Load and Supply Interaction Formalisms

The importance of the dynamical profile is revealed, when analyzing the effects of the load and source subsystems. Every practical converter is a part of an interconnected system; the dynamics of a stand-alone converter may be affected by external impedances e.g. EMI filters, cabling inductances, additional capacitors and closed-loop input and output impedances of the other converters in the system. The open-loop g-parameter set provides the basis for analyzing the interactions. However, the supply interaction formalism introduces two special admittance parameters that have to be considered also in the analysis. The two-port model of the interconnected system is shown in Fig. 2.6, where the load and supply systems are modeled as external impedances (i.e. the load as Z_L and the supply as Z_S).

The load interactions can be found by computing ˆi_o at the presence of the impedance-type load Z_L from Fig. 2.6 by applying basic circuit theory, yielding

ˆ ˆ ˆ

The load-affected open-loop transfer functions can be found by replacing ˆi_o in (2.4) by means of (2.16) giving a load-affected transfer function matrix as

ˆ ˆ

The two-port model of the interconnected system in closed-loop is similar to the open-loop system shown in Fig. 2.6. Only the subscript –o, which refers to open loop is changed to –c, referring to closed loop. Therefore, the corresponding load-affected set at closed loop assuming that the voltage reference u_r(i.e. 0uˆ_r o ) is constant can

Fig. 2.6. Two-port model of voltage-output converter with load and supply subsystems.

It is apparent that the load interactions on the output dynamics are directly reflected via the open-loop output impedance. According to (2.17) and (2.18), the internal output dynamics would stay intact if the open-loop output impedance is small.

The load interactions on the input dynamics are not as straightforward. The output impedance clearly has an effect on the internal transfer functions, but the open-loop forward (i.e. G_{io o} ) and reverse (i.e. T_{oi o} ) transfer functions have to be considered too. According to (2.17), a small open-loop output impedance would actually result only in intact reverse transfer functions, because

o o 0

The input admittance stays intact if G_{io o} or T_{oi o} is close to zero

Consequently, a zero T_{oi o} would make G_ci insensitive to load interactions (i.e. _ci ^{co oi o Toi o 0} _ci

The load-affected loop gain L^L_VO is a combination of the load affected control-to-output transfer function in (2.17) and the internal loop gain in (2.5), yielding

1

According to (2.20), the load interactions in the loop gain are reflected via the nominal open-loop output impedance Z_{o o} .

The supply or source interactions on the converter dynamics can be found by computing ˆu_in at the presence of the impedance-type source Z_S from Fig. 2.6, which

The source-affected open-loop transfer functions can be found by replacing ˆu_in in (2.4) by means of (2.21) giving a source-affected transfer function matrix as

ˆ 1 1 1 ˆ

Because the two-port model of the interconnected system in closed-loop is similar to the open-loop system shown in Fig. 2.6 (only the subscript –o, which refers to open

at closed loop, assuming that the voltage reference u_r(i.e. 0uˆ_r o ) is constant can be

They are the same at open and closed loop. Y_inf is the input admittance in a special condition, where both ˆu_o and ˆi_o are zero. Equation (2.24) can be obtained by letting

ˆ_o

u and ˆi_o zero in (2.1) and (2.2), and then solving cˆ from (2.2) and replacing it in (2.1) in order to compute ˆ / ˆi_in u_in Y_inf. Y_inf is also known as an ideal or infinite-bandwidth input admittance because of the closed-loop input admittance defined in (2.10). Therefore, it is symbolically the same for a converter with a certain topology regardless of the conduction and control modes as well as load [P5], [43] and [52].

For a buck converter it also is physically the same (i.e. I_in/U_in [P5]). In other converters the physical correspondence might be lost, because certain parameters can be dependent on the value of the actual circuit elements [52]. Y_{in sc} is known as the short-circuit input admittance and being dependent on the operation and control modes [P2] and [43] . In a short-circuited converter ˆu_ois zero. Therefore, Y_{in sc} can be

According to (2.22) and (2.23), the supply interactions on the input dynamics are reflected via the open-loop input admittance (i.e. Y_{in o} ). This applies also to the open-loop forward transfer function (i.e. G_{io o} ). However, on the open- and closed-loop output impedances the supply interactions are reflected via the open-loop input

admittance (i.e. Y_{in o} ) and the short circuit admittance Y_{in sc} . On the control-to-output transfer function (i.e. G_co) the supply interactions are reflected via the open-loop input admittance and the ideal input admittance Y_inf.

The source affected loop gain can be presented as 1

1

S s in

VO VO

s in o

L Z Y L

Z Y

f

(2.26)

Obviously, the source interactions in the loop gain are also reflected via the nominal open-loop input admittance Y_{in o} and the ideal input admittance Y_inf.

In order to have intact output impedance and loop gain, it is obvious (see (2.24) and (2.25)) that G_{io o} must be zero. This would make Y_inf and Y_{in sc} to be the same as

Yin o and Y_{in c} even if Y_{in o} or Y_{in c} exhibits a resonant behavior.

It is important to note that the load may also change the input-port parameters and hence the supply interactions by changing the input impedances [P11]. It is also evident that the source system may have effect on the load interactions by changing e.g. the output impedances in (2.22) (via Y_{in o} and Y_{in sc} ) and, hence in (2.17).

It should also be noted that it is not always necessary to analyze all the transfer functions of the profile. The four most meaningful and important transfer functions to be analyzed are Y_{in o} , Z_{o o} , G_{io o} and G_co. These four parameters are the key elements reflecting the load and source interactions as can be concluded e.g. from (2.17) and (2.22). If a complete understanding and characterization of a converter is needed, all the transfer functions in the dynamical profile are worth analyzing.