPZ around the loop. (For more information on this issue, please refer to The Right Half-Plane Zero and Its Effect on Stability.)

Put another way, the gain stage acts as an inverting amplifier for $$f < f_0$$, but as a noninverting amplifier for $$f > f_0$$, thus turning the overall feedback from negative to positive, and possibly destabilizing the loop.

To further evidence similarities and differences between a RHPZ and a LHPZ, consider the expressions

$$a_{RHPZ} = a_0 \frac{1-s/\omega_0}{(1+s/\omega_1)(1+s/\omega_2)} \; \; \; \; \; \; \; \; \; \; a_{LPHZ}= a_0 \frac{1+s/\omega_0}{(1+s/\omega_1)(1+s/\omega_2)}$$

**Equation 14**

**Equation 14**

which we are going to simulate via PSpice using the parameter values of Equation 14. To find expressions for magnitudes and phase angles in terms of frequency *ω*, we let *s → jω*. Then we write

$$Mag[a_{ORHPZ}]= Mag[a_{OLPHZ}] = \left | a_o \right |\sqrt{\frac{1+(\omega/\omega_0)^2}{(1+(\omega/\omega_1)^2)\times(1+(\omega/\omega_2)^2)}}$$

$$Ph[a_{ORPHZ}]=180^\circ - tan^{-1} (\omega/\omega_0)- tan^{-1}(\omega/\omega_1)- tan^{-1}(\omega/\omega_2)$$

$$Ph[a_{OLPHZ}]=180^\circ + tan^{-1} (\omega/\omega_0)- tan^{-1}(\omega/\omega_1)- tan^{-1}(\omega/\omega_2)$$

Magnitudes are the same because $$Mag[a \pm jb] = \sqrt{(a^2 + b^2)}$$. However, the phase angles differ because $$Ph[a \pm jb] = \pm tan^{-1} (b/a)$$. Magnitude similarity as well as phase-angle difference are evidenced in Figure 7.

**Figure 7.** Gain magnitude and phase plots for the PSpice circuits of Figure 6.

**Figure 7.**Gain magnitude and phase plots for the PSpice circuits of Figure 6.

It is important to realize that phase-margin estimations via the rate-of-closure (ROC) are not applicable in the present case because a system with a RHPZ is not a minimum-phase system.

#### References

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