diff --git a/interfaces/cython/cantera/test/test_thermo.py b/interfaces/cython/cantera/test/test_thermo.py
index 2fd0f7a68..ef4a66e2f 100644
--- a/interfaces/cython/cantera/test/test_thermo.py
+++ b/interfaces/cython/cantera/test/test_thermo.py
@@ -646,6 +646,12 @@ class TestThermo(utilities.CanteraTest):
         self.assertNear(self.gas.v, 3 * v1)
         self.assertTrue(self.gas.T < self.gas.min_temp)
 
+    def test_setSV_low_invalid(self):
+        self.gas.TPX = 450, 1e5, 'H2:1.0, O2:0.4, AR:3'
+        self.gas.SV = 4600, None
+        with self.assertRaises(ct.CanteraError):
+            self.gas.SV = -1000, None
+
     def test_setSV_highT(self):
         """
         Set state in terms of (s,v) when the end temperature is above the
@@ -675,6 +681,16 @@ class TestThermo(utilities.CanteraTest):
         self.assertNear(self.gas.P, p1)
         self.assertTrue(self.gas.T < self.gas.min_temp)
 
+    def test_setHP_low_invalid(self):
+        """
+        Set state in terms of (h,p) when the enthalpy would imply a negative
+        temperature
+        """
+
+        self.gas.TPX = 300, 101325, 'H2:1.0'
+        with self.assertRaises(ct.CanteraError):
+            self.gas.HP = -4e6, 101325
+
     def test_setHP_highT(self):
         """
         Set state in terms of (s,v) when the end temperature is above the
diff --git a/src/thermo/ThermoPhase.cpp b/src/thermo/ThermoPhase.cpp
index 8eb070efe..610ab79c5 100644
--- a/src/thermo/ThermoPhase.cpp
+++ b/src/thermo/ThermoPhase.cpp
@@ -409,9 +409,9 @@ void ThermoPhase::setState_HPorUV(doublereal Htarget, doublereal p,
         // Convergence in H
         double Herr = Htarget - Hnew;
         double acpd = std::max(fabs(cpd), 1.0E-5);
-        double denom = std::max(fabs(Htarget), acpd * dTtol);
+        double denom = std::max(fabs(Htarget), acpd * Tnew);
         double HConvErr = fabs((Herr)/denom);
-        if (HConvErr < 0.00001 *dTtol || fabs(dt) < dTtol) {
+        if (HConvErr < 0.00001 * dTtol || fabs(dt/Tnew) < 0.00001 * dTtol) {
             return;
         }
     }
@@ -595,9 +595,9 @@ void ThermoPhase::setState_SPorSV(doublereal Starget, doublereal p,
         // Convergence in S
         double Serr = Starget - Snew;
         double acpd = std::max(fabs(cpd), 1.0E-5);
-        double denom = std::max(fabs(Starget), acpd * dTtol);
+        double denom = std::max(fabs(Starget), acpd * Tnew);
         double SConvErr = fabs((Serr * Tnew)/denom);
-        if (SConvErr < 0.00001 *dTtol || fabs(dt) < dTtol) {
+        if (SConvErr < 0.00001 * dTtol || fabs(dt/Tnew) < 0.00001 * dTtol) {
             return;
         }
     }