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	   "source": [
       "# Chapter 12: Frequency Domain Analysis"
	   ]
	},
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		   "metadata": {},
		   "source": [
			"## Example 12.01: denominator_polynomial.sci"
		   ]
		  },
  {
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"//denominator polynomial//\n",
"s=poly(0,'s'); //Defines s as polynomial variable\n",
"F=syslin('c',[225/((s+6)*s)]); //Creates transfer function in forward path\n",
"B=syslin('c',(1+0*s)/(1+0*s)); //Creates transfer function in backward path\n",
"CL=F/.B //Calculates closed-loop transfer function\n",
"// compare CL with Wn^2/(s^2+2*zeta*Wn+Wn^2)\n",
"y=denom(CL) //extracting the denominator of CL\n",
"z=coeff(y)  //extracting the coefficients of the denominator polynomial\n",
"//Wn^2=z(1,1) ,comparing the coefficients\n",
"Wn=sqrt(z(1,1))            // Wn=natural frequency\n",
"//2*zeta*Wn=z(1,2)\n",
"zeta=z(1,2)/(2*Wn)             // zeta=damping factor\n",
"Mr=1/(2*zeta*sqrt(1-zeta^2))\n",
"Wr=Wn*sqrt(1-zeta^2)  \n",
"Wc=Wn*sqrt((1-2*zeta^2)+sqrt(4*zeta^4-4*zeta^2+2))\n",
"BW=Wc //BANDWIDTH"
   ]
   }
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		   "metadata": {},
		   "source": [
			"## Example 12.02: denominator_polynomial.sci"
		   ]
		  },
  {
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"//denominator polynomial//\n",
"s=poly(0,'s'); //Defines s as polynomial variable\n",
"F=syslin('c',[36/((s+8)*s)]); //Creates transfer function in forward path\n",
"B=syslin('c',(1+0*s)/(1+0*s)); //Creates transfer function in backward path\n",
"CL=F/.B //Calculates closed-loop transfer function\n",
"// compare CL with Wn^2/(s^2+2*zeta*Wn+Wn^2)\n",
"y=denom(CL) //extracting the denominator of CL\n",
"z=coeff(y)  //extracting the coefficients of the denominator polynomial\n",
"//Wn^2=z(1,1) ,comparing the coefficients\n",
"Wn=sqrt(z(1,1))             // Wn=natural frequency\n",
"//2*zeta*Wn=z(1,2)\n",
"zeta=z(1,2)/(2*Wn)             // zeta=damping factor\n",
"Mr=1/(2*zeta*sqrt(1-zeta^2))\n",
"Wr=Wn*sqrt(1-zeta^2)  \n",
"Wc=Wn*sqrt((1-2*zeta^2)+sqrt(4*zeta^4-4*zeta^2+2))\n",
"BW=Wc //BANDWIDTH"
   ]
   }
,
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		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 12.03: denominator_polynomial.sci"
		   ]
		  },
  {
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"//denominator polynomial//\n",
"s=poly(0,'s'); //Defines s as polynomial variable\n",
"F=syslin('c',[81/(s^2+7*s)]); //Creates transfer function in forward path\n",
"B=syslin('c',(1+0*s)/(1+0*s)); //Creates transfer function in backward path\n",
"CL=F/.B //Calculates closed-loop transfer function\n",
"// compare CL with Wn^2/(s^2+2*zeta*Wn+Wn^2)\n",
"y=denom(CL) //extracting the denominator of CL\n",
"z=coeff(y)  //extracting the coefficients of the denominator polynomial\n",
"//Wn^2=z(1,1) ,comparing the coefficients\n",
"Wn=sqrt(z(1,1))             // Wn=natural frequency\n",
"//2*zeta*Wn=z(1,2)\n",
"zeta=z(1,2)/(2*Wn)             // zeta=damping factor\n",
"Mr=1/(2*zeta*sqrt(1-zeta^2))\n",
"Wr=Wn*sqrt(1-zeta^2)  \n",
"Wc=Wn*sqrt((1-2*zeta^2)+sqrt(4*zeta^4-4*zeta^2+2))\n",
"BW=Wc //BANDWIDTH"
   ]
   }
,
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		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 12.04: denominator_polynomial.sci"
		   ]
		  },
  {
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"//denominator polynomial//\n",
"//Defines s as polynomial variable\n",
"s=poly(0,'s');\n",
"//Creates transfer function in forward path\n",
"F=syslin('c',[81/((s+18)*s)]); \n",
"//Creates transfer function in backward path\n",
"B=syslin('c',(1+0*s)/(1+0*s));\n",
"//Calculates closed-loop transfer function\n",
"CL=F/.B \n",
"// compare CL with Wn^2/(s^2+2*zeta*Wn+Wn^2)\n",
"y=denom(CL)//extracting the denominator of CL\n",
"//Extracting the coefficients of the denominator polynomial\n",
"z=coeff(y)  \n",
"//Wn^2=z(1,1) ,comparing the coefficients\n",
"Wn=sqrt(z(1,1)) \n",
"//2*zeta*Wn=z(1,2)\n",
"zeta=z(1,2)/(2*Wn)  \n",
"// zeta=damping factor\n",
"//NOTE= here sqrt(1-2zeta^2) becomes complex so the other solution is  Wr=0 & Mr=1\n",
"Mr=1 \n",
"Wr=0 \n",
"Wc=Wn*((1-2*zeta^2)+sqrt(4*zeta^4-4*zeta^2+2))\n",
"BW=Wc //BANDWIDTH"
   ]
   }
,
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		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 12.05: denominator_polynomial.sci"
		   ]
		  },
  {
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"//denominator polynomial//\n",
"ieee(2);\n",
"syms k a;\n",
"num=k ;\n",
"den=s*(a+s);\n",
"G=num/den;\n",
"disp(G,'G(s)=')\n",
"H=1;\n",
"CL=G/.H;\n",
"CL=simple(CL);\n",
"disp(CL,'C(s)/R(s)=')  //Calculates closed-loop transfer function\n",
"// compare CL with Wn^2/(s^2+2*zeta*Wn+Wn^2)\n",
"[num,den]=numden(CL); //extracts num & den of symbolic function (CL)\n",
"cof_a_0 = coeffs(den,'s',0); // coeff of den of symbolic function (CL)\n",
"cof_a_1 = coeffs(den,'s',1);\n",
"//Wn^2= cof_a_0,comparing the coefficients\n",
"Wn=sqrt(cof_a_0) \n",
"//cof_a_1=2*zeta*Wn\n",
"zeta=cof_a_1/(2*Wn)\n",
"Mr=1/(2*zeta*sqrt(1-zeta^2)) //----------1)\n",
"printf('Given ,Mr=1.25 \n');\n",
"//On solving eq (1) we get k=1.25a^2-------2\n",
"printf('k=1.25*a^2  \n')\n",
"Wr=Wn*sqrt(1-2*zeta^2)   //---------------3)\n",
"printf('Given, Wr=12.65  \n') ;\n",
"//on soving eq (3),we get 2k-a^2=320-----------4)\n",
"printf('2k-a^2=320 \n')\n",
"//now eq 2 &4 can be simultaneously soved to take out values of k &a\n",
"// Let k=x & a^2=y\n",
"A=[1,-1.25;2,-1]; // coefficient matrix\n",
"b=[0;320];\n",
"m=A\b;\n",
"x=m(1,1);\n",
"k=x\n",
"y=m(2,1);\n",
"a=sqrt(y)\n",
"Wn=dbl(eval(Wn)); \n",
"disp(Wn,'Wn=')\n",
"zeta=dbl(eval(zeta));\n",
"disp(zeta,'zeta=')\n",
"Ts=4/(zeta*Wn) ;\n",
"disp(Ts,'Settling Time (Ts)=')\n",
"Wc=Wn((1-(2*zeta^2))+sqrt(4*zeta^4-4*zeta^2+2))\n",
"disp(Wc,'BW=')"
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   }
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