IEEE Power Electronics Magazine - March 2021 - 56

NISM is applicable to both linear circuits and switchedmode power systems. Finding phase margin using impedance rather than loop gain addresses closed-form solutions
that provide no access to internal circuit nodes: The growing
family of sealed electronic modules includes fixed voltage/
current regulations, voltage references and op-amps. Now
they can be analyzed fully, thanks to the rapid interpretation
of NISM derived outcomes.

Equation [1] shows feedback factor {1 + G (s) . H (s)}
imprinted on the open loop output impedance of the system
to produce closed loop output impedance. The open loop output impedance {Zout(s)} is that of the " loaded " plant minus
the feedback: It encompasses the PRM output port, the output filter and load. Generally at low frequencies i.e., within
the loop gain bandwidth of the regulator, {G (s) . H (s) & 1}.
Zout CL .

3. Review of Classical Feedback Theory
The block diagram in figure 1 models a voltage regulator as
power train G(s) with feedback compensation network
H(s). We use (s = j~) to represent complex frequency. A
feedback control system for regulation must be designed in
such a way that {G (s) . H (s)} exhibits sufficient " margin " .
This prevents negative feedback turning into positive feedback sufficient to destabilize the loop.
The critical point is approached as loop gain magnitude
approaches (-1): both the closed loop gain and output
impedance of the feedback loop approach 3, implying that
external inputs no longer influence output voltage.
In Figure 1, {Vref(s)} is the reference voltage. This fixes
the output voltage level. This is achieved in spite of line,
load and environmental factors causing variation by using
negative feedback. In the simplified model, externally
imposed variations in Vin(s) are seen here multiplied by the
plant response. This is added to the output voltage, the load
transitions taking the form {Iout(s).Zout(s)} .
This model allows determination of output voltage as a
function of the system's parameters. With Vref(s) and Vin(s)
nulled, we can examine the loop and write an expression
for output impedance " on the diagram " that takes the form
Vout ^ s h = Iout . <

Zout ^ s h
F
1 + G^ s h . H^ s h

and because
def

	

Zout CL =

	

Zout CL =

Vout ^ s h
closed loop 
Iout ^ s h

Zout ^ s h
(1)
1 + G^ s h . H^ s h

V in(s ).G (s )

Vref(s)

+

Σ

+

V err(s)
G (s )

+

I out(s) . Zout(s)
+
Σ

V out(s)

-

H (s )

IEEE POWER ELECTRONICS MAGAZINE	

Using logarithmic scaling, this becomes
	

Zout CL , 20 . 6log Zout - log G . H @ " dB , (2)

To understand stability and its relation to closed loop
gain, we can examine a system that has a steady-state load,
this time with input voltage experiencing perturbations,
with the reference voltage and load held steady. Input line
regulation can be determined as {Vout(s)/Vin(s)}.
From Figure 1 we find that
Vout ^ s h = V in ^ s h .

G^ s h
1 + G^ s h . H^ s h

Leading to the well-known expression for closed loop gain
	

Av CL =

Vout ^ s h
G^ s h
=
(3)
Vin ^ s h
1 + G ^ s h .H ^ s h

Note that both the output impedance (equation [1])
and closed loop voltage gain (equation [3]) share the same
denominator term {1 + G (s) . H (s)} which . {G (s) . H (s)}
when {G (s) . H (s) & 1}. Product {G (s) . H (s)} is the feedback system's loop gain.
If loop gain with changing frequency moves too close to
(-1) in the frequency domain, it will result in its peaking
at crossover. If unchecked, the resonance results in oscillatory behavior.
Stability can be precisely determined within limits
on the basis of relative peaking of the output impedance.
Group delay maps to phase margin, with both parameters
being very sensitive to frequency variation. NISM is independent of absolute magnitude levels of output impedance;
one only has to assess the quality factor of this impedance
within the control loop bandwidth.
Direct series in-loop signal injection can be simulated
and measured, allowing comparisons to be made with
NISM (this likened to a parallel signal injection). The bandwidth of the system can be assessed with signal injection by
measuring the crossing frequency where the loop gain has
a unit magnitude. How close loop gain gets in the frequency
domain to the critical (-1) value can be accurately quantified by phase margin.

4.0 Mapping Output Impedance
to Phase Margin

FIG 1 Closed loop block diagram representation of a voltage
regulator.

56	

Zout ^ s h
G^ s h . H^ s h

z	March 2021

The detailed theory of the relationship between output
impedance and phase margin is beyond the scope of this
IEEE Power Electronics Magazine - March 2021

Table of Contents for the Digital Edition of IEEE Power Electronics Magazine - March 2021
