The block diagram for a power reactor appears in Fig. 5.2. The closed — loop transfer function is

G = GJ{ 1 + GjH)

Fig. 5.2. Power reactor block diagram where Gj(s) = <5n(s)/<5p(s) is the zero-power transfer function and H(s) = <5p(s)/<5n(s) the feedback transfer function.

or

G = 1/[(1/Gj) + H] (5.2.2)

From this, we can conclude

G = Gj when 11/G j I » |Я|

G = 1 /Н when 11/G! I « |H|

The effect of feedback on the closed-loop frequency response can be evaluated using a typical feedback transfer function and the zero-power transfer function (Eq. 5.1.3). The simplest form that might be used to model a feedback transfer function is a first-order lag.

H = Sp/Sn = A/(rs + 1) (5.2.3)

More detailed models would have a number of poles and zeroes, but there will always be a high-frequency break followed by a monotonic decrease in gain with increasing frequency. Typical values for this break are 1 rad/sec for light-water reactors and 10 rad/sec for liquid-metal-cooled fast reactors. (This difference is because fast-reactor fuel elements have smaller diameters and higher surface heat-transfer coefficients.)

Figure 5.3 shows gain plots for a zero-power frequency response and a typical feedback frequency response based on a first order lag. Figure 5.4 shows the closed-loop frequency response for this system. At low frequencies, ll/GJ « H, and G = 1 /Н. At high frequencies, |1 /GJ » |H|, and G ~ Gj. Figure 5.5 shows the effect of the break frequency for a first-order-lag feedback model on closed-loop system gain.

Of course, the feedback reactivity can have phase shifts that cause it to increase the net reactivity over the input reactivity at some frequencies.

c

в

(3

.1 1
Frequency (radians/sec.)

Q.

■o (3 .001

.001 .01 .1 1 10 100
Frequency (radians/sec.)

Fig. 5.3. (a) Amplitude for a zero-power reactor; (b) amplitude for a simple reactivity feedback model.

In this case the gain of the closed-loop frequency response is greater than the gain of the zero-power frequency response at these frequencies. This indicates that the feedback has a destabilizing effect. This often occurs in actual systems, and frequency response tests may be used to monitor this effect with changing system conditions (such as power level). Figure 5.6 shows closed-loop results for the molten-salt reactor experiment at several power levels(2). Clearly, the feedback causes the gain at some power levels to be

larger than the gain at zero power. Furthermore, stability for this system improves with increasing power level.