the average state space model system using the Euler method

The unit step response of the average state space model system using the Euler method differs slightly in amplitude from the step response of the Buck converter transfer function. Which method would you prefer to use instead of Euler, which is not as complicated as the differential equation?

Best regards,

% Buck converter parameters

L = 1e-3; % Inductance in Henry

C = 100e-6; % Capacitance in Farad

R = 5; % Load resistance in Ohm

Vin = 40; % Input voltage in Volt

Vref = 20; % Desired output voltage in Volt

f = 200e3; % Switching frequency in Hz

Tfinal = 0.01; % Simulation time in seconds

% Time vector

Ts = 1e-6; % Simulation step size

t = 0:Ts:Tfinal; % Time vector

N = length(t); % Number of simulation points

% State-space matrices (average model)

A = [ 0 -1/L;

1/C -1/(R*C) ];

B = [ Vin/L;

0 ];

% Initial conditions

x = zeros(2, N); % x(1,:) = iL (inductor current), x(2,:) = Vo (output voltage)

% Duty cycle (assumed constant for open-loop case)

D = Vref / Vin; % Ideal duty cycle to achieve Vref at steady state

% Simulation loop using Euler integration

for k = 1:N-1

dx = A * x(:,k) + B * D;

x(:,k+1) = x(:,k) + Ts * dx;

end

% Plotting results

figure;

subplot(2,1,1);

plot(t, x(1,:), 'b', 'LineWidth', 1.5);

ylabel('i_L (A)');

grid on;

title('Buck Converter - Average Model Simulation');

subplot(2,1,2);

plot(t, x(2,:), 'r', 'LineWidth', 1.5);

ylabel('V_o (V)');

xlabel('Time (s)');

grid on;

Gvd=tf([Vin/(L*C)],[1 1/(R*C) 1/(L*C)]);

step(Gvd);

Answer 1

[Sam Chak]

Hi @fatih

In the simulation of the transfer function Gvd, you forgot to multiply the transfer function with the Duty Cycle 'D'.

% Buck converter parameters
L       = 1e-3;     % Inductance in Henry
C       = 100e-6;   % Capacitance in Farad
R       = 5;        % Load resistance in Ohm
Vin     = 40;       % Input voltage in Volt
Vref    = 20;       % Desired output voltage in Volt
f       = 200e3;    % Switching frequency in Hz
Tfinal  = 0.01;     % Simulation time in seconds

% Time vector
Ts  = 1e-6;         % Simulation step size
t   = 0:Ts:Tfinal;  % Time vector
N   = length(t);    % Number of simulation points

% State-space matrices (average model)
A   = [  0      -1/L;
       1/C, -1/(R*C)];
B   = [Vin/L;
           0];

Cout= eye(2);
sys = ss(A, B, Cout, []);
G   = tf(sys)

G =
 
  From input to output...
         40000 s + 8e07
   1:  -------------------
       s^2 + 2000 s + 1e07
 
              4e08
   2:  -------------------
       s^2 + 2000 s + 1e07
 
Continuous-time transfer function.

% Initial conditions
x   = zeros(2, N);   % x(1,:) = iL (inductor current), x(2,:) = Vo (output voltage)

% Duty cycle (assumed constant for open-loop case)
D   = Vref / Vin;    % Ideal duty cycle to achieve Vref at steady state

% Simulation loop using Euler integration
for k = 1:N-1
    dx       = A*x(:,k) + B*D;   
    x(:,k+1) = x(:,k) + Ts*dx;
end

fun = @(T, X) A*X + B*D;
[T, X]  = ode45(fun, t, [0; 0]);

% Plotting results
figure;
subplot(2,1,1);
hold on
plot(t, x(1,:), 'b'); 
plot(T, X(:,1), 'b--', 'LineWidth', 1.5); 
hold off
legend('Euler', 'ode45', 'location', 'southeast')
ylabel('i_L (A)');
title('Buck Converter Current')
grid on;

subplot(2,1,2);
hold on
plot(t, x(2,:), 'r'); 
plot(T, X(:,2), 'r--', 'LineWidth', 1.5); 
hold off
legend('Euler', 'ode45', 'location', 'southeast')
ylabel('V_o (V)');
xlabel('Time (s)');
title('Buck Converter Voltage')
grid on;

sgtitle('Buck Converter - Average Model Simulation');

●

figure
subplot(211)
Gid = G(1)

Gid =
 
    40000 s + 8e07
  -------------------
  s^2 + 2000 s + 1e07
 
Continuous-time transfer function.

[y1, t1] = step(D*Gid, Tfinal);
plot(t1, y1, 'b'), grid on
title('Step Response of Buck Converter Current')
subplot(212)
Gvd = tf([Vin/(L*C)], [1 1/(R*C) 1/(L*C)])

Gvd =
 
         4e08
  -------------------
  s^2 + 2000 s + 1e07
 
Continuous-time transfer function.

[y2, t2] = step(D*Gvd, Tfinal);
plot(t2, y2, 'r'), grid on
title('Step Response of Buck Converter Voltage')

sgtitle('Buck Converter - Simulation of Transfer functions');

[fatih]

thanks